P-Adic Lie Groups (Grundlehren der mathematischen Wissenschaften)

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander A. Kupiainen G. Lebeau F.-H. Lin S. Mori B.C. Ngô M. Ratner D. Serre N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan

344

For further volumes: www.springer.com/series/138

Peter Schneider

p-Adic Lie Groups

Peter Schneider Institute of Mathematics University of Münster Einsteinstrasse 62 Münster 48149 Germany [email protected]

ISSN 0072-7830 ISBN 978-3-642-21146-1 e-ISBN 978-3-642-21147-8 DOI 10.1007/978-3-642-21147-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011930424 Mathematics Subject Classification: 22E20, 16S34 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Introduction This book presents a complete account of the foundations of the theory of p-adic Lie groups. It moves on to some of the important more advanced aspects. Although most of the material is not new, it is only in recent years that p-adic Lie groups have found important applications in number theory and representation theory. These applications constitute, in fact, an increasingly active area of research. The book is designed to give to the advanced, but not necessarily graduate, student a streamlined access to the basics of the theory. It is almost self contained. Only a few technical computations which are well covered in the literature will not be repeated. My hope is that researchers who see the need to take up p-adic methods also will find this book helpful for quickly mastering the necessary notions and techniques. The book comes in two parts. Part A on the analytic side grew out of a course which I gave at M¨ unster for the first time during the summer term 2001, whereas part B on the algebraic side is the content of a course given at the Newton Institute during September 2009. The original and proper context of p-adic Lie groups is p-adic analysis. This is the point of view in Part A. Of course, in a formal sense the notion of a p-adic Lie group is completely parallel to the classical notion of a real or complex Lie group. It is a manifold over a nonarchimedean field which carries a compatible group structure. The fundamental difference is that the p-adic notion has no geometric content. As we will see, a paracompact p-adic manifold is topologically a disjoint union of charts and therefore is, from a geometric perspective, completely uninteresting. The point instead is that, like for real Lie groups, manifolds and Lie groups in the p-adic world are a rich source, through spaces of functions and distributions, of interesting group representations as well as various kinds of important topological group algebras. We nevertheless find the geometric language very intuitive and therefore will use it systematically. In the first chapter we recall what a nonarchimedean field is and quickly discuss the elementary analysis over such fields. In particular, we carefully introduce the notion of a locally analytic function which is at the base for everything to follow. The second chapter then defines manifolds and establishes the formalism of their tangent spaces. As a more advanced topic we include the construction of the natural topology on vector spaces of locally analytic functions. This is due to F´eaux de Lacroix in his thesis. It is the starting point for the representation theoretic applications of the theory. In the third chapter we finally introduce p-adic Lie groups and we construct the corresponding Lie algebras. The main purpose of this chapter then is to understand how much informav

vi

Introduction

tion about the Lie group can be recovered from its Lie algebra. Here again lies a crucial difference to Lie groups over the real numbers. Since p-adic Lie groups topologically are totally disconnected they contain arbitrarily small open subgroups. Hence the Lie algebra determines the Lie group only locally around the unit element which is formalized by the notion of a Lie group germ. As the length of the chapter indicates this relation between Lie groups and Lie algebras is technically rather involved. It requires a whole range of algebraic concepts which we all will introduce. As said before, only for a few computations the reader will be referred to the literature. The key result is contained in the discussion of the convergence of the Hausdorff series. There are three existing books on the material in Part A: “Vari´et´es diff´erentielles et analytiques. Fascicule de r´esultats” and “Lie Groups and Lie Algebras” by Bourbaki and Serre’s lecture notes on “Lie Algebras and Lie groups”. The first one contains no proofs, the nature of the second one is encyclopedic, and the last one some times is a bit short on details. All three develop the real and p-adic case alongside each other which has advantages but makes a quick grasp of the p-adic case alone more difficult. The presentation in the present book places its emphasis instead on a streamlined but still essentially self contained introduction to exclusively the p-adic case. Lazard discovered in the 1960s a purely algebraic approach to p-adic Lie groups. Unfortunately his seminal paper is notoriously difficult to read. Part B of this book undertakes the attempt to give an account of Lazard’s work again in a streamlined form which is stripped of all inessential generalities and ramifications. Lazard proceeds in an axiomatic way starting from the notion of a p-valuation ω on a pro-p-group G. After some preliminaries in the fourth chapter this concept is explained in chapter five. It will not be too difficult to show that any p-adic Lie group has an open subgroup which carries a p-valuation. Lazard realized that, vice versa, any pro-p-group with a p-valuation (and satisfying an additional mild condition of being “of finite rank”) is a compact p-adic Lie group in a natural way. The technical tool to achieve this important result is the so called completed group ring Λ(G) of a profinite group G. It is the appropriate analog of the algebraic group ring of a finite (or, more generally, discrete) group in the context of profinite groups. In the presence of a p-valuation ω Lazard develops a technique of computation in Λ(G), which as such is a highly complicated and in general noncommutative algebra. All of this will be presented in the sixth chapter. In the last chapter seven we go back to Lie algebras. Being a p-adic Lie group a pro-p-group G with a p-valuation of finite rank ω has a Lie algebra Lie(G) over the field of p-adic numbers Qp . By inverting p and a further completion process the completed group ring Λ(G) can be enlarged to a Qp -Banach

Introduction

vii

algebra ΛQp (G, ω) which turns out to be naturally isomorphic to a certain completion of the universal enveloping algebra of Lie(G). This is another one of Lazard’s important results. It provides us with a different route to construct Lie(G) which is independent of any analysis. In fact, it does better than that since it leads to a natural Lie algebra over the ring over p-adic integers Zp associated with the pair (G, ω). This means that the algebraic theory, via this notion of a p-valuation, makes the connection between Lie group and Lie algebra much more precise than the analytic theory was able to do. The final question addressed in the last chapter is the question on the possibility of varying the p-valuation on the same group G. Using the newly established direct connection to the Lie algebra this problem can be transferred to the latter. There it eventually becomes a problem of convexity theory which is much easier to solve. This, in particular, allows to prove the very useful technical fact that there always exists a p-valuation with rational values. Its most important consequence is the result that the completed group ring Λ(G) of any (G, ω) of finite rank is a noetherian ring of finite global dimension. This is why completed group rings of p-adic Lie groups have become important in number theory (where they are applied to Galois groups G), and why they deserve further systematic study in noncommutative algebra. This is the first textbook in the proper sense on Lazard’s work. The book “Analytic Pro-p-Groups” by Dixon, du Sautoy, Mann, and Segal has a completely different perspective. It is written entirely from the point of view of abstract group theory. Moreover, it does not mention Lazard’s concept of a p-valuation at all but replaces it by an alternative axiomatic approach based on the notion of a uniformly powerful pro-p-group. This approach is very conceptual as well but also less flexible and more restrictive than the one by Lazard which we follow. It is a pleasure to thank J. Coates for persuading me to undertake this lecture series at the Newton Institute and to write it up in this book, the audience for the valuable feedback, the Newton Institute for its hospitality and support, and T. Schoeneberg for a careful reading of Part B. M¨ unster, February 2011

Peter Schneider

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Contents

A I

p-Adic Analysis and Lie Groups Foundations

1 3

1 Ultrametric Spaces

3

2 Nonarchimedean Fields

8

3 Convergent Series

14

4 Differentiability

17

5 Power Series

25

6 Locally Analytic Functions

38

II

45

Manifolds

7 Charts and Atlases

45

8 Manifolds

47

9 The Tangent Space

56

10 The Topological Vector Space C an (M, E), Part 1

74

11 Locally Convex K-Vector Spaces

79

12 The Topological Vector Space C an (M, E), Part 2

84

III

89

Lie Groups

13 Definitions and Foundations 14 The Universal Enveloping Algebra

89 101 ix

x

Contents

15 The Concept of Free Algebras

106

16 The Campbell-Hausdorff Formula

111

17 The Convergence of the Hausdorff Series

124

18 Formal Group Laws

132

B

155

IV

The Algebraic Theory of p-Adic Lie Groups Preliminaries

157

19 Completed Group Rings

157

20 The Example of the Group Zdp

163

21 Continuous Distributions

164

22 Appendix: Pseudocompact Rings

165

V

169

p-Valued Pro-p-Groups

23 p-Valuations

169

24 The Free Group on Two Generators

175

25 The Operator P

178

26 Finite Rank Pro-p-Groups

181

27 Compact p-Adic Lie Groups

192

VI

195

Completed Group Rings of p-Valued Groups

28 The Ring Filtration

195

29 Analyticity

201

30 Saturation

208

Contents

VII

The Lie Algebra

xi

219

31 A Normed Lie Algebra

219

32 The Hausdorff Series

232

33 Rational p-Valuations and Applications

243

34 Coordinates of the First and of the Second Kind

247

References

251

Index

253

Part A

p-Adic Analysis and Lie Groups

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Chapter I

Foundations 1

Ultrametric Spaces

We begin by establishing some very basic and elementary notions. Definition. A metric space (X, d) is called ultrametric if the strict triangle inequality d(x, z) ≤ max(d(x, y), d(y, z))

for any x, y, z ∈ X

is satisfied. Examples will be given later on. Remark. i. If (X, d) is ultrametric then (Y, d |Y × Y ), for any subset Y ⊆ X, is ultrametric as well. ii. If (X1 , d1 ), . . . , (Xm , dm ) are ultrametric spaces then the cartesian product X1 × · · · × Xm is ultrametric with respect to d((x1 , . . . , xm ), (y1 , . . . , ym )) := max(d1 (x1 , y1 ), . . . , dm (xm , ym )). Let (X, d) be an ultrametric space in the following. Lemma 1.1. For any three points x, y, z ∈ X such that d(x, y) = d(y, z) we have d(x, z) = max(d(x, y), d(y, z)). Proof. We may assume that d(x, y) < d(y, z). Then d(x, y) < d(y, z) ≤ max(d(y, x), d(x, z)) = max(d(x, y), d(x, z)). The maximum in question therefore necessarily is equal to d(x, z) so that d(x, y) < d(y, z) ≤ d(x, z). We deduce that d(x, z) ≤ max(d(x, y), d(y, z)) ≤ d(x, z).

P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8 1, © Springer-Verlag Berlin Heidelberg 2011

3

4

I

Foundations

Let a ∈ X be a point and ε > 0 be a positive real number. We call Bε (a) := {x ∈ X : d(a, x) ≤ ε} the closed ball and Bε− (a) := {x ∈ X : d(a, x) < ε} the open ball around a of radius ε. Any subset in X of one of these two kinds is simply referred to as a ball . As the following facts show this language has to be used with some care. Lemma 1.2.

i. Every ball is open and closed in X.

ii. For b ∈ Bε (a), resp. b ∈ Bε− (a), we have Bε (b) = Bε (a), resp. Bε− (b) = Bε− (a). Proof. Obviously Bε− (a) is open and Bε (a) is closed in X. We first consider the equivalence relation x ∼ y on X defined by d(x, y) < ε. The corresponding equivalence class of b is equal to Bε− (b) and hence is open. Since equivalence classes are disjoint or equal this implies Bε− (b) = Bε− (a) whenever b ∈ Bε− (a). It also shows that Bε− (a) as the complement of the other open equivalence classes is closed in X. Analogously we may consider the equivalence relation x ≈ y on X defined by d(x, y) ≤ ε. Its equivalence classes are the closed balls Bε (b), and we obtain in the same way as before the assertion ii. for closed balls. It remains to show that Bε (a) is open in X. But by what we have established already with any point b ∈ Bε (a) its open neighbourhood Bε− (b) is contained in Bε (b) = Bε (a). The assertion ii. in the above lemma can be viewed as saying that any point of a ball can serve as its midpoint. By way of an example we will see later on that also the notion of a radius is not well determined. Lemma 1.3. For any two balls B and B  in X such that B ∩ B  = ∅ we have B ⊆ B  or B  ⊆ B. Proof. Pick a point a ∈ B ∩ B  . As a consequence of Lemma 1.2.ii. the following four cases have to be distinguished: 1. B = Bε− (a), B  = Bδ− (a), 2. B = Bε− (a), B  = Bδ (a),

1

Ultrametric Spaces

5

3. B = Bε (a), B  = Bδ− (a), 4. B = Bε (a), B  = Bδ (a). Without loss of generality we may assume that ε ≤ δ. In cases 1, 2, and 4 we then obviously have B ⊆ B  . In case 3 we obtain B ⊆ B  if ε < δ and B  ⊆ B if ε = δ. Remark. If the ultrametric space X is connected then it is empty or consists of one point. Proof. Assuming that X is nonempty we pick a point a ∈ X. Lemma 1.2.i. then implies that X = Bε (a) for any ε > 0 and hence that X = {a}.  Lemma 1.4. Let U = i∈I Ui be a covering of an open subset U ⊆ X by open subsets Ui ⊆ X; moreover let ε1 > ε2 > · · · > 0 be a strictly descending sequence of positive real numbers which converges to zero; then there is a decomposition  U= Bj j∈J

of U into pairwise disjoint balls Bj such that: (a) Bj = Bεn(j) (aj ) for appropriate aj ∈ X and n(j) ∈ N, (b) Bj ⊆ Ui(j) for some i(j) ∈ I. Proof. For a ∈ U we put n(a) := min{n ∈ N : Bεn (a) ⊆ Ui for some i ∈ I}. The family of balls J := {Bεn(a) (a) : a ∈ U } by construction has the properties (a) and (b) and covers U (observe that for any point a in the open set Ui we find some sufficiently big n ∈ N such that Bεn (a) ⊆ Ui ). The balls in this family indeed are pairwise disjoint: Suppose that Bεn(a1 ) (a1 ) ∩ Bεn(a2 ) = ∅. By Lemma 1.3 we may assume that Bεn(a1 ) (a1 ) ⊆ Bεn(a2 ) (a2 ). But then Lemma 1.2.ii. implies that Bεn(a2 ) (a1 ) = Bεn(a2 ) (a2 ) and hence Bεn(a1 ) (a1 ) ⊆ Bεn(a2 ) (a1 ). Due to the minimality of n(a1 ) we must have n(a1 ) ≤ n(a2 ), resp. εn(a1 ) ≥ εn(a2 ) . It follows that Bεn(a1 ) (a1 ) = Bεn(a2 ) (a1 ) = Bεn(a2 ) (a2 ).

6

I

Foundations

As usual the metric space X is called complete if every Cauchy sequence in X is convergent. Lemma 1.5. A sequence (xn )n∈N in X is a Cauchy sequence if and only if limn→∞ d(xn , xn+1 ) = 0. For a subset A ⊆ X we call d(A) := sup{d(x, y) : x, y ∈ A} the diameter of A. Lemma 1.6. Let B ⊆ X be a ball with ε := d(B) > 0 and pick any point a ∈ B; we then have B = Bε− (a) or B = Bε (a). Proof. The inclusion B ⊆ Bε (a) is obvious. By Lemma 1.2.ii. the ball B is of the form B = Bδ− (a) or B = Bδ (a). The strict triangle inequality then implies ε = d(B) ≤ δ. If ε = δ there is nothing further to prove. If ε < δ we have B ⊆ Bε (a) ⊆ Bδ− (a) ⊆ B and hence B = Bε (a). Let us consider a descending sequence of balls B1 ⊇ B2 ⊇ · · · ⊇ Bn ⊇ · · · in X. If X is complete and if limn→∞ d(Bn ) = 0 then we claim that  Bn = ∅. n∈N

If we pick points xn ∈ Bn then (xn )n∈N is a Cauchy sequence. Put x := limn→∞  xn . Since each Bn is closed we must have x ∈ Bn and therefore x ∈ n Bn .  Without the condition on the diameters the intersection n Bn can be empty (compare the exercise further below). This motivates the following definition. Definition. The ultrametric space (X, d) is called spherically complete if any descending sequence of balls B1 ⊇ B2 ⊇ · · · in X has a nonempty intersection. Lemma 1.7.

i. If X is spherically complete then it is complete.

ii. Suppose that X is complete; if 0 is the only accumulation point of the set d(X × X) ⊆ R+ of values of the metric d then X is spherically complete.

1

Ultrametric Spaces

7

Proof. i. Let (xn )n∈N be any Cauchy sequence in X. We may assume that this sequence does not become constant after finitely many steps. Then the εn := max{d(xm , xm+1 ) : m ≥ n} are strictly positive real numbers satisfying εn ≥ εn+1 and εn ≥ d(xn , xn+1 ). Using Lemma 1.2.ii. we obtain  Bεn (xn ) = Bεn (xn+1 ) ⊇ Bεn+1 (xn+1 ). By assumption the intersection n Bεn (xn ) must contain a point x. We have d(x, xn ) ≤ εn for any n ∈ N. Since the sequence (εn )n converges to zero this implies that x = limx→∞ xn . ii. Let B1 ⊇ B2 ⊇ · · · be any decreasing sequence of balls in X. Obviously we have d(B1 ) ≥ d(B2 ) ≥ · · · . By our above discussion we only need to consider the case that inf n d(Bn ) > 0. Our assumption on accumulation points implies that d(Bn ) ∈ D(X × X) for any n ∈ N and then in fact that the sequence (d(Bn ))n must become constant after finitely many steps. Hence there exists an m ∈ N such that 0 < ε := d(Bm ) = d(Bm+1 ) = · · · . By Lemma 1.6 we have, for any n ≥ m and any a ∈ Bn , that Bn = Bε− (a)

or

Bn = Bε (a).

Moreover, which of the two equations holds is independent of the choice of a by Lemma 1.2.ii. Case 1: We have Bn = Bε (a) for any n ≥ m and any a ∈ Bn. It immediately follows that Bn = Bm for any n ≥ m and hence that n Bn = Bm . Case 2: There is an  ≥ m such that B = Bε− (a) B we then obtain for any a ∈ B . For any n ≥  and any a ∈ Bn ⊆  Bε− (a) = B ⊇ Bn ⊇ Bε− (a) so that B = Bn and hence n Bn = B . Exercise. Suppose that X is complete, and let B1 ⊃ B2 ⊃ · · · be a decreasing sequence of balls in X such  that d(B1 ) > d(B2 ) > · · · and inf n d(Bn ) > 0. Then the subspace Y := X\( n Bn ) is complete but not spherically complete. Lemma 1.8. Suppose that X is spherically complete; for any family (Bi )i∈I of  closed balls in X such that Bi ∩ Bj = ∅ for any i, j ∈ I we then have i∈I Bi = ∅. Proof. We choose a sequence (in )n∈N of indices in I such that: – d(Bi1 ) ≥ d(Bi2 ) ≥ · · · ≥ d(Bin ) ≥ · · · , – for any i ∈ I there is an n ∈ N with d(Bi ) ≥ d(Bin ). The proof of Lemma 1.6 shows that Bi = Bd(Bi ) (a) for any a ∈ Bi . Our assumption on the family (Bi )i therefore implies that:

8

I

Foundations

– Bi1 ⊇ Bi2 ⊇ · · · ⊇ Bin ⊇ · · · , – for any i ∈ I there is an n ∈ N with Bi ⊇ Bin . We see that



Bi =

i∈I

2



Bin = ∅.

n∈N

Nonarchimedean Fields

Let K be any field. Definition. A nonarchimedean absolute value on K is a function | | : K −→ R which satisfies: (i) |a| ≥ 0, (ii) |a| = 0 if and only if a = 0, (iii) |ab| = |a| · |b|, (iv) |a + b| ≤ max(|a|, |b|). Exercise.

i. |n · 1| ≤ 1 for any n ∈ Z.

ii. | | : K × −→ R× + is a homomorphism of groups; in particular, |1| = |−1| = 1. iii. K is an ultrametric space with respect to the metric d(a, b) := |b − a|; in particular, we have |a + b| = max(|a|, |b|) whenever |a| = |b|. iv. Addition and multiplication on the ultrametric space K are continuous maps. Definition. A nonarchimedean field (K, | |) is a field K equipped with a nonarchimedean absolute value | | such that: (i) | | is non-trivial, i. e., there is an a ∈ K with |a| = 0, 1, (ii) K is complete with respect to the metric d(a, b) := |b − a|.

2

Nonarchimedean Fields

9

The most important class of examples is constructed as follows. We fix a prime number p. Then |a|p := p−r

if a = pr m n with r, m, n ∈ Z and p  | mn

is a nonarchimedean absolute value on the field Q of rational numbers. The corresponding completion Qp is called the field of p-adic numbers. Of course, it is nonarchimedean as well. We note that |Qp |p = pZ ∪ {0}. Hence Qp is spherically complete by Lemma 1.7.ii. On the other hand we see that in the ultrametric space Qp we can have Bε (a) = Bδ (a) even if ε = δ. To have more examples we state without proof (compare [Se1] Chap. II §§1–2) the following fact. Let K/Qp be any finite extension of fields. Then  [K:Qp ] |NormK/Qp (a)|p |a| := is the unique extension of | |p to a nonarchimedean absolute value on K. The corresponding ultrametric space K is complete and spherically complete and, in fact, locally compact. In the following we fix a nonarchimedean field (K, | |). By the strict triangle inequality the closed unit ball oK := B1 (0) is a subring of K, called the ring of integers in K, and the open unit ball mK := B1− (0) is an ideal in oK . Because of o× K = oK \mK this ideal mK is the only maximal ideal of oK . The field oK /mK is called the residue class field of K. Exercise 2.1. i. If the residue class field oK /mK has characteristic zero then K has characteristic zero as well and we have |a| = 1 for any nonzero a ∈ Q ⊆ K. ii. If K has characteristic zero but oK /mK has characteristic p > 0 then we have −

|a| = |a|p

log |p| log p

for any a ∈ Q ⊆ K;

in particular, K contains Qp . A nonarchimedean field K as in the second part of Exercise 2.1 is called a p-adic field .

10

I

Foundations

Lemma 2.2. If K is p-adic then we have n−1

|n| ≥ |n!| ≥ |p| p−1

for any n ∈ N.

Proof. We may obviously assume that K = Qp . Then the reader should do this as an exercise but also may consult [B-LL] Chap. II §8.1 Lemma 1. Now let V be any K-vector space. Definition. A (nonarchimedean) norm on V is a function   : V −→ R such that for any v, w ∈ V and any a ∈ K we have: (i) av = |a| · v, (ii) v + w ≤ max(v, w), (iii) if v = 0 then v = 0. Moreover, V is called normed if it is equipped with a norm. Exercise.

i. v ≥ 0 for any v ∈ V and 0 = 0.

ii. V is an ultrametric space with respect to the metric d(v, w) := w −v; in particular, we have v + w = max(v, w) whenever v = w. +

iii. Addition V × V −−→ V and scalar multiplication K × V −→ V are continuous. Lemma 2.3. Let (V1 ,  1 ) and (V2 ,  2 ) let two normed K-vector spaces; a linear map f : V1 −→ V2 is continuous if and only if there is a constant c > 0 such that f (v)2 ≤ c · v1 for any v ∈ V1 . Proof. We suppose first that such a constant c > 0 exists. Consider any sequence (vn )n∈N in V1 which converges to some v ∈ V1 . Then (vn − v1 )n and hence (f (vn )−f (v)2 )n = (f (vn −v)2 )n are zero sequences. It follows that the sequence (f (vn ))n converges to f (v) in V2 . This means that f is continuous. Now we assume vice versa that f is continuous. We find a 0 < ε < 1 such that Bε (0) ⊆ f −1 (B1 (0)). Since | | is non-trivial we may assume that ε = |a| for some a ∈ K. In other words v1 ≤ |a| implies f (v)2 ≤ 1

2

Nonarchimedean Fields

11

for any v ∈ V1 . Let now 0 = v ∈ V1 be an arbitrary nonzero vector. We find an m ∈ Z such that |a|m+2 < v1 ≤ |a|m+1 . Setting c := |a|−2 we obtain f (v)2 = |a|m · f (a−m v)2 ≤ |a|m < c · v1 .

Definition. The normed K-vector space (V,  ) is called a K-Banach space if V is complete with respect to the metric d(v, w) := w − v. Examples. 1) K n with the norm (a1 , . . . , an ) := max1≤i≤n |ai | is a K-Banach space. 2) Let I be a fixed but arbitrary index set. A family (ai )i∈I of elements in K is called bounded if there is a c > 0 such that |ai | ≤ c for any i ∈ I. The set ∞ (I) := set of all bounded families (ai )i∈I in K with componentwise addition and scalar multiplication and with the norm (ai )i ∞ := sup |ai | i∈I

is a K-Banach space. 3) With I as above let c0 (I) := {(ai )i∈I ∈ ∞ (I) : for any ε > 0 we have |ai | ≥ ε for at most finitely many i ∈ I}. It is a closed vector subspace of ∞ (I) and hence a K-Banach space in its own right. Moreover, for (ai )i ∈ c0 (I) we have (ai )i ∞ = max |ai |. i∈I

For example, c0 (N) is the Banach space of all zero sequences in K. Remark. Any K-Banach space (V,  ) over a finite extension K/Qp which satisfies V  ⊆ |K| is isometric to some K-Banach space (c0 (I),  ∞ ); moreover, all such I have the same cardinality.

12

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Foundations

Proof. Compare [NFA] Remark 10.2 and Lemma 10.3. Let V and W be two normed K-vector spaces. From now on we denote, unless this causes confusion, all occurring norms indiscriminately by  . It is clear that L(V, W ) := {f ∈ HomK (V, W ) : f is continuous} is a vector subspace of HomK (V, W ). By Lemma 2.3 the operator norm  f  := sup

   f (v) f (v) : v ∈ V, v = 0 = sup : v ∈ V, 0 < v ≤ 1 v v

is well defined for any f ∈ L(V, W ). Lemma 2.4. L(V, W ) with the operator norm is a normed K-vector space. Proof. This is left to the reader as an exercise. Proposition 2.5. If W is a K-Banach space then so, too, is L(V, W ). Proof. Let (fn )n∈N be a Cauchy sequence in L(V, W ). Then, on the one hand, (fn )n is a Cauchy sequence in R and therefore converges, of course. On the other hand, because of fn+1 (v) − fn (v) = (fn+1 − fn )(v) ≤ fn+1 − fn  · v we obtain, for any v ∈ V , the Cauchy sequence (fn (v))n in W . By assumption the limit f (v) := limn→∞ fn (v) exists in W . Obviously we have f (av) = af (v) for any a ∈ K. For v, v  ∈ V we compute f (v) + f (v  ) = lim fn (v) + lim fn (v  ) = lim (fn (v) + fn (v  )) n→∞

n→∞ 

n→∞



= lim fn (v + v ) = f (v + v ). n→∞

This means that v −→ f (v) is a K-linear map which we denote by f . Since f (v) = lim fn (v) ≤ ( lim fn ) · v n→∞

n→∞

2

Nonarchimedean Fields

13

it follows from Lemma 2.3 that f is continuous. Finally the inequality   (f − fn )(v) f − fn  = sup : v = 0 v   limm→∞ fm (v) − fn (v) = sup : v = 0 v ≤ lim fm − fn  ≤ sup fm+1 − fm  m→∞

m≥n

shows that f indeed is the limit of the sequence (fn )n in L(V, W ). In particular,

V  := L(V, K)

always is a K-Banach space. It is called the dual space to V . Lemma 2.6. Let I be an index set; for any j ∈ I let 1j ∈ c0 (I) denote the family (ai )i∈I with ai = 0 for i = j and aj = 1; then ∼ =

c0 (I) −−→ ∞ (I)  −→ ((1i ))i∈I is an isometric linear isomorphism. Proof. We give the proof only in the case I = N. The general case follows the same line but requires the technical concept of summability (cf. [NFA] end of §3). Let us denote the map in question by ι. Because of |(1i )| ≤  · 1i ∞ =  it is well defined and satisfies ι()∞ ≤ 

for any  ∈ c0 (N) .

For trivial reasons ι is a linear map. Consider now an arbitrary nonzero vector v = (ai )i ∈ c0 (N). In the Banach space c0 (N) we then have the convergent series expansion v= ai · 1i . i∈N

Applying any  ∈ c0 (N) by continuity leads to ai (1i ). (v) = i∈N

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We obtain |(v)| supi |ai ||(1i )| ≤ sup |(1i )| = ι()∞ . ≤ v∞ supi |ai | i It follows that  ≤ ι()∞ and together with the previous inequality that ι in fact is an isometry and in particular is injective. For surjectivity let (ci )i ∈ ∞ (N) be any vector and put ε := (ci )i ∞ . We consider the linear form  : c0 (N) −→ K ai ci (ai )i −→ i

(note that the defining sum is convergent). Using Lemma 2.3 together with the inequality







ai ci ≤ sup |ai ||ci | ≤ sup |ai | · sup |ci | = ε · (ai )i ∞ |((ai )i )| =



i i i i

we see that  is continuous. It remains to observe that ι() = ((1i ))i = (ci )i .

3

Convergent Series

From now on throughout the book (K, | |) is a fixed nonarchimedean field. For the convenience of the reader we collect in this section the most basic facts about convergent series in Banach spaces (some of which we have used already in the proof of Lemma 2.6). Let (V,  ) be a K-Banach space. Lemma 3.1. Let (vn )n∈N be a sequence in V ; we then have:  i. The series ∞ n=1 vn is convergent if and only if limn→∞ vn = 0; ii. if the limit v := limn→∞ vn exists in V and is nonzero then vn  = v for all but finitely many n ∈ N;

3

Convergent Series

15

∞ iii. let σ : N → N be any bijection and suppose that the series v = n=1 vn ∞ is convergent in V ; then the series n=1 vσ(n) is convergent as well with the same limit v. Proof. i. This is immediate from Lemma 1.5. ii. If v = 0 then v = 0 and hence vn − v < v for any sufficiently big n ∈ N. For these n Lemma 1.1 then implies that vn  = (vn − v) + v = max(vn − v, v) = v. iii. We fix an ε > 0 and choose an m ∈ N such that s vn < ε for any s ≥ m. v − n=1

Then also   

s−1 s−1 s s vn − v− vn ≤ max v− vn , v− vn < ε vs  = v− n=1

n=1

n=1

n=1

for any s > m. Setting  := max{σ −1 (n) : n ≤ m} ≥ m we have {σ −1 (1), . . . , σ −1 (m)} ⊆ {1, . . . , } and hence, for any s ≥ , {σ(1), . . . , σ(s)} = {1, . . . , m} ∪ {n1 , . . . , ns−m } with appropriate natural numbers ni > m. We conclude that  s m vσ(n) = v − vn − vn1 − · · · − vns−m v − n=1 n=1 

m vn , vn1 , . . . , vns−m  ≤ max v − n=1

<ε for any s ≥ . The following identities between convergent series are obvious: ∞ ∞ – for any a ∈ K. n=1 avn = a · n=1 vn

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 ∞ +( ∞ n=1 wn ) = n=1 (vn + wn ). ∞  Lemma 3.2. Let ∞ n=1 an and  series in K and V , ∞ n=1 vn be convergent respectively; then the series n=1 wn with wn := +m=n a vm is convergent, and  ∞ 

∞ ∞ wn = an vn . – (

∞

n=1 vn )

n=1

n=1

n=1

Proof. Let A := supn |an | and C := supn vn . The other cases being trivial we will assume that A, C > 0. For any given ε > 0 we find an N ∈ N such that ε ε |an | < and vn  < for any n ≥ N. C A Then   wn  ≤ max |a | · vm  ≤ max C · max |a |, A · max vm  < ε +m=n

m≥N

≥N

 for any n ≥ 2N . By Lemma 3.1.i. the series ∞ n=1 wn therefore is convergent. To establish the asserted identity we note that its left hand side is the limit of the sequence s Ws := wn = a vm n=1

+m≤s

whereas its right hand side is the limit of the sequence  s 

s  an vn = a vm . Ws := n=1

n=1

,m≤s

It therefore suffices to show that the differences Ws − Ws converge to zero. But we have Ws − Ws  = a vm ≤ max |a | · vm  ,m≤s ,m≤s +m>s

+m>s

 |a |, A · maxs vm  . ≤ max C · max s 

> 2

m> 2

 Analogous assertions hold true for series ∞ n1 ,...,nr =1 vn1 ,...,nr indexed by multi-indices in N × · · · × N. But we point out the following additional fact.

4

Differentiability

17

Lemma 3.3. Let (vm,n )m,n∈N be a double sequence in V such that lim

m+n→∞

we then have

∞ ∞

vm,n = 0;

vm,n =

m=1 n=1

∞ ∞

vm,n

n=1 m=1

which, in particular, means that all series involved are convergent. Proof. It is  immediate from  the assumption and Lemma 3.1.i. that all “in∞ v and ner” series ∞ n=1 m,n m=1 vm,n are convergent. Let ε > 0. By assumption we find an N ∈ N such that vm,n  ≤ ε whenever ∞m + n > N . This implies that  ∞ v  ≤ ε for any m > N and  m,n n=1 m=1 vm,n  ≤ ε for any n > N . Again using Lemma 3.1.i. we obtain that the “outer” series in the asserted identity are convergent as well. But we also see that ∞ ∞ ∞ vm,n − vm,n = vm,n ≤ ε. m=1 n=1

m+n≤N

m=1 n>N −m

∞   By symmetry we also have  ∞ n=1 m=1 vm,n − m+n≤N vm,n  ≤ ε and hence ∞ ∞ ∞ ∞ vm,n − vm,n ≤ ε. m=1 n=1

n=1 m=1

Since ε was arbitrary this implies the asserted identity.

4

Differentiability

Let V and W be two normed K-vector spaces, let U ⊆ V be an open subset, and let f : U −→ W be some map. Definition. The map f is called differentiable in the point v0 ∈ U if there exists a continuous linear map Dv0 f : V −→ W such that for any ε > 0 there is an open neighbourhood Uε = Uε (v0 ) ⊆ U of v0 with f (v) − f (v0 ) − Dv0 f (v − v0 ) ≤ εv − v0 

for any v ∈ Uε .

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We may of course assume in the above definition that the open neighbourhood Uε is of the form Uε = Bδ(ε) (v0 ) for some sufficiently small radius δ(ε) > 0. We claim that the linear map Dv0 f is uniquely determined. Fix an ε > 0, and choose a basis {vj }j∈J of the vector space V . By scaling we may assume that vj  ≤ δ(ε ). We put vj := vj + v0 . Then vj ∈ Bδ(ε ) (v0 ) = Uε . More generally, for any 0 < ε ≤ ε we pick a tε ∈ K × such that |tε |δ(ε ) ≤ δ(ε). Then tε (vj − v0 ) + v0 ∈ Bδ(ε) (v0 ) = Uε

for any j ∈ J.

It follows that f (tε (vj − v0 ) + v0 ) − f (v0 ) − Dv0 f (tε (vj − v0 )) ≤ εtε (vj − v0 ) and hence that f (tε (vj − v0 ) + v0 ) − f (v0 ) − Dv0 f (vj − v0 ) ≤ εvj − v0 . tε By letting ε tend to zero we obtain Dv0 f (vj ) = Dv0 f (vj − v0 ) =

f (t(vj − v0 ) + v0 ) − f (v0 ) t t∈K × ,t→0 lim

for any j ∈ J. But as a linear map Dv0 f is uniquely determined by its value on the basis vectors vj . The continuous linear map Dv0 f : V −→ W is called (if it exists) the derivative of f in the point v0 ∈ U . In case V = K we also write f  (a0 ) := Da0 f (1). Remark 4.1.

i. If f is differentiable in v0 then it is continuous in v0 .

ii. (Chain rule) Let V, W1 , and W2 be normed K-vector spaces, U ⊆ V and U1 ⊆ W1 be open subsets, and f : U −→ U1 and g : U1 −→ W2 be maps; suppose that f is differentiable in some v0 ∈ U and g is differentiable in f (v0 ); then g ◦ f is differentiable in v0 and Dv0 (g ◦ f ) = Df (v0 ) g ◦ Dv0 f.

4

Differentiability

19

iii. A continuous linear map u : V −→ W is differentiable in any v0 ∈ V and Dv0 u = u; in particular, in the situation of ii. we have Dv0 (u ◦ f ) = u ◦ Dv0 f. iv. (Product rule) Let V, W1 , . . . , Wm , and W be normed K-vector spaces, let U ⊆ V be an open subset with maps fi : U −→ Wi , and let u : W1 × · · · × Wm −→ W be a continuous multilinear map; suppose that f1 , . . . , fm all are differentiable in some point v0 ∈ U ; then u(f1 , . . . , fm ) : U −→ W is differentiable in v0 and Dv0 (u(f1 , . . . , fm )) =

m

u(f1 (v0 ), . . . , Dv0 fi , . . . , fm (v0 )).

i=1

Proof. These are standard arguments of which we only recall the proof of ii. Let ε > 0 and choose a δ > 0 such that δ 2 , δDv0 f , δDf (v0 ) g ≤ ε (here   refers to the operator norm of course). By assumption on g we have g(w) − g(f (v0 )) − Df (v0 ) g(w − f (w0 )) ≤ δw − f (v0 )

(1)

for any w ∈ Uδ (f (v0 )). By the differentiability and hence continuity of f in v0 there exists an open neighbourhood U (v0 ) ⊆ U of v0 such that f (U (v0 )) ⊆ Uδ (f (v0 )) and f (v) − f (v0 ) − Dv0 f (v − v0 ) ≤ δv − v0 

(2)

for any v ∈ U (v0 ). In particular (3)

f (v) − f (v0 ) = Dv0 f (v − v0 ) + f (v) − f (v0 ) − Dv0 f (v − v0 ) ≤ max(Dv0 f  · v − v0 , δv − v0 )

for any v ∈ U (v0 ). We now compute g(f (v)) − g(f (v0 )) − Df (v0 ) g ◦ Dv0 f (v − v0 ) = g(f (v)) − g(f (v0 )) − Df (v0 ) g(f (v) − f (v0 ))   + Df (v0 ) g f (v) − f (v0 ) − Dv0 f (v − v0 )    ≤ max δf (v) − f (v0 ), Df (v0 ) g · f (v) − f (v0 ) − Dv0 f (v − v0 )

(1)

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  ≤ max δf (v) − f (v0 ), Df (v0 ) g · δv − v0 

(2)

  ≤ max δDv0 f , δ 2 , δDf (v0 ) g · v − v0 

(3)

≤ εv − v0  for any v ∈ U (v0 ). Suppose that the vector space V = V1 ⊕ · · · ⊕ Vm is the direct sum of finitely many vector spaces V1 , . . . , Vm and that the norm on V is the maximum of its restrictions to the Vi . Write a vector v0 ∈ U as v0 = v0,1 + · · · + v0,m with v0,i ∈ Vi . For each 1 ≤ i ≤ m there is an open neighbourhood Ui ⊆ Vi of v0,i such that U1 + · · · + Um ⊆ U. Therefore the maps fi : Ui −→ W vi −→ f (v0,1 + · · · + vi + · · · + v0,m ) are well defined. If it exists the continuous linear map f := Dv0,i fi : Vi −→ W Dv(i) 0 is called the i-th partial derivative of f in v0 . We recall that differentiability of f in v0 implies the existence of all partial derivatives together with the identity m Dv0 f = Dv(i) f. 0 i=1 f

Let us go back to our initial situation V ⊇ U −→ W . Definition. The map f is called strictly differentiable in v0 ∈ U if there exists a continuous linear map Dv0 f : V −→ W such that for any ε > 0 there is an open neighbourhood Uε ⊆ U of v0 with f (v1 ) − f (v2 ) − Dv0 f (v1 − v2 ) ≤ εv1 − v2 

for any v1 , v2 ∈ Uε .

Exercise. Suppose that f is strictly differentiable in every point of U . Then the map U −→ L(V, W ) v −→ Dv f is continuous.

4

Differentiability

21

Our goal for the rest of this section is to discuss the local invertibility properties of strictly differentiable maps. Lemma 4.2. Let Bδ (v0 ) be a closed ball in a K-Banach space V and let f : Bδ (v0 ) −→ V be a map for which there exists a 0 < ε < 1 such that (4)

f (v1 ) − f (v2 ) − (v1 − v2 ) ≤ εv1 − v2 

for any v1 , v2 ∈ Bδ (v0 );

then f induces a homeomorphism

Bδ (v0 ) −−→ Bδ (f (v0 )). Proof. By Lemma 1.1 we have (5)

f (v1 ) − f (v2 ) = v1 − v2 

for any v1 , v2 ∈ Bδ (v0 ).

In particular, f is a homeomorphism onto its image which satisfies f (Bδ (v0 )) ⊆ Bδ (f (v0 )). It remains to show that this latter inclusion in fact is an equality. Let w ∈ Bδ (f (v0 )) be an arbitrary but fixed vector. For any v  ∈ Bδ (v0 ) we put v  := w + v  − f (v  ). We compute v  − v0  ≤ max(v  − v  , v  − v0 ) ≤ max(w − f (v  ), δ) ≤ max(w − f (v0 ), f (v0 ) − f (v  ), δ) ≤ max(δ, f (v0 ) − f (v  )) (5)

≤ max(δ, v0 − v  )

≤δ which means that v  ∈ Bδ (v0 ). Hence we may define inductively a sequence (vn )n≥0 in Bδ (v0 ) by vn+1 := w + vn − f (vn ). Using (4) we see that vn+1 − vn  = vn − f (vn ) − (vn−1 − f (vn−1 )) = f (vn−1 ) − f (vn ) − (vn−1 − vn ) ≤ εvn − vn−1 

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and therefore vn+1 − vn  ≤ εn v1 − v0  ≤ εn δ for any n ≥ 1. It follows that (vn )n is a Cauchy sequence and, since V is complete, is convergent. Because Bδ (v0 ) is closed in V the limit v := limn→∞ vn lies in Bδ (v0 ). By passing to the limit in the defining equation and using the continuity of f we finally obtain that f (v) = w. Proposition 4.3. (Local invertibility) Let V and W be K-Banach spaces, U ⊆ V be an open subset, and f : U −→ W be a map which is strictly ∼ = differentiable in the point v0 ∈ U ; suppose that the derivative Dv0 f : V − →W is a topological isomorphism; then there are open neighbourhoods U0 ⊆ U of v0 and U1 ⊆ W of f (v0 ) such that:

i. f : U0 −−→ U1 is a homeomorphism; ii. the inverse map g : U1 −→ U0 is strictly differentiable in f (v0 ), and Df (v0 ) g = (Dv0 f )−1 . Proof. We consider the map f1 := (Dv0 f )−1 ◦ f : U −→ V. As a consequence of the chain rule it is strictly differentiable in v0 and Dv0 f1 = (Dv0 f )−1 ◦ Dv0 f = idV . Hence, fixing some 0 < ε0 < 1 we find a neighbourhood Bδ0 (v0 ) ⊆ U of v0 such that the condition (4) in Lemma 4.2 is satisfied. The lemma then says that f1 : U0 := Bδ0 (v0 ) −−→ Bδ0 (f1 (v0 )) is a homeomorphism. Since Dv0 f is a homeomorphism by assumption U1 := → U1 Dv0 f (Bδ0 (f1 (v0 ))) is an open neighbourhood of f (v0 ) in W and f : U0 − is a homeomorphism. To prove ii. let ε > 0. We have (Dv0 f )−1  > 0 since (Dv0 f )−1 is bijective. Hence we find a δ > 0 such that δ(Dv0 f )−1  < 1

and

δ(Dv0 f )−1 2 ≤ ε.

By the strict differentiability of f in v0 we have f (v1 ) − f (v2 ) − Dv0 f (v1 − v2 ) ≤ δv1 − v2 

for any v1 , v2 ∈ Uδ .

4

Differentiability

23

Applying (Dv0 f )−1 gives (Dv0 f )−1 (f (v1 ) − f (v2 )) − (v1 − v2 ) ≤ δ(Dv0 f )−1  · v1 − v2 . By our choice of δ and Lemma 1.1 we deduce (Dv0 f )−1 (f (v1 ) − f (v2 )) = v1 − v2 . Combining the last two formulas we obtain v1 − v2 − (Dv0 f )−1 (f (v1 ) − f (v2 )) ≤ δ(Dv0 f )−1  · (Dv0 f )−1 (f (v1 ) − f (v2 )) ≤ δ(Dv0 f )−1 2 · f (v1 ) − f (v2 ) ≤ εf (v1 ) − f (v2 ) for any v1 , v2 ∈ Uδ . It follows that g(w1 ) − g(w2 ) − (Dv0 f )−1 (w1 − w2 ) ≤ εw1 − w2  for any w1 , w2 ∈ f (U0 ∩ Uδ ). Since f (U0 ∩ Uδ ) is an open neighbourhood of f (v0 ) in U1 this establishes ii. In regard to the assumption on the derivative in the above proposition it is useful to have in mind the open mapping theorem (cf. [NFA] Cor. 8.7) which says that any continuous linear bijection between K-Banach spaces necessarily is a topological isomorphism. We also point out the trivial fact that any linear map between two finite dimensional K-Banach spaces is continuous. Corollary 4.4. Let U ⊆ K n be an open subset and f : U −→ K m be a map which is strictly differentiable in v0 ∈ U ; suppose that Dv0 f is injective; then there are open neighbourhoods U0 ⊆ U of v0 and U1 ⊆ K m of f (v0 ) as well as a ball Bε (0) ⊆ K m−n around zero and linearly independent vectors w1 , . . . , wm−n ∈ K m such that the map

U0 × Bε (0) −−→ U1 (v, (a1 , . . . , am−n )) −→ f (v) + a1 w1 + · · · + am−n wm−n is a homeomorphism.

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Proof. We choose the vectors wi in such a way that K m = im(Dv0 f ) ⊕ Kw1 ⊕ · · · ⊕ Kwm−n . Then the linear map ∼ =

K n × K m−n −−→ K m

u:

(v, (a1 , . . . , am−n )) −→ (Dv0 f )(v) + a1 w1 + · · · + am−n wm−n is a topological isomorphism. One checks that the map f˜ :

U × K m−n −→ K m (v, (a1 , . . . , am−n )) −→ f (v) + a1 w1 + · · · + am−n wm−n

is strictly differentiable in (v0 , 0) with D(v0 ,0) f˜ = u. Now apply Prop. 4.3.i. Corollary 4.5. Let U ⊆ K n be an open subset and f : U −→ K m be a map which is strictly differentiable in v0 ∈ U ; suppose that Dv0 f is surjective; then there are open neighbourhoods U0 ⊆ U of v0 and U1 ⊆ K m of f (v0 ) as well as a ball Bε (0) ⊆ K n−m around zero and a linear map p : K n −→ K n−m such that the map

U0 −−→ U1 × Bε (0) v −→ (f (v), p(v) − p(v0 )) is a homeomorphism; in particular, the restricted map f : U0 −→ K m is open. Proof. We choose a decomposition K n = ker(Dv0 f ) ⊕ C and let

p : K n −→ ker(Dv0 f ) ∼ = K n−m

be the corresponding projection map. Then u : K n −→ K m × K n−m v −→ ((Dv0 f )(v), p(v)) is a topological isomorphism. One checks that f˜ : U −→ K m × K n−m v −→ (f (v), p(v) − p(v0 )) is strictly differentiable in v0 with Dv0 f˜ = u. Now apply Prop. 4.3.i.

5

Power Series

25

We finish this section with a trivial observation. A map f : X −→ A from some topological space X into some set A is called locally constant if f −1 (a) is open (and closed) in X for any a ∈ A. Lemma 1.4 implies that in our standard situation of two normed Kvector spaces V and W and an open subset U ⊆ W there are plenty of locally constant maps f : U −→ W . They all are strictly differentiable in any v0 ∈ U with Dv0 f = 0.

5

Power Series

Let V be a K-Banach space. By a power series f (X) in r variables X = (X1 , . . . , Xr ) with coefficients in V we mean a formal series X α vα with vα ∈ V. f (X) = α∈Nr0

Here and in the following we use the usual conventions for multi-indices X α := X1α1 · . . . · Xrαr

and |α| := α1 + · · · + αr

if α = (α1 , . . . , αr ) ∈ Nr0 (with N0 := N ∪ {0}).  For any ε > 0 the power series f (X) = α X α vα is called ε-convergent if lim ε|α| vα  = 0. |α|→∞

Remark. If f (X) is ε-convergent then it also is δ-convergent for any 0 < δ ≤ ε. The K-vector space Fε (K r ; V ) := all ε-convergent power series f (X) =



X α vα

α∈Nr0

is normed by

f ε := max ε|α| vα . α

By a straightforward generalization of the argument for c0 (N) it is shown that Fε (K r ; V ) is a Banach space. By the way, in case ε = |c| for some c ∈ K × the map ∼ =

c0 (Nr0 ) −−→ F|c| (K r ; K) aα Xα (aα )α −→ |α| c α is an isometric linear isomorphism.

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Remark. The vector space Fε (K r ; V ) together with its topology only depends on the topology of V (and not on its specific norm). Proof. Let   be a second norm on V which induces the same topology as  . Applying Lemma 2.3 to the identity map idV we obtain two constants c1 , c2 > 0 such that c1   ≤   ≤ c2  . Then obviously lim ε|α| vα  = 0 if and only if

lim ε|α| vα  = 0.

|α|→∞

|α|→∞

This means that using   instead of   leads to the same vector space Fε (K r ; V ) but which carries the two norms  ε and f ε := maxα ε|α| vα  . The above inequalities immediately imply the analogous inequalities c1  ε ≤  ε ≤ c2  ε . By Lemma 2.3 for the identity map on Fε (K r ; V ) this means that  ε and  ε induce the same topology. Consider a convergent series f=

∞

fi

i=0

in the Banach space Fε (K r ; V ). Suppose that f (X) =



X α vα

and fi (X) =

α



X α vi,α .

α

We have

 n n n fi = X α vα − vi,α = max ε|α| vα − vi,α . f − α α i=0

i=0

ε

This shows that vα =

∞

vi,α

ε

i=0

for any α ∈ Nr0

i=0

which means that limits in Fε (K r ; V ) can be computed coefficientwise.

5

Power Series

27

Let Bε (0) denote the closed ball around zero in K r of radius ε. We recall that K r always is equipped with the norm (a1 , . . . , ar ) = max1≤i≤r |ai |. By Lemma 3.1.i. we have the K-linear map Fε (K r ; V ) −→ K-vector space of maps Bε (0) −→ V X α vα −→ f˜(x) := xα vα . f (X) = α

α

Remark 5.1. For any x ∈ Bε (0) the linear evaluation map Fε (K r ; V ) −→ V f −→ f˜(x) is continuous of operator norm ≤ 1. Proof. We have

α x vα ≤ max ε|α| vα  = f ε . f˜(x) = α α

Proposition 5.2. Let u : V1 × V2 −→ V be a continuous bilinear map between K-Banach spaces; then U : Fε (K r ; V1 ) × Fε (K r ; V2 ) −→ Fε (K r ; V ) 

 α α α X vα , X wα −→ X u(vβ , wγ ) α

α

α

β+γ=α

is a continuous bilinear map satisfying U (f, g)∼ (x) = u(f˜(x), g˜(x))

for any x ∈ Bε (0)

and any f ∈ Fε (K r ; V1 ) and g ∈ Fε (K r ; V2 ). Proof. By a similar argument as for Lemma 2.3 the bilinear map u is continuous if and only if there is a constant c > 0 such that u(v1 , v2 ) ≤ cv1  · v2 

for any v1 ∈ V1 , v2 ∈ V2 .

We therefore have   ε|α| u(vβ , wγ ) ≤ c max ε|β| vβ  · ε|γ| vγ  . β+γ=α β+γ=α

28

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This shows (compare the proof of Lemma 3.2) that with f and g also U (f, g) is ε-convergent and that U (f, g)ε ≤ cf ε gε . Hence U is well defined, bilinear, and continuous. The asserted identity between evaluations is an immediate generalization of Lemma 3.2. Proposition 5.3. Fε (K r ; K) is a commutative K-algebra with respect to the multiplication

 

 bα X α cα X α := bβ cγ X α ; α

α

α

β+γ

in addition we have g (x) (f g)∼ (x) = f˜(x)˜

for any x ∈ Bε (0)

as well as f gε = f ε gε for any f, g ∈ Fε (K r ; K). Proof. Apart from the norm identity this is a special case of Prop. 5.2 (and its proof) for the multiplication in K as the bilinear map. It remains to show that f gε ≥ f ε gε . Let ≥ denote the lexicographic order on Nr0 , and let μ and ν be lexicographically minimal multi-indices such that |bμ |e|μ| = f ε

and |cν |ε|ν| = gε , respectively.

Put λ := μ + ν and consider any equation of the form λ = β + γ. Claim: β ≤ μ or γ ≤ ν. Otherwise we would have β > μ and γ > ν. This means that there are 1 ≤ i, j ≤ r such that β1 = μ1 , . . . , bi−1 = μi−1 , and βi > μi as well as γ1 = ν1 , . . . , γj−1 = νj−1 , and γj > νj .

5

Power Series

29

By symmetry we may assume that i ≤ j. We then obtain the contradiction λi = μi + νi < βi + γi = λi . This establishes the claim. We of course have |bβ |ε|β| ≤ f ε

and |cγ |ε|γ| ≤ gε .

But in case (β, γ) = (μ, ν) the fact that β < μ or γ < ν together with the minimality property of μ and ν implies that |bβ |ε|β| < f ε

or

|cγ |ε|γ| < gε .

It follows that |bβ cγ |ε|λ| = |bβ |ε|β| · |cγ |ε|γ| < f ε gε whenever β + γ = λ but (β, γ) = (μ, ν). We conclude that







bβ cγ ε|λ| = |bμ cν |ε|λ| = f ε gε . f gε ≥



β+γ=λ

Proposition 5.4. Let g ∈ Fδ (K r ; K n ) such that gδ ≤ ε; then Fε (K n ; V ) −→ Fδ (K r ; V ) Y β vβ −→ f ◦ g(X) := g(X)β vβ f (Y ) = β

β

is a continuous linear map of operator norm ≤ 1 which satisfies (f ◦ g)∼ (x) = f˜(˜ g (x))

for any x ∈ Bδ (0) ⊆ K r .

Proof. Using the obvious identification Fδ (K ; K ) = r

n

n 

Fδ (K r ; K)

i=1

g = (g1 , . . . , gn )

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we have maxi gi δ = gδ ≤ ε. The Prop. 5.3 therefore implies that g(X)β ∈ Fδ (K r ; K) for each β and n n   βi β g(X) δ = gi = gi βδ i ≤ ε|β| . i=1

δ

i=1

It follows that g(X)β vβ ∈ Fδ (K r ; V ) for each β with g(X)β vβ δ = g(X)β δ vβ  ≤ ε|β| vβ . Since the right  hand side goes to zero by the ε-convergence of f the series f ◦ g(X) = β g(X)β vβ is convergent in the Banach space Fδ (K r ; V ). Moreover, we have f ◦ gδ ≤ max g(X)β vβ δ ≤ max ε|β| vβ  = f ε . β

β

To see the asserted identity between evaluations we first note that by Remark 5.1 the map g˜ indeed maps the ball Bδ (0) ⊆ K r into the ball Bε (0) ⊆ K n . Using Remark 5.1 together with Prop. 5.3 we obtain (g β )∼ (x)vβ = g˜(x)β vβ = f˜(˜ g (x)). (f ◦ g)∼ (x) = β

β

As a consequence of the discussion before Remark 5.1 and of Prop. 5.3 the power series f ◦ g can be computed by formally inserting g into f . The reader is warned that although, for any g ∈ Fδ (K r ; K n ), we have, by Remark 5.1, the inequality sup ˜ g (x) ≤ gδ x∈Bδ (0)

it is, in general, not an equality. This means that we may have g˜(Bδ (0)) ⊆ Bε (0) even if ε < gδ . Then, for any f ∈ Fε (K n ; V ), the composite of maps f˜ ◦ g˜ exists but the composite of power series f ◦ g ∈ Fδ (K r ; V ) may not. Exercise. An example of such a situation is g(X) := X p − X ∈ F1 (Qp ; Qp )

and

f (Y ) :=

∞ n=0

Y n ∈ F p1 (Qp ; Qp ).

5

Power Series

31

Corollary 5.5. (Point of expansion) Let f ∈ Fε (K r ; V ) and y ∈ Bε (0); then there exists an fy ∈ Fε (K r ; V ) such that fy ε = f ε and f˜(x) = f˜y (x − y)

for any x ∈ Bε (0) = Bε (y).

r Prop. 5.4 Proof. Let e1 , . . . , er denote the standard r basis of K . Applying to the power series g(X) := X + y = i=1 Xi ei + y ∈ Fε (K r ; K r ) which satisfies gε ≤ ε we obtain the existence of fy (X) := f (X + y) satisfying

fy ε ≤ f ε

and f˜y (x) = f˜(x + y)

for x ∈ Bε (0).

By symmetry we also have f ε = (fy )−y ε ≤ fy ε .

It will be convenient in the following to use the short notation i := (0, . . . , 1, . . . , 0) for the multi-index whose only nonzero entry  is a 1 in the i-th place. Suppose that the power series f (X) = α X α vα is ε-convergent. Then also, for any 1 ≤ i ≤ r, its i-th formal partial derivative ∂f (X) := X α−i αi vα ∂Xi α is ε-convergent (since αi vα  ≤ vα ). In case our field K has characteristic zero it follows inductively that   α1 αr ∼ ∂ ∂ 1 vα = ··· f (0) for any α. αi ! · . . . · α r ! ∂X1 ∂Xr Proposition 5.6. The map f˜ is strictly differentiable in every point z ∈ Bε (0) and satisfies   ∂f ∼ (i) ˜ Dz f (1) = (z). ∂Xi Proof. Case 1: We assume that z = 0, and introduce the continuous linear map D:

K r −→ V r (a1 , . . . , ar ) −→ ai vi . i=1

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Let δ > 0 and choose a 0 < δ  < ε such that δ

f ε ≤ δ. ε2

By induction with respect to |α| one checks that |xα − y α | ≤ (δ  )|α|−1 x − y

for any x, y ∈ Bδ (0).

We now compute α α ˜ ˜ (x − y )vα f (x) − f (y) − D(x − y) = |α|≥2

≤ max |xα − y α | · vα  |α|≥2

|xα − y α | |α|≥2 ε|α|

≤ f ε · max

(δ  )|α|−1 · x − y |α|≥2 ε|α| δ = f ε · 2 · x − y ε ≤ δx − y ≤ f ε · max

for any x, y ∈ Bδ (0). This proves that f˜ is strictly differentiable in 0 with D0 f˜ = D and hence   ∂f ∼ (i) ˜ (0). D0 f (1) = vi = ∂Xi Case 2: Let z ∈Bε (0) be an arbitrary point. By Cor. 5.5 we find a power series fz (X) = α X α vα (z) in Fε (K r ; V ) such that f˜(x) = f˜z (x − z)

for any x ∈ Bε (0).

Using the chain rule together with the first case we see that f˜ is strictly differentiable in z with (i) Dz(i) f˜(1) = D0 f˜z (1) = vi (z).

Since fz (X) can be computed by formally inserting X +z into f (X) we have X α vα (z) = (X + z)α vα α

α

5

Power Series

33

and hence vi (z) =



 z

α−i

α i vα =

α

∂f ∂Xi

∼ (z).

By Prop. 5.6 the map ∂ f˜ : Bε (0) −→ V ∂xi x −→ Dx(i) f˜(1) is well defined and satisfies ∂ f˜ = ∂xi



∂f ∂Xi

∼ .

Corollary 5.7. (Taylor expansion) If K has characteristic zero then we have      ∂ α1 ∂ αr ˜ 1 α X ··· f (0). f (X) = α1 ! · . . . · α r ! ∂x1 ∂xr α Corollary 5.8. (Identity theorem for power series) If K has characteristic zero then for any nonzero f ∈ Fε (K r ; V ) there is a point x ∈ Bε (0) such that f˜(x) = 0. In fact much stronger results than Cor. 5.8 hold true. In particular, the assumption on the characteristic of K is superfluous. But this requires a different method of proof (cf. [BGR] 5.1.4 Cor. 5 and subsequent comment). In any case the map Fε (K r ; V ) −→ strictly differentiable maps Bε (0) −→ V f −→ f˜ is injective and commutes with all the usual operations as considered above. We therefore will simplify notations in the following and write very often f for the power series as well as the corresponding map. Proposition 5.9. (Invertibility for power series) Let f (X) ∈ Fε (K r ; K r ) such that f (0) = 0, and suppose that D0 f is bijective; we fix a 0 < δ < ε2 ; then δ < f ε , and there is a uniquely determined g(Y ) ∈

f ε (D0 f )−1 2 r r Fδ (K ; K ) such that g(0) = 0, gδ < ε, and f ◦ g(Y ) = Y ;

34

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in particular, the diagram Bδ (0) ⊆ g

Bε (0)

B f ε (0)

f

is commutative. Proof. Case 1: We assume that D0 f = idK r . Let ai,α X α . f = (f1 , . . . , fr ) and fi (X) = α

Since f (0) = 0 we have ai,0 = 0. Moreover, by Prop. 5.6, the matrix of D0 f in the standard basis of K r is equal to   ∂fi (0) = (ai,j )i,j . ∂Xj i,j But we are in the special case that this matrix is the identity matrix. Hence ai,j = 0 for i = j and ai,i = 1. We therefore see that (6)

fi (X) = Xi +



ai,α X α .

|α|≥2

It follows in particular that f ε ≥ ε and hence (7)

δ<

ε2 ε2 = ≤ ε ≤ f ε . f ε (D0 f )−1 2 f ε

In case 1 of the proof of Prop. 5.6 we have computed that f (x) − f (y) − (x − y) ≤ δ

f ε x − y ε2

for any x, y ∈ Bδ (0).

As δ fε2 ε < 1 this is the condition (4) in Lemma 4.2. We therefore conclude that f : Bδ (0) −−→ Bδ (0)

5

Power Series

35

is a homeomorphism. Furthermore, for |α| ≥ 2 we have |ai,j |δ |α| ≤ f ε

  |α|  2  δ δ f ε ≤ f ε =δ δ 2 < δ. ε ε ε

Hence it follows from (6) that f δ = δ. In a next step we establish the existence of a formal power series g = (g1 , . . . , gr ) with bi,β Y β gi (Y ) = |β|≥1

such that f (g(Y )) = Y. First of all let us check that formally inserting any such g into f is a well defined operation. We formally compute fi (g(Y )) = ai,α g1 (Y )α1 · . . . · gr (Y )αr |α|≥1

=

|α|≥1

=

|γ|≥1

ai,α



α1 b1,β Y

· ... ·

β

|β|≥1





αr br,β Y

|β|≥1

β



ai,α b1,β(1) · . . . · b1,β(α1 ) b2,β(α1 +1) · . . . · br,β(α1 +···+αr ) X γ

···

where in the last expression the multi-indices in the inner sum run over all α, β(1), . . . , β(α1 + · · · + αr ) such that |α|, |β(1)|, . . . , |β(α1 + · · · + αr )| ≥ 1 and β(1) + · · · + β(α1 ) + β(α1 + 1) + · · · + β(α1 + α2 ) + · · · + β(α1 + · · · + αr ) = γ. Because of |β(ν)| ≥ 1 this condition enforces |α| ≤ |γ| so that these inner sums in fact are finite. We now set Yi = fi (g(Y )) and compare coefficients. For γ = i we obtain 1=

r =1

ai, b,i = bi,i .

36

I

For γ = i we have ai,α b1,β(1) · . . . = bi,γ + 0= ···



Foundations

ai,α C(α, γ)

2≤|α|≤|γ|

where C(α, γ) is a (finite) sum of products of the form b1,β(1) · . . . · br,β(α1 +···+αr )

with |β(ν)| ≥ 1 and



β(ν) = γ.

ν

In particular,



|β(ν)| = |γ| and |β(ν)| ≥ 1.

ν

Since the number of summands |β(ν)| is equal to |α| ≥ 2 it follows that |β(ν)| < |γ|. We see that on the right hand side of the equation bi,γ = − ai,α C(α, γ) 2≤|α|≤γ

only coefficients b,β appear with |β| < |γ|. This means that the coefficients βi,γ can be computed recursively from these equations. Hence g exists and is uniquely determined. In addition we check inductively that   f ε |γ|−1 |bi,γ | ≤ ε2 holds true. For |γ| = 1 we have βi,γ = 0 or 1 and the inequality is trivial. If |γ| ≥ 2 then the induction hypothesis implies |C(α, γ)| ≤ max |b1,β(1) | · . . . · |br,β(α1 +···+αr ) | ···   f ε (|β(1)|−1)+···+(|β(α1 +···+αr )|−1) ≤ max ··· ε2   f ε |γ|−|α| = . ε2 Hence we have f ε |C(α, γ)| 2≤|α|≤|γ| 2≤|α|≤|γ| ε|α|    |α|−2   f ε |γ|−|α| f ε |γ|−1 ε f ε ≤ max · = max · ε2 ε2 2≤|α|≤|γ| ε|α| 2≤|α|≤|γ| f ε |γ|−1  f ε = ε2

|bi,γ | ≤

max |ai,α | · |C(α, γ)| ≤

max

5

Power Series

37

(the last identity since ε ≤ f ε ). We deduce that |bi,γ |δ |γ| ≤

 δ

f ε ε2

|γ|−1 δ.

Because of δ fε2 ε < 1 this shows that g is δ-convergent with gδ ≤ δ. But bi,i = 1 then implies that gδ = δ. Altogether we have shown so far that: – f is δ-convergent with f δ = δ; – there is a uniquely determined δ-convergent g with g(0) = 0, gδ = δ < ε, and f ◦ g(Y ) = Y. Using Prop. 5.4 we see that f˜ ◦ g˜ = id so that f˜

Bδ (0)

Bδ (0) g˜

are homeomorphisms which are inverse to each other. But we also conclude that g ◦ f (X) exists as a δ-convergent power series as well and satisfies (g ◦ f )∼ = g˜ ◦ f˜ = id. The identity theorem Cor. 5.8 then implies that g ◦ f (X) = X. Case 2: Let D0 f be arbitrary bijective. If (aij )i,j is the matrix of (D0 f )−1 in the standard basis of K r then the operator norm is given by (D0 f )−1  = max |aij |. i,j

Viewed as a power series (D0 f )−1 is ε -convergent for any ε > 0 with (D0 f )−1 ε = ε · max |ai,j | = ε (D0 f )−1 . i,j

In the following we put ε := Prop. 5.4 we obtain

ε .

(D0 f )−1

Then (D0 f )−1 ε = ε, and from

f0 := f ◦ (D0 f )−1 ∈ Fε (K r ; K r ) and f0 ε ≤ f ε . Any δ as in the assertion then satisfies δ<

ε2 ε2 ε2 = ≤ . −1 2 f ε (D0 f )  f ε f0 ε

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Obviously f0 (0) = 0, and D0 f0 = idK r by the chain rule. So we may apply the first case to f0 and obtain a uniquely determined g0 ∈ Fδ (K r ; K r ) such that g0 (0) = 0, g0 δ = δ, and f0 ◦ g0 (Y ) = Y as well as

ε2 ε ≤ ε = ≤ f0 ε ≤ f ε f0 ε (D0 f )−1 

δ< by (7). We define

g := (D0 f )−1 ◦ g0 ∈ Fδ (K r ; K r ). Then g(0) = 0 and f (g(Y )) = f ((D0 f )−1 ◦ g0 (Y )) = f ◦ (D0 f )−1 (g0 (Y )) = f0 (g0 (Y )) = Y. In addition, Prop. 5.4 implies gδ ≤ (D0 f )−1 δ = δ(D0 f )−1  < ε. The unicity of g easily follows from the unicity of g0 . The next result is more or less obvious. Proposition 5.10. Let u : V −→ W be a continuous linear map between K-Banach spaces; then Fε (K r ; V ) −→ Fε (K r ; W ) X α vα −→ u ◦ f (X) := X α u(vα ) f (X) = α

α

is a continuous linear map of operator norm ≤ u which satisfies u ◦ f (x) = u(f (x))

6

for any x ∈ Bε (0).

Locally Analytic Functions

Let U ⊆ K r be an open subset and V be a K-Banach space. Definition. A function f : U −→ V is called locally analytic if for any point x0 ∈ U there is a ball Bε (x0 ) ⊆ U around x0 and a power series F ∈ Fε (K r ; V ) such that f (x) = F (x − x0 )

for any x ∈ Bε (x0 ).

6

Locally Analytic Functions

39

The set C an (U, V ) := all locally analytic functions f : U −→ V is a K-vector space with respect to pointwise addition and scalar multiplication. For f1 , f2 ∈ C an (U, V ) and x0 ∈ U let Fi ∈ Fεi (K r ; V ) such that fi (x) = Fi (x − x0 ) for any x ∈ Bεi (x0 ). Put ε := min(ε1 , ε2 ). Then F1 + F2 ∈ Fε (K r ; V ) and (f1 + f2 )(x) = (F1 + F2 )(x − x0 )

for any x ∈ Bε (x0 ).

The vector space C an (U, V ) carries a natural topology which we will discuss later on in a more general context. Example. By Cor. 5.5 we have F˜ ∈ C an (Bε (0), V ) for any F ∈ Fε (K r ; V ). Proposition 6.1. Suppose that f : U −→ V is locally analytic; then f is strictly differentiable in every point x0 ∈ U and the function x −→ Dx f is locally analytic in C an (U, L(K r , V )). Proof. Let F ∈ Fε (K r ; V ) such that f (x) = F˜ (x − x0 )

for any x ∈ Bε (x0 ).

From Prop. 5.6 and the chain rule we deduce that f is strictly differentiable in every x ∈ Bε (x0 ) and   r r ∂F ∼ (i) ˜ Dx f ((a1 , . . . , ar )) = ai Dx−x0 F (1) = ai (x − x0 ). ∂Xi i=1

Let

i=1

∂F (X) = X α vi,α . ∂Xi α

For any multi-index α we introduce the continuous linear map K r −→ V

Lα :

(a1 , . . . , ar ) −→ a1 v1,α + · · · + ar vr,α . Because of Lα  ≤ maxi vi,α  we have X α Lα ∈ Fε (K r ; L(K r , V )) G(X) := α

and ˜ − x0 ) Dx f = G(x

for any x ∈ Bε (x0 ).

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Remark 6.2. If K has characteristic zero then, for any function f : U −→ V , the following conditions are equivalent: i. f is locally constant; ii. f is locally analytic with Dx f = 0 for any x ∈ U . Proof. This is an immediate consequence of the Taylor formula in Cor. 5.7. We now give a list of more or less obvious properties of locally analytic functions. 1) For any open subset U  ⊆ U we have the linear restriction map C an (U, V ) −→ C an (U  , V ) f −→ f |U  . 2) For any open and closed subset U  ⊆ U we have the linear map C an (U  , V ) −→ C an (U, V )  f −→ f! (x) :=

f (x) if x ∈ U  , 0 otherwise

called extension by zero.  3) If U = i∈I Ui is a covering by pairwise disjoint open subsets then  C an (Ui , V ) C an (U, V ) ∼ = i∈I

f −→ (f |Ui )i . 4) For any two K-Banach spaces V and W we have C an (U, V ⊕ W ) ∼ = C an (U, V ) ⊕ C an (U, W ) f −→ (prV ◦f, prW ◦f ). In particular C (U, K ) ∼ = an

n

n 

C an (U, K).

i=1

5) For any continuous bilinear map u : V1 × V2 −→ V between K-Banach spaces we have the bilinear map

6

Locally Analytic Functions

41

C an (U, V1 ) × C an (U, V2 ) −→ C an (U, V ) (f, g) −→ u(f, g) (cf. Prop. 5.2). In particular, C an (U, K) is a K-algebra (cf. Prop. 5.3), and C an (U, V ) is a module over C an (U, K). 6) For any continuous linear map u : V −→ W between K-Banach spaces we have the linear map C an (U, V ) −→ C an (U, W ) f −→ u ◦ f (cf. Prop. 5.10). Lemma 6.3. Let U  ⊆ K n be an open subset and let g ∈ C an (U, K n ) such that g(U ) ⊆ U  ; then the map C an (U  , V ) −→ C an (U, V ) f −→ f ◦ g is well defined and K-linear. Proof. Let x0 ∈ U and put y0 := g(x0 ) ∈ U  . We choose a ball Bε (y0 ) ⊆ U  and a power series F ∈ Fε (K n ; V ) such that f (y) = F (y − y0 )

for any y ∈ Bε (y0 ).

We also choose a ball Bδ (x0 ) ⊆ U and a power series G ∈ Fδ (K r ; K n ) such that g(x) = G(x − x0 ) for any x ∈ Bδ (x0 ). Observing that G − G(0)δ ≤

δ G − G(0)δ δ

for any 0 < δ  ≤ δ

we may decrease δ so that G − y0 δ = G − G(0)δ ≤ ε (and, in particular, g(Bδ (x0 )) ⊆ Bε (y0 )) holds true. It then follows from Prop. 5.4 that F ◦ (G − y0 ) ∈ Fδ (K r ; V ) and (F ◦ (G − y0 ))∼ (x − x0 ) = F (G(x − x0 ) − y0 ) = F (g(x) − y0 ) = f (g(x)) for any x ∈ Bδ (x0 ).

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The last result can be expressed by saying that the composite of locally analytic functions again is locally analytic. Proposition 6.4. (Local invertibility) Let U ⊆ K r be an open subset and let f ∈ C an (U, K r ); suppose that Dx0 f is bijective for some x0 ∈ U ; then there are open neighbourhoods U0 ⊆ U of x0 and U1 ⊆ K r of f (x0 ) such that:

i. f : U0 −−→ U1 is a homeomorphism; ii. the inverse map g : U1 −→ U0 is locally analytic, i. e., g ∈ C an (U1 , K r ). Proof. According to Prop. 4.3 we find open neighbourhoods U0 ⊆ U of x0 and U1 ⊆ K r of f (x0 ) such that

f : U0 −−→ U1

is a homeomorphism.

We choose a ball Bε (x0 ) ⊆ U0 and a power series F ∈ Fε (K r ; K r ) such that for any x ∈ Bε (x0 ).

f (x) = F (x − x0 )

The power series F1 (X) := F (X) − f (x0 ) ∈ Fε (K r ; K r ) satisfies F1 (0) = 0. Moreover, D0 F1 = Dx0 f is invertible. By Prop. 5.9 we therefore find, for a sufficiently small 0 < δ < ε, a power series G1 (Y ) ∈ Fδ (K r ; K r ) such that G1 (0) = 0, G1 δ < ε, and F1 ◦ G1 (Y ) = Y. In particular, G1 : Bδ (0) −→ Bε (0) is locally analytic. Hence the composite −f (x0 )

G

+x

1 0 Bε (0) −−−→ Bε (x0 ) g : U1 := Bδ (f (x0 )) −−−−−→ Bδ (0) −−→

is locally analytic and satisfies f ◦ g(y) = f (G1 (y − f (x0 )) + x0 ) = F (G1 (y − f (x0 ))) = F1 (G1 (y − f (x0 ))) + f (x0 ) = y − f (x0 ) + f (x0 ) = y for any y ∈ U1 . By further decreasing δ we may assume that U1 ⊆ U1 , and by setting U0 := g(U1 ) we obtain the commutative diagram U0

f

U1

⊆ ⊆

Bε (x0 ) ⊆ f

U0

g

U1

6

Locally Analytic Functions

43

in which the two lower horizontal arrows both are locally analytic and are inverse to each other. There are “locally analytic versions” of the Corollaries 4.4 and 4.5 the formulation of which we leave to the reader. We have done so already and we will systematically continue to call a map f : U −→ U  between open subsets U ⊆ K r and U  ⊆ K n locally f

⊆

analytic if the composite U −→ U  −−→ K n is a locally analytic function.

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Chapter II

Manifolds We continue to fix the nonarchimedean field (K, | |). But we change our system of notations insofar as from now on we will denote K-Banach spaces by letters like E whereas we reserve letters like U and V for open subsets in a topological space.

7

Charts and Atlases

Let M be a Hausdorff topological space. Definition. i. A chart for M is a triple (U, ϕ, K n ) consisting of an open subset U ⊆ M and a map ϕ : U −→ K n such that: (a) ϕ(U ) is open in K n , 

(b) ϕ : U −−→ ϕ(U ) is a homeomorphism. ii. Two charts (U1 , ϕ1 , K n1 ) and (U2 , ϕ2 , K n2 ) for M are called compatible if both maps ϕ1 (U1 ∩ U2 )

ϕ2 ◦ϕ−1 1 ϕ1 ◦ϕ−1 2

ϕ2 (U1 ∩ U2 )

are locally analytic. We note that the condition in part ii. of the above definition makes sense since ϕ1 (U1 ∩ U2 ) is open in K ni . If (U, ϕ, K n ) is a chart then the open subset U is called its domain of definition and the integer n ≥ 0 its dimension. Usually we omit the vector space K n from the notation and simply write (U, ϕ) instead of (U, ϕ, K n ). If x is a point in U then (U, ϕ) is also called a chart around x. Lemma 7.1. Let (Ui , ϕi , K ni ) for i = 1, 2 be two compatible charts for M ; if U1 ∩ U2 = ∅ then n1 = n2 . Proof. Let x ∈ U1 ∩ U2 and put xi := ϕi (x). We consider the locally analytic maps ϕ1 (U1 ∩ U2 )

f :=ϕ2 ◦ϕ−1 1 g:=ϕ1 ◦ϕ−1 2

ϕ2 (U1 ∩ U2 ).

P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8 2, © Springer-Verlag Berlin Heidelberg 2011

45

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They are differentiable and inverse to each other, and x2 = f (x1 ). Hence, by the chain rule, the derivatives K n1

Dx1 f

K n2

Dx2 g

are linear maps inverse to each other. It follows that n1 = n2 . Definition. i. An atlas for M is a set A = {(Ui , ϕi , K ni )}i∈I of charts for M any two of which are compatible and which cover M in the sense that M = i∈I Ui . ii. Two atlases A and B for M are called equivalent if A ∪ B also is an atlas for M . iii. An atlas A for M is called maximal if any equivalent atlas B for M satisfies B ⊆ A. Remark 7.2. lation.

i. The equivalence of atlases indeed is an equivalence re-

ii. In each equivalence class of atlases there is exactly one maximal atlas. Proof. i. Let A, B, and C be three atlases such that A is equivalent to B and B is equivalent to C. Then A is equivalent to C if we show that any chart (U1 , ϕ1 ) in A is compatible with any chart (U2 , ϕ2 ) in C. By symmetry it suffices to show that the map ϕ2 ◦ ϕ−1 1 : ϕ1 (U1 ∩ U2 ) −→ ϕ2 (U1 ∩ U2 ) is locally analytic in a sufficiently small open neighbourhood of ϕ1 (x) for any point x ∈ U1 ∩ U2 . Since B covers M we find a chart (V, ψ) around x in B. By assumption (V, ψ) is compatible with both (U1 , ϕ1 ) and (U2 , ϕ2 ). Then ϕ1 (U1 ∩ V ∩ U2 ) is an open neighbourhood of ϕ1 (x) in ϕ1 (U1 ∩ U2 ) on which −1 the map ϕ2 ◦ ϕ−1 1 is the composite of the two locally analytic maps ϕ2 ◦ ψ −1 and ψ ◦ ϕ1 . Hence it is locally analytic by Lemma 6.3. ii. If the given equivalence class consists of the atlases Aj for j ∈ J then A := j∈J Aj is the unique maximal atlas in this class. Lemma 7.3. If A is a maximal atlas for M the domains of definition of all the charts in A form a basis of the topology of M . Proof. Let U ⊆ M be an open subset. We have to show that U is the union of the domains of definition of the charts in some subset of A, or equivalently that for any point x ∈ U we find a chart (Ux , ϕx ) around x in A such that

8

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47

Ux ⊆ U . Since A covers M we at least find a chart (Ux , ϕx ) around x in A. We put Ux := Ux ∩ U and ϕx := ϕx |Ux . Clearly (Ux , ϕx ) is a chart around x for M such that Ux ⊆ U . We claim that (Ux , ϕx ) is compatible with any chart (V, ψ) in A. But we do have the locally analytic maps ϕx (Ux

∩V)

ψ◦ϕ−1 x ϕx ◦ψ −1

ψ(Ux ∩ V )

which restrict to the locally analytic maps ϕx (Ux ∩ V )

ψ◦ϕ−1 x ϕx ◦ψ

ψ(Ux ∩ V ).

Hence B := A ∪ {(Ux , ϕx )} is an atlas equivalent to A. The maximality of A then implies that B ⊆ A and a fortiori (Ux , ϕx ) ∈ A. Definition. An atlas A for M is called n-dimensional if all the charts in A with nonempty domain of definition have dimension n. Remark 7.4. Let A be an n-dimensional atlas for M ; then any atlas B equivalent to A is n-dimensional as well. Proof. Let (V, ψ) be any chart in B and choose a point x ∈ V . We find a chart (U, ϕ) in A around x. Since A and B are equivalent these two charts have to be compatible. It then follows from Lemma 7.1 that both have the same dimension u.

8

Manifolds

Definition. A (locally analytic) manifold (M, A) (over K) is a Hausdorff topological space M equipped with a maximal atlas A. The manifold is called n-dimensional (we write dim M = n) if the atlas A is n-dimensional. By abuse of language we usually speak of a manifold M while considering A as given implicitly. A chart for M will always mean a chart in A. Example. K n will always denote the n-dimensional manifold whose maximal atlas is equivalent to the atlas {(U, ⊆, K n ) : U ⊆ K n open}. Remark 8.1. Let (U, ϕ, K n ) be a chart for the manifold M ; if V ⊆ U is an open subset then (V, ϕ|V, K n ) also is a chart for M .

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Proof. This was shown in the course of the proof of Lemma 7.3. Let (M, A) be a manifold and U ⊆ M be an open subset. Then AU := {(V, ψ, K n ) ∈ A : V ⊆ U }, by Lemma 7.3, is an atlas for U . We claim that AU is maximal. Let (V0 , ψ0 ) be a chart for U which is compatible with any chart in AU . To see that (V0 , ψ0 ) ∈ AU it suffices, by the maximality of A, to show that (V0 , ψ0 ) is compatible with any chart (V, ψ) in A. The Remark 8.1 implies that (V ∩ U, ψ|V ∩ U ) is a chart in A and hence in AU . By assumption (V0 , ψ0 ) is compatible with (V ∩ U, ψ|V ∩ U ). Since V0 ∩ V ⊆ V ∩ U the compatibility of (V0 , ψ0 ) with (V, ψ) follows trivially. The manifold (U, AU ) is called an open submanifold of (M, A). As a nontrivial example of a manifold we discuss the d-dimensional projective space Pd (K) over K. We recall that Pd (K) = (K d+1 \ {0})/ ∼ is the set of equivalence classes in K d+1 \ {0} for the equivalence relation (a1 , . . . , ad+1 ) ∼ (ca1 , . . . , cad+1 ) for any c ∈ K × . As usual we write [a1 : . . . : ad+1 ] for the equivalence class of (a1 , . . . , ad+1 ). With respect to the quotient topology from K d+1 \ {0} the projective space Pd (K) is a Hausdorff topological space. For any 1 ≤ j ≤ d + 1 we have the open subset Uj := {[a1 : . . . : ad+1 ] ∈ Pd (K) : |ai | ≤ |aj | for any 1 ≤ i ≤ d + 1} together with the homeomorphism 

Uj −−→ B1 (0) ⊆ K d   aj−1 aj+1 a1 ad+1 [a1 : . . . : ad+1 ] −→ ,..., , ,..., . aj aj aj aj  The (Uj , ϕj , K d ) are charts for Pd (K) such that j Uj = Pd (K). We claim that they are pairwise compatible. For 1 ≤ j < k ≤ d + 1 the composite ϕj :

ϕ−1 j

ϕ

f : V := {x ∈ B1 (0) : |xk−1 | = 1} −−−→ Uj ∩ Uk −−k→ {y ∈ B1 (0) : |yj | = 1} is given by  f (x1 , . . . , xd ) =

 xj−1 xj x1 1 xk−2 xk xd ,..., , , ,..., , ,..., . xk−1 xk−1 xk−1 xk−1 xk−1 xk−1 xk−1

8

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49

Let a ∈ V be a fixed but arbitrary point and choose a 0 < ε < 1. Then Bε (a) ⊆ V . We consider the power series Fj (X) :=

1 ak−1



−

n≥0

1 ak−1

n n Xk−1

and  (Xi + ai ) Fi (X) := Fj (X) · (Xi−1 + ai−1 )

if 1 ≤ i < j or k ≤ i ≤ d, if j < i < k.

Because of |ak−1 | = 1 we have F := (F1 , . . . , Fd ) ∈ Fε (K d ; K d ). For x ∈ Bε (a) we compute Fj (x − a) =

1 ak−1

  xk−1 − ak−1 n 1 = · − ak−1 ak−1 1 + n≥0

1 xk−1 −ak−1 ak−1

=

1 xk−1

and then f (x) = F (x − a). Hence f is locally analytic. In case j > k the argument is analogous. The above charts therefore form a d-dimensional atlas for Pd (K). Exercise. Let (M, A) and (N, B) be two manifolds. Then A × B := {(U × V ), ϕ × ψ, K m+n ) : (U, ϕ, K m ) ∈ A, (V, ψ, K n ) ∈ B} is an atlas for M × N with the product topology. We call M × N equipped with the equivalent maximal atlas the product manifold of M and N . Let M be a manifold and E be a K-Banach space. Definition. A function f : M −→ E is called locally analytic if f ◦ ϕ−1 ∈ C an (ϕ(U ), E) for any chart (U, ϕ) for M . Remark 8.2. uous.

i. Every locally analytic function f : M −→ E is contin-

ii. Let B be any atlas consisting of charts for M ; a function f : M −→ E is locally analytic if and only if f ◦ ϕ−1 ∈ C an (ϕ(U ), E) for any (U, ϕ) ∈ B.

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The set C an (M, E) := all locally analytic functions f : M −→ E is a K-vector space with respect to pointwise addition and scalar multiplication. It is easy to see that a list of properties 1)–6) completely analogous to the one given in Sect. 6 holds true. In a later section we will come back to a more detailed study of this vector space. Let now M and N be two manifolds. The following result is immediate. Lemma 8.3. For a map g : M −→ N the following assertions are equivalent: i. g is continuous and ψ ◦ g ∈ C an (g −1 (V ), K n ) for any chart (V, ψ, K n ) for N ; ii. for any point x ∈ M there exist a chart (U, ϕ, K m ) for M around x and a chart (V, ψ, K n ) for N around g(x) such that g(U ) ⊆ V and ψ ◦ g ◦ ϕ−1 ∈ C an (ϕ(U ), K n ). Definition. A map g : M −→ N is called locally analytic if the equivalent conditions in Lemma 8.3 are satisfied. Lemma 8.4. i. If g : M −→ N is a locally analytic map and E is a K-Banach space then C an (N, E) −→ C an (M, E) f −→ f ◦ g is a well defined K-linear map. f

g

ii. With L −→ M −→ N also g ◦ f : L −→ N is a locally analytic map of manifolds. Proof. This follows from Lemma 6.3. Examples 8.5.

1) For any open submanifold U of M the inclusion map

⊆

U −−→ M is locally analytic. 2) Let g : M −→ N be a locally analytic map; for any open submanifold g V ⊆ N the induced map g −1 (V ) −→ V is locally analytic.

8

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51

3) The two projection maps pr1 : M × N −→ M

and

pr2 : M × N −→ N

are locally analytic. 4) For any pair of locally analytic maps g : L −→ M and f : L −→ N the map (g, f ) : L −→ M × N x −→ (g(x), f (x)) is locally analytic. For the remainder of this section we will discuss a certain technical but useful topological property of manifolds. First let X be an arbitrary Hausdorff topological space. We recall:   – Let X = i∈I Ui and X = j∈J Vj be two open coverings of X. The second one is called a refinement of the first if for any j ∈ J there is an i ∈ I such that Vj ⊆ Ui .  – An open covering X = i∈I Ui of X is called locally finite if every point x ∈ X has an open neighbourhood Ux such that the set {i ∈ I : Ux ∩ Ui = ∅} is finite. – The space X is called paracompact, resp. strictly paracompact, if any open covering of X can be refined into an open covering which is locally finite, resp. which consists of pairwise disjoint open subsets. Remark 8.6.

i. Any ultrametric space X is strictly paracompact.

ii. Any compact space X is paracompact. Proof. i. This follows from Lemma 1.4. ii. This is trivial. Proposition 8.7. For a manifold M the following conditions are equivalent: i. M is paracompact; ii. M is strictly paracompact; iii. the topology of M can be defined by a metric which satisfies the strict triangle inequality.

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Proof. The implication iii. =⇒ ii. is Lemma 1.4, and the implication ii. =⇒ i. is trivial. i. =⇒ ii. We suppose that M is paracompact. From general topology we recall the following property of paracompact Hausdorff spaces (cf. [B-GT] Chap. IX §4.4 Cor. 2). Let A ⊆ U ⊆ M be subsets with A closed and U open. Then there is another open subset V ⊆ M such that A ⊆ V ⊆ V¯ ⊆ U. Step 1: We show that the open and closed subsets of M form a basis of the topology. Given a point x in an open subset U ⊆ M we have to find an open and closed subset W ⊆ M such that x ∈ W ⊆ U . By Lemma 7.3 we may assume that U is the domain of definition of a chart (U, ϕ, K n ) for M . As recalled above there is an open neighbourhood V ⊆ M of x such that V¯ ⊆ U . We then have the vertical homeomorphisms ⊆

V 

ϕ(V )

⊆

V¯ 

⊆

ϕ(V¯ )

U

⊆

M

 ⊆

ϕ(U )

⊆

K n.

Since ϕ(V ) is open in K n there is a ball B := Bε (ϕ(x)) ⊆ ϕ(V ) around ϕ(x). We put W := ϕ−1 (B) ⊆ V . Clearly x ∈ W ⊆ U . The ball B is open and hence B is open in V and M . But the ball B also is closed in K n . Hence W is closed in V¯ and therefore in M . This finishes step 1.  Let now M = i∈I Ui be a fixed but arbitrary open covering. By Lemma 7.3 we may assume, after refinement, that any Ui is the domain of definition of some chart for M . By the first step and Remark 8.1 we may even assume, after a further refinement, that each Ui is open and closed in M and is the domain of definition of some chart for M . In particular, each Ui has the topology of an ultrametric space. By assumption we may pick a locally finite refinement (Vj )j∈J of (Ui )i∈I . So we have the locally finite open cover ing M = j∈J Vj , and for each j ∈ J there is an i(j) ∈ I such that Vj ⊆ Ui(j) .  Step 2: We construct a covering M = j∈J Wj by open and closed subsets Wj ⊆ M such that Wj ⊆ Vj for any j ∈ J. For this purpose we equip J with a well-order ≤ (recall that this is a total order on J with the property that each nonempty subset of J has a minimal element—by the axiom of choice such a well-order always exists). We now use transfinite induction to find open and closed subsets Wj ⊆ M such that

8

Manifolds

53

(a) Wj ⊆ Vj for any j ∈ J, and     (b) M = j≤k Wj ∪ j>k Vj

for any k ∈ J.

We fix a k ∈ J and  suppose that  the Wj for j < k are constructed already. Claim: M = j jr and  the induction hypothesis (property (b) for jr ) implies x ∈  j≤jr Wj ⊆ j


W := M \ Wj ∪ Vj j
j>k

of M satisfies W ⊆ Vk ⊆ Ui(k) . Claim: Let (X, d) be an ultrametric space; for any subsets A ⊆ U ⊆ X with A closed and U open there exists an open and closed subset V ⊆ X such that A ⊆ V ⊆ U. For any subset D ⊆ X and any x ∈ X we put d(x, D) := inf d(x, y). y∈D

The strict triangle inequality implies that the function d(., D) on X is continuous and that D(ε) := {x ∈ X : d(x, D) = ε}, ¯ The closed subsets A for any ε > 0, is open in X. Moreover, D(0) = D. and B := X \ U of X satisfy A ∩ B = ∅. By the continuity of the functions d(., A) and d(., B) the subset V := {x ∈ X : d(x, A) < d(x, B)} therefore is open in X and satisfies A ⊆ V ⊆ U . Similarly V  := {x ∈ X : d(x, A) > d(x, B)} is open  in X. It follows that  V as the complement in X of the open subset V  ∪ (A(ε) ∩ B(ε)) is closed. This establishes the ε>0 claim.

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We apply this claim to W ⊆ Vk ⊆ Ui(k) and obtain an open and closed subset Wk ⊆ Ui(k) such that W ⊆ Wk ⊆ Vk . With Ui(k) also Wk is open and closed in M . As W ⊆ Wk the index k has the property (b). It remains to show that the Wj for j ∈ J actually cover M . Let x ∈ M . As argued before the set {j ∈ J : x ∈ Vj } = {j1 < · · · < jr } is finite. Then x ∈ Vj for any  j > jr . The property (b) for the index jr therefore implies that x ∈ j≤jr Wj . This finishes step 2. Step 3: At this point we have constructed a locally finite refinement (Wj )j∈J of our initial covering which consists of open and closed subsets Wj ⊆ M .  Claim: WL := j∈L Wj , for any subset L ⊆ J, is open and closed in M . Obviously WL is open. To see that its complement M \ WL is open as well let x ∈ M \ WL be any point. In particular, x ∈ Wj for any j ∈ L. Since the covering (Wj )j is locally finite we find an open neighbourhood Ux ⊆ M of x such that the set {j ∈ L : Ux ∩ Wj = ∅} = {j1 , . . . , js } is finite. Then Ux \ (Wj1 ∪ · · · ∪ Wjs ) is an open neighbourhood of x in M \ WL . This establishes the claim. We finally define a new index set P by P := all nonempty finite subsets of J, and for any L ∈ P we put   

 WL := Wj \ Wj = Wj \ WJ\L . j∈L

j∈J\L

j∈L

Clearly any WL is contained in some Wj . By the above claim each WL is open and closed in M . To check that

M= WL L∈P

holds true let x ∈ M be any point. Then x ∈ WL for the finite set L := {j ∈ J : x ∈ Wj }. Moreover, the WL are pairwise disjoint: Let L1 = L2 be two different indices in P . By symmetry we may assume that there is a j ∈ L1 \ L2 . Then WL1 ⊆ Wj and WL2 ⊆ M \ Wj . It follows that (WL )L∈P is a refinement of our initial covering by pairwise disjoint open subsets. This proves that M is strictly paracompact. ii. =⇒ iii. We start with an open covering of M by domains of definition  of charts for M . By assumption we may refine it into a covering M = i∈I Ui

8

Manifolds

55

by pairwise disjoint open subsets. According to Remark 8.1 each Ui also is the domain of definition of some chart for M . In particular, the topology of Ui can be defined by a metric di which satisfies the strict triangle inequality. We put di (x, y) di (x, y) := for any x, y ∈ Ui . 1 + di (x, y) Obviously we have di (x, y) = di (y, x) and di (x, y) = 0 if and only if x = y. To see that di satisfies the strict triangle inequality we compute di (x, z) =

di (x, z) max(di (x, y), di (y, z)) ≤  1 + di (x, z) 1 + max(di (x, y), di (y, z))    di (x, y) di (y, z) = max , 1 + di (x, y) 1 + di (y, z) = max(di (x, y), di (y, z)).

Here we have used the simple fact that t ≥ s ≥ 0 implies t(1 + s) = t + ts ≥ t s ≥ 1+s . For trivial reasons we have di ≤ di . s + st = s(1 + t) and hence 1+t On the other hand di di = 1 − di and hence, for 0 < ε ≤ 1, di (x, y) ≤ ε

if

ε di (x, y) ≤ . 2

This shows that the metrics di and di define the same topology on Ui . We note that di (x, y) < 1 for any x, y ∈ Ui . We now define d : M × M −→ R≥0  di (x, y) (x, y) −→ 1

if x, y ∈ Ui for some i ∈ I, otherwise.

This is a metric on d. The strict triangle equality d(x, z) ≤ max(d(x, y), d(y, z)) only needs justification if not all three points lie in the same subset Ui . But then the right hand side is ≥ 1 whereas the left hand side is ≤ 1. We claim that this metric d defines the topology of M . First consider any ball Bε (x)

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with respect to d in M . If ε ≥ 1 then Bε (x) = M , and if ε < 1 then Bε (x) is open in some Ui . Hence Bε (x) is open in M . Vice versa let V ⊆ M be any open subset and let x ∈ V . We choose an i ∈ I such that x ∈ Ui . Then V ∩Ui is an open neighbourhood of x in Ui . Hence, for some 0 < ε < 1, the ball Bε (x) with respect to d (or equivalently di ) is contained in V ∩ Ui ⊆ V . Corollary 8.8. Open submanifolds and product manifolds of paracompact manifolds are paracompact.

9

The Tangent Space

Let M be a manifold, and fix a point a ∈ M . We consider pairs (c, v) where – c = (U, ϕ, K m ) is a chart for M around a and – v ∈ K m. Two such pairs (c, v) and (c , v  ) are called equivalent if we have Dϕ(a) (ϕ ◦ ϕ−1 )(v) = v  . It follows from the chain rule that this indeed defines an equivalence relation. Definition. A tangent vector of M at the point a is an equivalence class [c, v] of pairs (c, v) as above. We define Ta (M ) := set of all tangent vectors of M at a. Lemma 9.1. Let c = (U, ϕ, K m ) and c = (U  , ϕ , K m ) be two charts for M around a; we then have: i. The map ∼

θc : K m −−→ Ta (M ) v −→ [c, v] is bijective. ∼ =

m − −→ K m is a K-linear isomorphism. ii. θc−1  ◦ θc : K

9

The Tangent Space

57

Proof. (We recall from Lemma 7.1 that the dimensions of two charts around the same point necessarily coincide.) i. Surjectivity follows from [c , v  ] = [c, Dϕ (a) (ϕ ◦ ϕ−1 )(v  )]. If [c, v] = [c, v  ] then v  = Dϕ(a) (ϕ ◦ ϕ−1 )(v) = v. This proves the injectivity. ii. From [c, v] = [c , Dϕ(a) (ϕ ◦ ϕ−1 )(v)] we deduce that  −1 ). θc−1  ◦ θc = Dϕ(a) (ϕ ◦ ϕ

The set Ta (M ), by Lemma 9.1.i., has precisely one structure of a topological K-vector space such that the map θc is a K-linear homeomorphism. Because of Lemma 9.1.ii. this structure is independent of the choice of the chart c around a. Definition. The K-vector space Ta (M ) is called the tangent space of M at the point a. Remark. The manifold M has dimension m if and only if dimK Ta (M ) = m for any a ∈ M . Let g : M −→ N be a locally analytic map of manifolds. By Lemma 8.3.ii. we find charts c = (U, ϕ, K m ) for M around a and c˜ = (V, ψ, K n ) for N around g(a) such that g(U ) ⊆ V . The composite θ−1

Dϕ(a) (ψ◦g◦ϕ−1 )

θ

c ˜ Tg(a) (N ) Ta (g) : Ta (M ) −−c−→ K m −−−−−−−−−−−→ K n −−→

is a continuous K-linear map. We claim that Ta (g) does not depend on the particular choice of charts. Let c = (U  , ϕ ) and c˜ = (V  , ψ  ) be other charts around a and g(a), respectively. Using the identity in the proof of Lemma 9.1.ii. as well as the chain rule we compute θc˜ ◦ Dϕ(a) (ψ ◦ g ◦ ϕ−1 ) ◦ θc−1 = θc˜ ◦ Dψ(g(a)) (ψ  ◦ ψ −1 ) ◦ Dϕ(a) (ψ ◦ g ◦ ϕ−1 ) ◦ Dϕ(a) (ϕ ◦ ϕ−1 )−1 ◦ θc−1  = θc˜ ◦ Dϕ (a) (ψ  ◦ g ◦ ϕ−1 ) ◦ θc−1  . Definition. Ta (g) is called the tangent map of g at the point a. Remark. Ta (idM ) = idTa (M ) .

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Manifolds g

Lemma 9.2. For any locally analytic maps of manifolds L −→ M −→ N we have Ta (g ◦ f ) = Tf (a) (g) ◦ Ta (f ) for any a ∈ L. Proof. This is an easy consequence of the chain rule. Proposition 9.3. (Local invertibility) Let g : M −→ N be a locally analytic ∼ = map of manifolds, and suppose that Ta (g) : Ta (M ) −−→ Tg(a) (N ) is bijective for some a ∈ M ; then there are open neighbourhoods U ⊆ M of a and V ⊆ N of g(a) such that g restricts to a locally analytic isomorphism 

g : U −−→ V of open submanifolds. Proof. By Lemma 8.3.ii. we find charts c = (U  , ϕ, K m ) for M around a and c˜ = (V  , ψ, K n ) for N around g(a) such that g(U  ) ⊆ V  . We consider the locally analytic function ϕ−1

g

ψ

ϕ(U  ) −−−→ U  −→ V  −−→ ψ(V  ) ⊆ K n . By assumption the derivative Dϕ(a) (ψ ◦ g ◦ ϕ−1 ) = θc˜−1 ◦ Ta (g) ◦ θc is bijective. Prop. 6.4 therefore implies the existence of open neighbourhoods W0 ⊆ ϕ(U  ) of ϕ(a) and W1 ⊆ ψ(V  ) of ψ(g(a)) such that 

ψ ◦ g ◦ ϕ−1 : W0 −−→ W1 is a locally analytic isomorphism. Hence 

g : U := ϕ−1 (W0 ) −−→ V := ψ −1 (W1 ) is a locally analytic isomorphism as well (observe the subsequent exercise). 

Exercise. Let (U, ϕ, K m ) be a chart for the manifold M ; then ϕ : U −−→ ϕ(U ) is a locally analytic isomorphism between the open submanifolds U of M and ϕ(U ) of K m .

9

The Tangent Space

59

Let M be a manifold, E be a K-Banach space, f ∈ C an (M, E), and a ∈ M . If c = (U, ϕ, K m ) is a chart for M around a then f ◦ϕ−1 ∈ C an (ϕ(U ), E). Hence θ−1

Dϕ(a) (f ◦ϕ−1 )

c da f : Ta (M ) −− → K m −−−−−−−−−→ E

[c, v] −−−−−−−−−−−→ Dϕ(a) (f ◦ ϕ−1 )(v) is a continuous K-linear map. If c = (U  , ϕ , K m ) is another chart around a then Dϕ(a) (f ◦ ϕ−1 ) ◦ θc−1 = Dϕ(a) (f ◦ ϕ−1 ) ◦ Dϕ(a) (ϕ ◦ ϕ−1 )−1 ◦ θc−1  = Dϕ (a) (f ◦ ϕ−1 ) ◦ θc−1  . This shows that da f does not depend on the choice of the chart c. Definition. da f is called the derivative of f in the point a. Remark 9.4. For E = K r viewed as a manifold and for the chart c0 = (K r , id, E) for E we have Ta (f ) = θc0 ◦ da f. Obviously the map C an (M, E) −→ L(Ta (M ), E) f −→ da f is K-linear. Lemma 9.5. (Product rule) i. Let u : E1 ×E2 −→ E be a continuous bilinear map between K-Banach spaces; if fi ∈ C an (M, Ei ) for i = 1, 2 then u(f1 , f2 ) ∈ C an (M, E) and da (u(f1 , f2 )) = u(f1 (a), da f2 ) + u(da f1 , f2 (a))

for any a ∈ M.

ii. For g ∈ C an (M, K) and f ∈ C an (M, E) we have da (gf ) = g(a) · da f + da g · f (a)

for any a ∈ M.

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Proof. i. It is a straightforward consequence of Prop. 5.2 that the function u(f1 , f2 ) is locally analytic (compare the property 5) of the list in Sect. 6). Let c = (U, ϕ) be a chart of M around a. Using the product rule in Remark 4.1.iv. we compute da (u(f1 , f2 ))([c, v]) = Dϕ(a) (u(f1 , f2 ) ◦ ϕ−1 )(v) = Dϕ(a) (u(f1 ◦ ϕ−1 , f2 ◦ ϕ−1 ))(v) = u(f1 ◦ ϕ−1 (ϕ(a)), Dϕ(a) (f2 ◦ ϕ−1 )(v)) + u(Dϕ(a) (f1 ◦ ϕ−1 )(v), f2 ◦ ϕ−1 (ϕ(a))) = u(f1 (a), da f2 ([c, v])) + u(da f1 ([c, v]), f2 (a)). ii. This is a special case of the first assertion. Let c = (U, ϕ, K m ) be a chart for M . On the one hand, by definition, we have da ϕ = θc−1 for any a ∈ U ; in particular ∼ =

da ϕ : Ta (M ) −−→ K m is a K-linear isomorphism. On the other hand viewing ϕ = (ϕ1 , . . . , ϕm ) as a tuple of locally analytic functions ϕi : U −→ K we have da ϕ = (da ϕ1 , . . . , da ϕm ). This means that {da ϕi }1≤i≤m is a K-basis of the dual vector space Ta (M ) . Let  

 ∂ (a) ∂ϕi 1≤i≤m denote the corresponding dual basis of Ta (M ), i. e.,  da ϕi

  ∂ (a) = δij ∂ϕj

for any a ∈ U

where δij is the Kronecker symbol. For any f ∈ C an (M, E) we define the functions ∂f : U −→ E ∂ϕi a −→ da f



  ∂ (a) . ∂ϕi

9

The Tangent Space

Lemma 9.6.

∂f ∂ϕi

61

∈ C an (U, E) for any 1 ≤ i ≤ m, and

da f =

m 

da ϕi ·

i=1

∂f (a) ∂ϕi

for any a ∈ U.

Proof. We have ∂f (a) = Dϕ(a) (f ◦ ϕ−1 ) ◦ θc−1 ∂ϕi



  ∂ (a) ∂ϕi

= Dϕ(a) (f ◦ ϕ−1 )(ei ) where e1 , . . . , em denotes the standard basis of K m . Hence posite x →Dx (f ◦ϕ−1 )

ϕ

∂f ∂ϕi

is the com-

D →D(ei )

U −→ ϕ(U ) −−−−−−−−−−→ L(K m , E) −−−−−−−→ E. The function in the middle is locally analytic by Prop. 6.1. Clearly, D −→ D(ei ) is a continuous K-linear map. Hence the composite of the right two maps is locally analytic by the property 6) in Sect. 6. That the full composite ∂f ∂ϕi is locally analytic now follows from Lemma 8.4.i. Let t=

  m  ∂ (a) ∈ Ta (M ) ci ∂ϕi i=1

be an arbitrary vector. By the definition of the dual basis we have ci = da ϕi (t). We now compute da f (t) =

m 

 ci · da f

i=1

   m ∂ ∂f da ϕi (t) · (a). (a) = ∂ϕi ∂ϕi i=1

In a next step we want to show that the disjoint union

Ta (M ) T (M ) := a∈M

in a natural way is a manifold again. We introduce the projection map pM : T (M ) −→ M t −→ a if t ∈ Ta (M ). Hence Ta (M ) = p−1 M (a).

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Consider any chart c = (U, ϕ, K m ) for M . By Lemma 9.1.i. the map ∼

τc : U × K m −−→ p−1 M (U ) (a, v) −→ [c, v] viewed in Ta (M ) is bijective. Hence the composite τ −1

ϕ×id

ϕc : p−1 −c−→ U × K m −−−−→ K m × K m = K 2m M (U ) − is a bijection onto an open subset in K 2m . The idea is that   2m cT := p−1 M (U ), ϕc , K should be a chart for the manifold T (M ) yet to be constructed. Clearly we have

T (M ) = p−1 M (U ). c=(U,ϕ)

(V, ψ, K m )

Let c˜ = composed map

be another chart for M such that U ∩ V = ∅. The

ϕ−1 c

ψ

c ˜ −1 −1 m ϕ(U ∩ V ) × K m −−−→ p−1 M (U ∩ V ) = pM (U ) ∩ pM (V ) −−→ ψ(U ∩ V ) × K

is given by (8)

(x, v) −→ (ψ ◦ ϕ−1 (x), Dx (ψ ◦ ϕ−1 )(v)).

The first component ψ ◦ϕ−1 of this map is locally analytic on ϕ(U ∩V ) since M is a manifold. The second component can be viewed as the composite ϕ(U ∩ V ) × K m −→ L(K m , K m ) × K m −→ (x, v) −→ (Dx (ψ ◦ ϕ−1 ), v) (u, v) −→

Km u(v).

The left function is locally analytic by Prop. 6.1. The right bilinear map is continuous. Hence the composite is locally analytic by Lemma 9.5.i. This shows that, once cT and c˜T are recognized as charts for T (M ) with respect to a topology yet to be defined, they in fact are compatible, and hence that the set {cT : c a chart for M } is an atlas for T (M ). We have shown in particular that the composed map (9)

τ −1

c (U ∩ V ) × K m −→ p−1 −c˜−→ (U ∩ V ) × K m M (U ∩ V ) −

is a homeomorphism.

τ

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The Tangent Space

63

Definition. A subset X ⊆ T (M ) is called open if τc−1 (X ∩ p−1 M (U )) is open m m in U × K for any chart c = (U, ϕ, K ) for M . This defines the finest topology on T (M ) which makes all composed maps ⊆

c p−1 −→ T (M ) U × K m −−→ M (U ) −

τ

continuous. 

Lemma 9.7. i. The map τc : U × K m −−→ p−1 M (U ) is a homeomorphism with respect to the subspace topology induced by T (M ) on p−1 M (U ). ii. The map pM is continuous. iii. The topological space T (M ) is Hausdorff. Proof. i. The continuity of τc holds by construction. Let Y ⊆ U × K m be an open subset. We will show that τc (Y ) is open in T (M ), i. e., that m for any chart c τc˜−1 (τc (Y ) ∩ p−1 ˜ = (V, ψ, K n ) for M . M (V )) is open in V × K We may of course assume that U ∩ V = ∅ so that n = m. Clearly the subset Y ∩ ((U ∩ V ) × K m ) is open in (U ∩ V ) × K m . By (9) the subset τc˜−1 (τc (Y ∩ ((U ∩ V ) × K m ))) = τc˜−1 (τc (Y ) ∩ p−1 M (U ∩ V )) −1 = τc˜−1 (τc (Y ) ∩ p−1 M (U ) ∩ pM (V ))

= τc˜−1 (τc (Y ) ∩ p−1 M (V )) is open in (U ∩ V ) × K m and therefore in V × K m . ii. The above reasoning for Y = U × K m shows that τc (Y ) = p−1 M (U ) is open in T (M ) where U is the domain of definition of any chart for M . It then follows from Lemma 7.3 that pM is continuous. iii. Let s and t be two different points in T (M ). Case 1: We have pM (s) = pM (t). Since M is Hausdorff we find open neighbourhoods U ⊆ M of pM (s) −1 and V ⊆ M of pM (t) such that U ∩V = ∅. By ii. then p−1 M (U ) and pM (V ) are −1 open neighbourhoods of s and t, respectively, such that pM (U ) ∩ p−1 M (V ) = −1 pM (U ∩ V ) = ∅. Case 2: We have a := pM (s) = pM (t). We choose a chart c = (U, ϕ, K m ) for M around a. The two points s and t lie in the open (by −1 ii.) subset p−1 M (U ) of T (M ). But by i. the subspace pM (U ) is homeomorphic, via the map τc , to the Hausdorff space U × K m . Hence p−1 M (U ) is Hausdorff and s and t can be separated by open neighbourhoods in p−1 M (U ) and a fortiori in T (M ).

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The Lemma 9.7 in particular says that cT indeed is a chart for T (M ). Altogether we now have established that {cT : c a chart for M } is an atlas for T (M ). We always view T (M ) as a manifold with respect to the equivalent maximal atlas. Definition. The manifold T (M ) is called the tangent bundle of M . Remark. If M is m-dimensional then T (M ) is 2m-dimensional. Lemma 9.8. The map pM : T (M ) −→ M is locally analytic. Proof. Let c = (U, ϕ, K m ) be a chart for M . It suffices to contemplate the commutative diagram T (M )

⊇

p−1 M (U )

m ϕc (p−1 M (U )) = ϕ(U ) × K

K 2m

pr1

pM

M

⊆

⊇

ϕ

U

⊆

ϕ(U )

K m.

Let g : M −→ N be a locally analytic map of manifolds. We define the map T (g) : T (M ) −→ T (N ) by T (g)|Ta (M ) := Ta (g)

for any a ∈ M.

In particular, the diagram T (M )

T (g)

pM

M

T (N ) pN

g

N

is commutative. Proposition 9.9.

i. The map T (g) is locally analytic. f

g

ii. For any locally analytic maps of manifolds L −→ M −→ N we have T (g ◦ f ) = T (g) ◦ T (f ).

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65

Proof. i. We choose charts c = (U, ϕ, K m ) for M and c˜ = (V, ψ, K n ) for N such that g(U ) ⊆ V . The composite ϕ−1 c

T (g)

ψ

c ˜ −1 n ϕ(U ) × K m −−−→ p−1 M (U ) −−−→ pN (V ) −−→ ψ(V ) × K

is given by (x, v) −→ (ψ ◦ g ◦ ϕ−1 (x), Dx (ψ ◦ g ◦ ϕ−1 )(v)). It is locally analytic by the same argument as for (8). ii. This is a restatement of Lemma 9.2. The following is left to the reader as an exercise. Remark 9.10. i. If U ⊆ M is an open submanifold then T (⊆) induces an isomorphism between T (U ) and the open submanifold p−1 M (U ). ii. For any two manifolds M and N the map 

T (pr1 ) × T (pr2 ) : T (M × N ) −−→ T (M ) × T (N ) is an isomorphism of manifolds. Now let M be a manifold and E be a K-Banach space. For any f ∈ we define

C an (M, E)

df : T (M ) −→ E t −→ dpM (t) f (t). Lemma 9.11. We have df ∈ C an (T (M ), E). Proof. Let c = (U, ϕ, K m ) be a chart for M . The composed map ϕ−1 c

df

−→ E ϕ(U ) × K m −−−→ p−1 M (U ) − is given by

(x, v) −→ Dx (f ◦ ϕ−1 )(v)

and hence is locally analytic by the same argument as for (8). Lemma 9.12. Let g : M −→ N be a locally analytic map of manifolds; for any f ∈ C an (N, E) we have d(f ◦ g) = df ◦ T (g).

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Proof. This is a consequence of the chain rule. Exercise. The map d : C an (M, E) −→ C an (T (M ), E) f −→ df is K-linear. Remark 9.13. If K has characteristic zero then a function f ∈ C an (M, E) is locally constant if and only if df = 0. Proof. Let c = (U, ϕ) be any chart for M . As can be seen from the proof of −1 Lemma 9.11 we have df |p−1 M (U ) = 0 if and only if Dx (f ◦ ϕ ) = 0 for any −1 x ∈ ϕ(U ). By Remark 6.2 the latter is equivalent to f ◦ ϕ being locally constant on ϕ(U ) which, of course, is the same as f being locally constant on U . Definition. Let U ⊆ M be an open subset; a vector field ξ on U is a locally analytic map ξ : U −→ T (M ) which satisfies pM ◦ ξ = idU . We define Γ(U, T (M )) := set of all vector fields on U. It follows from Remark 9.10.i. that Γ(U, T (M )) = Γ(U, T (U )). Suppose that U is the domain of definition of some chart c = (U, ϕ, K m ) for M . Because of the commutative diagram p−1 M (U )

 ϕc

pM

ϕ(U ) × K m (x,v) →ϕ−1 (x)

U the map ∼

C an (U, K m ) −−→ Γ(U, T (M )) f −→ ξf (a) := ϕ−1 c ((ϕ(a), f (a))) = τc (a, f (a)) is bijective. The left hand side is a K-vector space. On the right hand side this vector space structure corresponds to the pointwise addition and scalar

9

The Tangent Space

67

multiplication of maps which makes sense since each Ta (M ) is a K-vector space. The latter we can define on any open subset U ⊆ M . For any c ∈ K and ξ, η ∈ Γ(U, T (M )) we define (c · ξ)(a) := c · ξ(a)

and

(ξ + η)(a) := ξ(a) + η(a).

Obviously the result are again maps ? : U −→ T (M ) satisfying pM ◦ ? = idU . But since U can be covered by domains of definition of charts for M the above discussion actually implies that these maps are locally analytic again. We see that Γ(U, T (M )) is a K-vector space. We have the bilinear map Γ(M, T (M )) × C an (M, E) −→ C an (M, E) (ξ, f ) −→ Dξ (f ) := df ◦ ξ. Lemma 9.14. Let u : E1 × E2 −→ E be a continuous bilinear map between K-Banach spaces; for any ξ ∈ Γ(M, T (M )) and fi ∈ C an (M, Ei ) we have Dξ (u(f1 , f2 )) = u(Dξ (f1 ), f2 ) + u(f1 , Dξ (f2 )). Proof. This follows from the product rule in Lemma 9.5.i. Corollary 9.15. For any vector field ξ ∈ Γ(M, T (M )) the map Dξ : C an (M, K) −→ C an (M, K) is a derivation, i. e.: (a) Dξ is K-linear, (b) Dξ (f g) = Dξ (f )g + f Dξ (g) for any f, g ∈ C an (M, K). Proposition 9.16. Suppose that M is paracompact; then for any derivation D on C an (M, K) there is a unique vector field ξ on M such that D = Dξ . The proof requires some preparation. In the following we always assume M to be paracompact. At first we fix a point a ∈ M . A K-linear map Δ : C an (M, K) −→ K will be called an a-derivation if Δ(f g) = Δ(f )g(a) + f (a)Δ(g)

for any f, g ∈ C an (M, K).

The a-derivations form a K-vector subspace Dera (M, K) of the dual vector space C an (M, K)∗ .

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Lemma 9.17. Suppose that M is paracompact, and let Δ be an a-derivation; if f ∈ C an (M, K) is constant in a neighbourhood of the point a then Δ(f ) = 0. Proof. Case 1: We assume that f vanishes in the neighbourhood U ⊆ M of a. By Prop. 8.7 we may assume that U is open and closed in M . Then the function  1 if x ∈ U, g(x) := 0 if x ∈ U lies in C an (M, K) and satisfies gf = f . It follows that Δ(f ) = Δ(gf ) = Δ(g)f (a) + g(a)Δ(f ) = 0. Case 2: We assume that f is constant on M with value c. Let 1M denote the constant function with value one on M . Then f = c1M and hence Δ(f ) = cΔ(1M ) = cΔ(1M 1M ) = cΔ(1M ) + cΔ(1M ) = 2cΔ(1M ) = 2Δ(f ) which means Δ(f ) = 0. Case 3: In general we write f = f (a)1M + (f − f (a)1M ) and use the K-linearity of Δ together with the first two cases. As a consequence of the product rule Lemma 9.5.ii. we have the K-linear map (10)

Ta (M ) −→ Dera (M, K) t −→ Δt (f ) := da f (t).

Proposition 9.18. If M is paracompact then (10) is an isomorphism. Proof. We fix a chart c = (U, ϕ, K m ) for M around a point a and write ϕ = (ϕ1 , . . . , ϕm ). Since M is paracompact we may assume by Prop. 8.7 that U is open and closed in M . Then each ϕi extends by zero to a function ϕi! ∈ C an (M, K). In the discussion before Lemma 9.6 we had seen that ∼ =

da ϕ = (da ϕ1 , . . . , da ϕm ) : Ta (M ) −−→ K m is an isomorphism, and we had introduced the K-basis ti :=

∂ ∂ϕi (a)

of Ta (M ).

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69

Injectivity: Let t ∈ Ta (M ) such that Δt (f ) = 0 for any f ∈ C an (M, K). In particular for any 1 ≤ i ≤ m.

0 = Δt (ϕi! ) = da ϕi! (t) = da ϕi (t)

This means da ϕ(t) = 0 and hence t = 0. Surjectivity: From the injectivity which we just have established we deduce that the Δti are linearly independent in Dera (M, K). It therefore suffices to write an arbitrarily given Δ ∈ Dera (M, K) as a linear combination of the Δti . In fact, we claim that Δ=

m 

Δ(ϕi! ) · Δti

i=1

holds true. Let f ∈ C an (M, K). We find an open and closed neighbourhood V ⊆ U of a such that ϕ(V ) = Bε (ϕ(a)) and a power series F (X) =  α m α cα X ∈ Fε (K ; K) such that f (x) = F (ϕ(x) − ϕ(a))

for any x ∈ V.

We may write f (x) =



cα (ϕ(x) − ϕ(a))α = f (a) +

m 

α

(ϕi (x) − ϕi (a))gi (x)

i=1

for any x ∈ V where the gi ∈ C an (V, K) are appropriate functions satisfying gi (a) = ci (recall that i = (0, . . . , 1, . . . , 0)). From the proof of Prop. 6.1 we know that ∂F Dϕ(a) (f ◦ ϕ−1 )(ei ) = (0) = ci ∂Xi where e1 , . . . , em denotes, as usual, the standard basis of K m . We therefore obtain gi (a) = ci = Dϕ(a) (f ◦ ϕ−1 )(ei ) = da f (θc (ei )). By the construction of the ti we have θc (ei ) = ti . It follows that gi (a) = da f (ti ). On the other hand we extend each gi by zero to a function gi! ∈ C an (M, K). The function m  f− (ϕi! − ϕi (a))gi! i=1

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is constant (with value f (a)) in a neighbourhood of a. Using Lemma 9.17 we compute  m  (ϕi! − ϕi (a))gi! Δ(f ) = Δ i=1 m    Δ ϕi! − ϕi (a) gi (a) = i=1

= =

m  i=1 m 

Δ(ϕi! ) · da f (ti ) Δ(ϕi! ) · Δti (f ).

i=1

Since f was arbitrary this establishes our claim. Proof of Proposition 9.16: First of all we note that the relation between derivations and a-derivations on C an (M, K) is given by the formula (11)

Dξ (f )(a) = df (ξ(a)) = da f (ξ(a)) = Δξ(a) (f ).

Therefore, if Dξ = 0 then Δξ(a) = 0 for any a ∈ M . The Prop. 9.18 then implies that ξ(a) = 0 for any a ∈ M , i. e., that ξ = 0. This shows that the ξ in our assertion is unique if it exists. For the existence we first fix a point a ∈ M and consider the a-derivation Δ(f ) := D(f )(a). By Prop. 9.18 there is a tangent vector ξ(a) ∈ Ta (M ) such that Δ = Δξ(a) . For varying a ∈ M this gives a map ξ : M −→ T (M ) which satisfies pM ◦ ξ = idM . It remains to show that ξ is locally analytic, since D = Dξ then is a formal consequence of (11). So let c = (U, ϕ, K m ) be a chart for M . In the proof of Prop. 9.18 we have seen that m  D(f )(a) = D(ϕi! )(a) · Δθc (ei ) (f ). i=1

It follows that m    D ϕi! )(a) · θc (ei ) = θc ((D(ϕ1! )(a), . . . , D(ϕm! )(a)) . ξ(a) = i=1

Using the commutative diagram Km

v −→(a,v)

θc ∼ =

Ta (M )

U × Km  τc

⊆

p−1 M (U )

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71

we rewrite this as ξ(a) = τc (a, (D(ϕ1! )(a), . . . , D(ϕm! )(a))). This means that under the identification discussed after the definition of vector fields we have ξ|U = ξf

with f := (D(ϕ1! ), . . . , D(ϕm! )) ∈ C an (U, K m ). 

Hence ξ is locally analytic.

Lemma 9.19. For any derivations B, C, D : C an (M, K) −→ C an (M, K) we have: i. [B, C] := B ◦ C − C ◦ B again is a derivation; ii. [ , ] is K-bilinear ; iii. [B, B] = 0 and [B, C] = −[C, B]; iv. (Jacobi identity) [[B, C], D] + [[C, D], B] + [[D, B], C] = 0. Proof. These are straightforward completely formal computations. Definition. A K-vector space g together with a K-bilinear map [ , ] : g × g −→ g which is antisymmetric (i. e., [z, z] = 0 for any z ∈ g) and satisfies the Jacobi identity is called a Lie algebra over K. If M is paracompact then, using Prop. 9.16 and Lemma 9.19, we may define the Lie product [ξ, η] of two vector fields ξ, η ∈ Γ(M, T (M )) by the requirement that D[ξ,η] = Dξ ◦ Dη − Dη ◦ Dξ holds true. This makes Γ(M, T (M )) into a Lie algebra over K. Proposition 9.20. Suppose that M is paracompact, and let E be a KBanach space and ξ, η ∈ Γ(M, T (M )) be two vector fields; on C an (M, E) we then have D[ξ,η] = Dξ ◦ Dη − Dη ◦ Dξ .

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Proof. Let f ∈ C an (M, E). We have to show equality of the functions D[ξ,η] (f ) = df ◦ [ξ, η] and (Dξ ◦ Dη − Dη ◦ Dξ )(f ) = d(Dη (f )) ◦ ξ − d(Dξ (f )) ◦ η = d(df ◦ η) ◦ ξ − d(df ◦ ξ) ◦ η. This, of course, can be done after restriction to the domain of definition U of any chart c = (U, ϕ, K m ) for M . Since M is paracompact we furthermore need only to consider charts for which U is open and closed in M . Let ϕ = (ϕ1 , . . . , ϕm ) and denote, as before, by ϕi! ∈ C an (M, K) the extension by zero of ϕi . We now make use of the following identifications. If ? denotes the restriction to U of any of the vector fields ξ, η, and [ξ, η] then, as discussed after the definition of vector fields, we have a commutative diagram  ϕc

p−1 M (U )

ϕ(U ) × K m x →(x,g? (x))

?  ϕ

U

ϕ(U )

with g? ∈ C an (ϕ(U ), K m ). On the other hand, as noticed already in the proof of Lemma 9.11, we have, for any function ? ∈ C an (U, E), the commutative diagram E (x,v) →Dx (?◦ϕ−1 )(v)

d?

p−1 M (U )

 ϕc

ϕ(U ) × K m .

These identifications reduce us to proving the equality of the following two functions of x ∈ ϕ(U ) given by (12)

Dx (f ◦ ϕ−1 )(g[ξ,η] (x))

and (13) Dx (df ◦ η ◦ ϕ−1 )(gξ (x)) − Dx (df ◦ ξ ◦ ϕ−1 )(gη (x))     = Dx D.(f ◦ ϕ−1 )(gη (.)) (gξ (x)) − Dx D.(f ◦ ϕ−1 )(gξ (.)) (gη (x)),

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73

respectively. By viewing D.(f ◦ ϕ−1 ), resp. gη (.) and gξ (.), as functions from ϕ(U ) into L(K m , E), resp. into K m , we may apply the product rule Remark 4.1.iv. for the continuous bilinear map L(K m , E) × K m −→ E (u, v) −→ u(v) to both summands in the last expression for (13) and rewrite it as (14)

= [Dx (D.(f ◦ ϕ−1 ))(gξ (x))](gη (x)) + Dx (f ◦ ϕ−1 )[Dx gη (gξ (x))] − [Dx (D.(f ◦ ϕ−1 )(gη (x))](gξ (x)) − Dx (f ◦ ϕ−1 )[Dx gξ (gη (x))].

To simplify this further we establish the following general Claim: For any open subset V ⊆ K m , any point x ∈ V , any vectors v = (v1 , . . . , vm ) and w = (w1 , . . . , wm ) in K m , and any function h ∈ C an (V, E) we have Dx (D.h(v))(w) = Dx (D.h(w))(v). (Note that the function D.h(v) is the composite y →Dy h

u →u(v)

V −−−−−→ L(K m , E) −−−−−→ E.) We expand h around the point x into a power series h(y) = H(y − x). By the proof of Prop. 6.1 we then have Dy h(v) =

m  i=1

and

vi

∂H (y − x) ∂Yi

 ∂H Dx (D.h(v))(w) = vi D x (. − x) (w) ∂Yi i=1   m m   ∂ ∂ = vi wj H (0) ∂Yj ∂Yi i=1 j=1   m m   ∂ ∂ wj vi H (0) = ∂Yi ∂Yj m 

j=1



i=1

.. . = Dx (D.h(w))(v).

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Applying this claim to (14) we see that the expression for the function (13) simplifies to Dx (f ◦ ϕ−1 )[Dx gη (gξ (x)) − Dx gξ (gη (x))]. Comparing this with (12) we are reduced to showing that the identity (15)

g[ξ,η] (x) = Dx gη (gξ (x)) − Dx gξ (gη (x))

holds true in C an (ϕ(U ), K m ). But in case E = K our assertion and the whole computation above holds by construction. In particular we have Dx (ϕi! ◦ ϕ−1 )(g[ξ,η] (x)) = Dx (ϕi! ◦ ϕ−1 )[Dx gη (gξ (x)) − Dx gξ (gη (x))] for any 1 ≤ i ≤ m. Since Dx (ϕi! ◦ ϕ−1 ) :

K m −→ K (v1 , . . . , vm ) −→ vi

the identity (15) follows immediately. Remarks 9.21. i. The identity (15) shows that C an (V, K m ), for any open subset V ⊆ K m , is a Lie algebra with respect to [f, g](x) := Dx f (g(x)) − Dx g(f (x)). ii. The identity (15) can be made into a definition of which one then can show that it is compatible with any change of charts for M . In this way a Lie product [ξ, η] can be obtained and Prop. 9.20 can be proved even for manifolds which are not paracompact.

10

The Topological Vector Space C an (M, E), Part 1

Throughout this section M is a paracompact manifold and E is a K-Banach space. Following [Fea] we will show that C an (M, E) in a natural way is a topological vector space. To motivate the later construction we first consider a fixed function f ∈ C an (M, E). Since, by Prop. 8.7, M is strictly paracompact we find a , ϕj , K mj ), for j ∈ J, for M such that the Uj are pairwise family of charts (Uj  disjoint and M = j∈J Uj . According to Remark 8.2.ii. the function f is : ϕj (Uj ) −→ E, for j ∈ J, are locally analytic if and only if all f ◦ ϕ−1 j

10

The Topological Vector Space C an (M, E), Part 1

75

locally analytic. For each ϕj (Uj ) we find balls Bεj,ν (xj,ν ) ⊆ K mj and power series Fj,ν ∈ Fεj,ν (K mj ; E) such that

(16) ϕj (Uj ) = Bεj,ν (xj,ν ) ν

and

f ◦ ϕ−1 j (x) = Fj,ν (x − xj,ν )

for any x ∈ Bεj,ν (xj,ν ).

By Lemma 1.4 the covering (16) can be refined into a covering by pairwise disjoint balls Bδj,α (yj,α ). Consider a fixed α. We find a ν such that Bδj,α (yj,α ) ⊆ Bεj,ν (xj,ν ). In fact we then have Bmin(δj,α ,εj,ν ) (yj,α ) = Bδj,α (yj,α ) ⊆ Bεj,ν (xj,ν ) = Bεj,ν (yj,α ). Hence we may assume that δj,α ≤ εj,ν . Using Cor. 5.5 we may change Fj,ν into a power series Fj,α ∈ Fδj,α (K mj ; E) such that f ◦ ϕ−1 j (x) = Fj,α (x − yj,α )

for any x ∈ Bδj,α (yj,α ).

mj ) again are charts for M We put Uj,α := ϕ−1 j (Bδj,α (yj,α )). The (Uj,α , ϕj , K such that the Uj,α cover M and are pairwise disjoint.

Resume: Given f ∈ C an (M, E) there is a family of charts (Ui , ϕi , K mi ), for i ∈ I, for M together with real numbers εi > 0 such that:  (a) M = i∈I Ui , and the Ui are pairwise disjoint; (b) ϕi (Ui ) = Bεi (xi ) for one (or any) xi ∈ ϕi (Ui ); (c) there is a power series Fi ∈ Fεi (K mi ; E) with f ◦ ϕ−1 i (x) = Fi (x − xi )

for any x ∈ ϕi (Ui ).

We note that by Cor. 5.5 the existence of Fi as well as its norm Fi εi is independent of the choice of the point xi . Let (c, ε) be a pair consisting of a chart c = (U, ϕ, K m ) for M and a real number ε > 0 such that ϕ(U ) = Bε (a) for one (or any) a ∈ ϕ(U ). As a consequence of the identity theorem for power series Cor. 5.8 the K-linear map Fε (K m ; E) −→ C an (U, E) F −→ F (ϕ(.) − a)

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is injective. Let F(c,ε) (E) denote its image. It is a K-Banach space with respect to the norm f  = F ε

if

f (.) = F (ϕ(.) − a).

By Cor. 5.5 the pair (F(c,ε) (E),  ) is independent of the choice of the point a. Definition. An index for M is a family I = {(ci , εi )}i∈I of charts ci = (Ui , ϕi , K mi ) for M and real numbers εi > 0 such that the above conditions (a) and (b) are satisfied. For any index I for M we have   FI (E) := F(ci ,εi ) (E) ⊆ C an (Ui , E) = C an (M, E). i∈I

i∈I

Our above resume says that C an (M, E) =



FI (E)

I

where I runs over all indices for M . Hence C an (M, E) is a union of direct products of Banach spaces. This is the starting point for the construction of a topology on C an (M, E). But first we have to discuss the inclusion relations between the subspaces FI (E) for varying I. Let I = {(ci = (Ui , ϕi , K mi ), εi )}i∈I and J = {(dj = (Vj , ψj , K nj ), δj )}j∈J be two indices for M . Definition. The index I is called finer than the index J if for any i ∈ I there is a j ∈ J such that: (i) Ui ⊆ Vj , (ii) there is an a ∈ ϕi (Ui ) and a power series Fi,j ∈ Fεi (K mi ; K nj ) with Fi,j − Fi,j (0)εi ≤ δj and ψj ◦ ϕ−1 i (x) = Fi,j (x − a)

for any x ∈ ϕi (Ui ).

We observe that if the condition (ii) holds for one point a ∈ ϕi (Ui ) then it holds for any other point b ∈ ϕi (Ui ) as well. This follows from Cor. 5.5 which implies that Gi,j (X) := Fi,j (X + b − a) ∈ Fεi (K mi ; K nj ) with ψj ◦ ϕ−1 i (x) = Gi,j (x − b)

for any x ∈ ϕi (Ui )

The Topological Vector Space C an (M, E), Part 1

10

77

and Gi,j − Gi,j (0)εi = (Fi,j − Fi,j (0))(X + b − a) + Fi,j (0) − Gi,j (0)εi ≤ max((Fi,j − Fi,j (0))(X + b − a)εi , δj ) = max(Fi,j − Fi,j (0)εi , δj ) = δj . Lemma 10.1. If I is finer than J then we have FJ (E) ⊆ FI (E). Proof. Let f ∈ FJ (E). We have to show that f |Ui ∈ F(ci ,εi ) (E) for any i ∈ I. In the following we fix an i ∈ I. We have ϕi (Ui ) = Bεi (a). By assumption we find a j ∈ J and an Fi,j ∈ Fεi (K mi ; K nj ) such that – Ui ⊆ Vj , – Fi,j − Fi,j (0)εi ≤ δj , and – ψj ◦ ϕ−1 i (x) = Fi,j (x − a) for any x ∈ ϕi (Ui ). We put b := ψj ◦ ϕ−1 i (a) = Fi,j (0) ∈ ψj (Vj ). Since f ∈ FJ (E) we also find a Gj ∈ Fδj (K nj ; E) such that f ◦ ψj−1 (y) = Gj (y − b)

for any y ∈ ψj (Vj ) = Bδj (b).

As a consequence of Prop. 5.4 then the power series Fi := Gj ◦ (Fi,j − Fi,j (0)) ∈ Fεi (K mi ; E) exists and satisfies −1 Fi (x − a) = Gj (Fi,j (x − a) − b) = f ◦ ψj−1 (ψj ◦ ϕ−1 i (x)) = f ◦ ϕi (x)

for any x ∈ ϕi (Ui ). The relation of being finer only is a preorder. If the index I is finer than the index J and J is finer than I one cannot conclude that I = J . But it does follow that FI (E) = FJ (E) which is sufficient for our purposes. Lemma 10.2. For any two indices J1 and J2 for M there is a third index I for M which is finer than J1 and J2 .

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Proof. Let J1 = {((Ui , ϕi , K ni ), εi )}i∈I and J2 = {((Vj , ψj , K mj ), δj )}j∈J . We have the covering

M= Ui ∩ V j i,j

by pairwise disjoint open subsets. For any pair (i, j) ∈ I × J the function mj ψj ◦ ϕ−1 i : ϕi (Ui ∩ Vj ) −→ K

is locally analytic. Hence we may cover ϕi (Ui ∩ Vj ) by a family of balls Bi,j,k = Bβi,j,k (ai,j,k ) such that – βi,j,k ≤ min(εi , δj ), and – there is a power series Fi,j,k ∈ Fβi,j,k (K ni ; K mj ) with ψj ◦ ϕ−1 i (x) = Fi,j,k (x − ai,j,k )

for any x ∈ Bi,j,k .

Using the fact that Fi,j,k − Fi,j,k (0)α ≤

α Fi,j,k − Fi,j,k (0)βi,j,k βi,j,k

for any 0 < α ≤ βi,j,k

together with Cor. 5.5 we may, after possibly decreasing the βi,j,k , assume in addition that – Fi,j,k − Fi,j,k (0)βi,j,k ≤ δj . After a possible further refinement based on Lemma 1.4 (compare the argument for the resume at the beginning of this section) we finally achieve that the Bi,j,k are pairwise disjoint. We put Wi,j,k := ϕ−1 i (Bi,j,k ) and obtain the index I := {((Wi,j,k , ϕi , K ni ), βi,j,k )}i,j,k for M . By construc⊆

tion I is finer than J2 . Moreover, observing that ϕi ◦ϕ−1 −→ K ni i : ϕi (Wi,j,k ) − is the inclusion map and that βi,j,k ≤ εi we see that I is finer than J1 for trivial reasons.  Given any index I for M we consider FI (E) = i∈I F(ci ,εi ) (E) from now on as a topological K-vector space with respect to the product topology of the Banach space topologies on the F(ci ,εi ) (E). Obviously FI (E) is Hausdorff. But it is not a Banach space if I is infinite. Suppose that the topology of FI (E) can be defined by a norm. The corresponding unit ball

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Locally Convex K-Vector Spaces

79

B1 (0) is open. By the definition of the product topology there exist finitely many indices i1 , . . . , ir ∈ I such that  F(ci ,εi ) (E) × {0} × · · · × {0} ⊆ B1 (0). i=i1 ,...,ir

As a vector subspace the left hand side then necessarily is contained in any ball Bε (0) for ε > 0. The intersection of the latter being equal to {0} it follows that I is finite. ⊆

Lemma 10.3. If I is finer than J then the inclusion map FJ (E) −−→ FI (E) is continuous. Proof. For any i ∈ I there exists, by assumption, a j(i) ∈ J such that the conditions (i) and (ii) in the definition of “finer” are satisfied. The inclusion map in question can be viewed as the map   F(dj ,δj ) (E) −→ F(ci ,εi ) (E) j

i

(fj )j −→ (fj(i) |Ui )i . Hence it suffices to show that each individual restriction map F(dj(i) ,δj(i) ) (E) −→ F(ci ,εi ) (E) is continuous. But we even know from Prop. 5.4 that the operator norm of this map is ≤ 1. We point out that, for I finer than J , the topology of FJ (E) in general is strictly finer than the subspace topology induced by FI (E). In the present situation there is a certain universal procedure to construct from the topologies on all the FI (E) a topology on their union C an (M, E) =  I FI (E). Since this construction takes place within the class of locally convex topologies we first need to review this concept in the next section.

11

Locally Convex K-Vector Spaces

This section serves only as a brief introduction to the subject. The reader who is interested in more details is referred to [NFA]. Let E be any K-vector space. Definition. A (nonarchimedean) seminorm on E is a function q : E −→ R such that for any v, w ∈ E and any a ∈ K we have:

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(i) q(av) = |a| · q(v), (ii) q(v + w) ≤ max(q(v), q(w)). It follows immediately that a seminorm q also satisfies: (iii) q(0) = |0| · q(0) = 0; (iv) q(v) = max(q(v), q(−v)) ≥ q(v − v) = q(0) = 0 for any v ∈ E; (v) q(v + w) = max(q(v), q(w)) for any v, w ∈ E such that q(v) = q(w) (compare the proof of Lemma 1.1); (vi) −q(v − w) ≤ q(v) − q(w) ≤ q(v − w) for any v, w ∈ E. Let (qi )i∈I be a family of seminorms on E. We consider the coarsest topology on E such that: (1) All maps qi : E −→ R, for i ∈ I, are continuous, (2) all translation maps v + . : E −→ E, for v ∈ E, are continuous. It is called the topology defined by (qi )i∈I . For any finitely many qi1 , . . . , qir and any w ∈ E and ε > 0 we define Bε (qi1 , . . . , qir ; w) := {v ∈ E : qir (v − w), . . . , qir (v − w) ≤ ε}. The following properties are obvious: (a) Bε (qi1 , . . . , qir ; w) = Bε (qi1 ; w) ∩ · · · ∩ Bε (qir ; w);  (b) Bε1 (qi1 ; w1 ) ∩ Bε2 (qi2 ; w2 ) = w Bmin(ε1 ,ε2 ) (qi1 , qi2 ; w) where w runs over all points in the left hand side; (c) Bε (qi1 , . . . , qir ; w) = w + Bε (qi1 , . . . , qir ; 0); (d) a · Bε (qi1 , . . . , qir ; w) = B|a|ε (qi1 , . . . , qir ; aw) for any a ∈ K × . Lemma 11.1. The subsets Bε (qi1 , . . . , qir ; w) form a basis for the topology on E defined by (qi )i∈I . Proof. The Bε (qi1 , . . . , qir ; w), by (a) and (b), do form a basis for a (unique) topology T  on E. On the other hand let T denote the topology defined by (qi )i∈I . We first show that T  ⊆ T . By (a), (c), and (2) it suffices to check

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Locally Convex K-Vector Spaces

81

that Bε (qi ; 0) ∈ T for any i ∈ I and ε > 0. As a consequence of (1) and (2) we certainly have that Bδ− (qi ; w) := {v ∈ E : qi (v − w) < δ} ∈ T for any w ∈ E and δ > 0. But using (ii) we see that

Bε (qi ; 0) = Bε− (qi ; 0) ∪ Bε− (qi ; w). qi (w)=ε

To conclude that actually T  = T holds true it now suffices to show that T  satisfies (1) and (2). The continuity property (2) follows immediately from (c). To establish (1) for T  we have to show that qi−1 ((α, β)) ∈ T  for any i ∈ I and any open interval (α, β) ⊆ R. Because of (iv) we may assume that β > 0. Let w ∈ qi−1 ((α, β)) be any point. Case 1: We have qi (w) > 0. Choose any 0 < ε < qi (w). It then follows from (v) that Bε (qi ; w) ⊆ qi−1 (qi (w)) ⊆ qi−1 ((α, β)). Case 2: We have qi (w) = 0. Choose any 0 < ε < β. We obtain Bε (qi ; w) ⊆ qi−1 ([0, ε]) ⊆ qi−1 ((α, β)) since necessarily α < 0 in this case. Lemma 11.2. E is a topological K-vector space, i. e., addition and scalar multiplication are continuous, with respect to the topology defined by (qi )i∈I . Proof. Using Lemma 11.1 this easily follows from the following inclusions: – Bε (qi1 , . . . , qir ; w1 ) + Bε (qi1 , . . . , qir ; w2 ) ⊆ Bε (qi1 , . . . , qir ; w1 + w2 ); – Bδ (a) · B|a|−1 ε (qi1 , . . . , qir ; w) ⊆ Bε (qi1 , . . . , qir ; aw) provided δ ≤ |a| and δ · max(qi1 (w), . . . , qir (w)) ≤ ε; – Bδ (0) · Bε (qi1 , . . . , qir ; w) ⊆ Bε (qi1 , . . . , qir ; 0) provided δ ≤ 1 and δ · max(qi1 (w), . . . , qir (w)) ≤ ε. The details are left to the reader as an exercise. Exercise. The topology on E defined by (qi )i∈I is Hausdorff if and only if for any vector 0 = v ∈ E there is an index i ∈ I such that qi (v) = 0. Definition. A topology on a K-vector space E is called locally convex if it can be defined by a family of seminorms. A locally convex K-vector space is a K-vector space equipped with a locally convex topology. Obviously any normed K-vector space and in particular any K-Banach space is locally convex.

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Remark 11.3. Let {Ej }j∈J be a family of locally convex K-vector spaces;  then the product topology on E := j∈J Ej is locally convex. Proof. Let (qj,i )i be a family of seminorms which defines the locally convex topology on Ej . Moreover, let prj : E −→ Ej denote the projection maps. Using Lemma 11.1 one checks that the family of seminorms (qj,i ◦ prj )i,j defines the product topology on E. Exercise 11.4.  Let {Ej }j∈J be a family of locally convex K-vector spaces and let E := j∈J Ej with the product topology; for any continuous seminorm q on E there is a unique minimal finite subset Jq ⊆ J such that   Ej × {0} × · · · × {0} = {0}. q j∈J\Jq

For our purposes the following construction is of particular relevance. Let E be a any K-vector space, and suppose that there is given a family {Ej }j∈J of vector subspaces Ej ⊆ E each of which is equipped with a locally convex topology. Lemma 11.5. There is a unique finest locally convex topology T on E such ⊆ that all the inclusion maps Ej −−→ E, for j ∈ J, are continuous. Proof. Let Q be the set of all seminorms q on E such that q|Ej is continuous for any j ∈ J, and let T be the topology on E defined by Q. It follows ⊆ immediately from Lemma 11.1 that all the inclusion maps Ej −−→ (E, T ) are continuous. On the other hand, let T  be any topology on E defined ⊆ by a family of seminorms (qi )i∈I such that Ej −−→ (E, T  ) is continuous for any j ∈ J. Obviously we then have (qi )i∈I ⊆ Q. This implies, using again Lemma 11.1, that T  ⊆ T . The topology T on E in the above Lemma is called the locally convex final topology with respect to the family {Ej }j∈J . Suppose that the family {Ej }j∈J has the additional properties:  – E = j∈J Ej ; – the set J is partially ordered by ≤ such that for any two j1 , j2 ∈ J there is a j ∈ J such that j1 ≤ j and j2 ≤ j; ⊆

– whenever j1 ≤ j2 we have Ej1 ⊆ Ej2 and the inclusion map Ej1 −−→ Ej2 is continuous.

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Locally Convex K-Vector Spaces

83

In this case the locally convex K-vector space (E, T ) is called the locally convex inductive limit of the family {Ej }j∈J . ˜ into any locally convex KLemma 11.6. A K-linear map f : E −→ E ˜ is continuous (with respect to T ) if and only if the restrictions vector space E f |Ej , for any j ∈ J, are continuous. Proof. It is trivial that with f all restrictions f |Ej are continuous. Let us therefore assume vice versa that all f |Ej are continuous. Let (˜ qi )i∈I be a ˜ family of seminorms which defines the topology of E. Then all seminorms qi := q˜i ◦f , for i ∈ I, lie in the set of seminorms Q which defines the topology T of E. It follows that f −1 (Bε (˜ qi1 , . . . , q˜ir ; f (w)) = Bε (qi1 , . . . , qir ; w) is open in E. Because of Lemma 11.1 this means that f is continuous. Lemma 11.7.  Let {Ej }j∈J be a family of locally convex K-vector spaces and let E := j∈J Ej with the product topology; suppose that each Ej has the locally convex final topology with respect  to a family of locally convex K-vector spaces {Ej,k } k∈Ij and that Ej = k Ej,k ; for any k = (kj )j ∈ I :=  I we put E := E with the product topology; then the topology k j∈J j j∈J j,kj of E is the locally convex final topology with respect to the family {Ek }k∈I . Proof. By the proof of Lemma 11.5 the locally convex topology of Ej is defined by the set Qj of all seminorms q such that q|Ej,k is continuous for any k ∈ Ij . Let prj : E −→ Ej denote the projection maps.  By Remark 11.3 the topology of E is defined by the set of seminorms Q := j∈J {q◦prj : q ∈ Qj }. For any q ∈ Qj and any k ∈ I we have the commutative diagram Ek

⊆

prj

Ej,kj

E prj

⊆

Ej

q◦prj q

R.

Hence the restriction of any seminorm in Q to any Ek is continuous. This means that the locally convex final topology on E with respect to the family {Ek }k is finer than the product topology. Vice versa, let q be any seminorm on E such that q|Ek , for any k ∈ I, is continuous. We have to show that q is continuous. By Exercise 11.4 we find, for any k ∈ I, a unique minimal finite

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subset Jq,k ⊆ J such that the restriction q|Ek factorizes into pr

Ek

 j∈Jq,k

Ej,kj

q

R

.

In particular, q|Ej,kj = 0 for any j ∈ Jq,k . We claim that the set

Jq := Jq,k k∈I

is finite. We define  = (j )j ∈ I in the following way. If j ∈ Jq we choose a k ∈ I such that j ∈ Jq,k and we put j := kj ; in particular, q|Ej,j = q|Ej,kj = 0. For j ∈ J \ Jq we pick any j ∈ Ij . By construction we have Jq ⊆ Jq, so that Jq necessarily is finite. This means that the seminorm q on E factorizes into  pr E j∈Jq Ej q

R

.

It follows that q(v) ≤ max (q|Ej ) ◦ prj (v) j∈Jq

for any v ∈ E.

Since each q|Ej is continuous by assumption we conclude that q is continuous.

12

The Topological Vector Space C an (M, E), Part 2

As in Sect. 10 we let M be a paracompact manifold and E be a K-Banach space. We have seen that

FI (E) C an (M, E) = I

where I runs over all indices for M . Each FI (E) by Remark 11.3 is locally convex as a product of Banach spaces. By Lemmas 10.1–10.3 we may and always will view C an (M, E) as the locally convex inductive limit of the family {FI (E)}I (where I ≤ J if J is finer than I). All our earlier constructions involving C an (M, E) are compatible with this topology. In the following we briefly discuss the most important ones.

12

The Topological Vector Space C an (M, E), Part 2

85

Proposition 12.1. For any a ∈ M the evaluation map δa : C an (M, E) −→ E f −→ f (a) is continuous. Proof. It suffices, by Lemma 11.6, to show that the restriction δa |FI (E) is continuous for any index I for M . Let I = {(ci = (Ui , ϕi , K mi ), εi )}i∈I . There is a unique i ∈ I such that a ∈ Ui . Then ϕi (Ui ) = Bεi (ϕi (a)), and we have the commutative diagram FI (E)

δa

F →F (0)

pr

F(ci ,εi ) (E)

E

∼ = F (ϕi (.)−ϕi (a))←F

Fεi (K mi ; E).

The left vertical projection map clearly is continuous. The lower horizontal map is a topological isomorphism by construction. By Remark 5.1 the right vertical evaluation map is continuous of operator norm ≤ 1. Corollary 12.2. The locally convex vector space C an (M, E) is Hausdorff. Proof. Let f = g be two different functions in C an (M, E). We find a point a ∈ M such that f (a) = g(a). Since E is Hausdorff there are open neighbourhoods Vf of f (a) and Vg of g(a) in E such that Vf ∩ Vg = ∅. Using Prop. 12.1 we see that Uf := δa−1 (Vf ) and Ug := δa−1 (Vg ) are open neighbourhoods of f and g, respectively, in C an (M, E) such that Uf ∩ Ug = ∅. Remark 12.3. With M also its tangent bundle T (M ) is paracompact. Proof. Since M is strictly paracompact by Prop. 8.7 we find a family of charts {ci =(Ui , ϕi , K mi )}i∈I for M such that the Ui are pairwise disjoint 2mi ) form a family of and M = i Ui . Then the ci,T = (p−1 M (Ui ), ϕi,ci , K charts for T (M ) such that T (M ) is the disjoint union of the open subsets −1 2mi carp−1 M (Ui ). Each pM (Ui ) being homeomorphic to an open subset in K ries the topology of an ultrametric space. The construction in the proof of implication ii. =⇒ iii. in Prop. 8.7 then shows that the topology of T (M ) can be defined by a metric which satisfies the strict triangle inequality. Hence T (M ) is paracompact by Lemma 1.4.

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Manifolds

i. The map d : C an (M, E) −→ C an (T (M ), E) is

ii. For any locally analytic map of paracompact manifolds g : M −→ N the map C an (N, E) −→ C an (M, E) f −→ f ◦ g is continuous. iii. For any vector field ξ on M the map Dξ : C an (M, E) −→ C an (M, E) is continuous. Proof. i. By Lemma 11.6 we have to show that d|FI (E) is continuous for any index I = {(ci = (Ui , ϕi , K mi ), εi )}i∈I for M . Let f ∈ FI (E). We have the commutative diagrams E df

p−1 M (Ui )

ϕi,ci

(x,v) →Dx (f ◦ϕ−1 i )(v)

ϕi (Ui ) × K mi

pM

pr

Ui

ϕi

ϕi (Ui )

(cf. the proof of Lemma 9.11). We also have power series Fi ∈ Fεi (K mi ; E) such that f ◦ ϕ−1 i (x) = Fi (x − ai )

for any x ∈ ϕi (Ui ) = Bεi (ai ).

From the proof of Prop. 6.1 we recall the formula Dx (f ◦ ϕ−1 i )(v) =

mi  j=1

vj

∂Fi (x − ai ) ∂Xj

for any x ∈ ϕi (Ui ) and any v = (v1 , . . . , vmi ) ∈ K mi . We now cover K mi by (i) (i) (i) (i) pairwise disjoint balls Bεi (wk ) where wk = (wk,1 , . . . , wk,mi ) runs over an appropriate family of vectors in K mi , and we put mi  (i) ∂Fi (Yj + wk,j ) ∈ Fεi (K 2mi ; E). Gi,k (X1 , . . . , Xmi , Y1 , . . . , Ymi ) := ∂Xj j=1

12

The Topological Vector Space C an (M, E), Part 2

Then

87

−1 df ◦ ϕ−1 i,ci (x, v) = Dx (f ◦ ϕi )(v) = Gi,k (x − ai , v − wk ) (i)

(i)

(i)

for any (x, v) ∈ ϕi (Ui ) × Bεi (wk ) = Bεi (ai , wk ). This means that df ∈ FJ (E) ⊆ C an (T (M ), E) for the index 2mi ), εi )}i,k . J := {((ϕ−1 i,ci (Bεi (ai , wk )), ϕi,ci , K (i)

In other words we have the commutative diagram d

C an (M, E)

C an (T (M ), E)

⊆

⊆

FJ (E).

FI (E)

Since the vertical inclusion maps are continuous by construction this reduces us to showing the continuity of the lower horizontal map FI (E) −→ FJ (E). But this easily follows from the inequalities  Gi,k εi ≤ max 1,

(i)

|wk,1 | εi

(i)

,...,

|wk,mi |



εi

· Fi εi .

ii. We only sketch the argument and leave the details to the reader. Let I = {((Ui , ϕi , K ni ), εi )}i∈I be an index for N . We refine the covering M =   −1 i g (Ui ) into a covering M = j∈J Vj which underlies an appropriate index J = {((Vj , ψj , K mj ), δj )}j∈J and such that, for any i ∈ I and j ∈ J with Vj ⊆ g −1 (Ui ), there is a power series Gi,j ∈ Fδj (K mj ; K ni ) with Gi,j − Gi,j (0)δj ≤ εi and ϕi ◦ g ◦ ψj−1 (x) = Gi,j (x − aj )

for any x ∈ ψj (Vj ) = Bδj (aj ).

In this situation we have the commutative diagram C an (N, E)

f →f ◦g

C an (M, E)

⊆

FI (E)

⊆

FJ (E)

where the lower horizontal arrow in terms of power series is given by the maps Fεi (K ni ; E) −→ Fδj (K mj ; E) F −→ F ◦ (Gi,j − Gi,j (0))

88

II

Manifolds

whose continuity was established in Prop. 5.4. iii. follows from i. and ii.  Proposition 12.5. For any covering M = i∈I Ui by pairwise disjoint open subsets Ui we have  C an (Ui , E) C an (M, E) = i∈I

as topological vector spaces. Proof. Using Lemma 10.2 one checks that in the construction of C an (M, E) as a locally convex inductive limit it suffices to consider indices  for M whose underlying covering of M refines the given covering M = i Ui . Then the assertion is a formal consequence of Lemma 11.7.

Chapter III

Lie Groups As before we fix the nonarchimedean field (K, | |).

13

Definitions and Foundations

Definition. A Lie group G (over K) is a manifold (over K) which also carries the structure of a group such that the multiplication map m = mG : G × G −→ G (g, h) −→ gh is locally analytic. In the following let G be a Lie group, and let e ∈ G denote the unit element. Lemma 13.1. For any h ∈ G the maps 

h : G −−→ G



rh : G −−→ G

and

g −→ hg

g −→ gh

are locally analytic isomorphisms (of manifolds). Proof. By symmetry we only need to consider the case of the left multiplication h . This map can be viewed as the composite m

G −→ G × G −−→ G g −→ (h, g). The left arrow is locally analytic by Example 8.5.4 and the right arrow by assumption. Hence the map h is locally analytic by Lemma 8.4.ii. We obviously have h ◦ h−1 = hh−1 = e = idG and then also h−1 ◦ h = idG . It follows that −1 h = h−1 is locally analytic as well. Corollary 13.2. For any two elements g, h ∈ G the map ∼ =

Tg (hg−1 ) : Tg (G) −−→ Th (G) is a K-linear isomorphism; in particular, ∼ =

Te (g ) : Te (G) −−→ Tg (G) is an isomorphism for any g ∈ G. P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8 3, © Springer-Verlag Berlin Heidelberg 2011

89

90

III

Lie Groups

Proof. Use Lemma 9.2. Corollary 13.3. Every Lie group is n-dimensional for some n ≥ 0. Proof. We have dim G = dimK Te (G) = dimK Tg (G) for any g ∈ G. Examples. 1) K n and more generally any ball Bε (0) (as an open submanifold of K n ) with the addition is a Lie group. 2) K × and more generally B1− (1) and Bε (1) for any 0 < ε < 1 (as open submanifolds of K) with the multiplication (observe that ab − 1 = (a − 1)(b − 1) + (a − 1) + (b − 1)) are Lie groups. 2

3) GLn (K) viewed as the open submanifold in K n defined by “det = 0” with the matrix multiplication is a Lie group. Let g, h ∈ G. We know from Remark 9.10.ii. that the map ∼ =

T (pr1 ) × T (pr2 ) : T(g,h) (G × G) −−→ Tg (G) × Th (G) is a K-linear isomorphism. In order to describe its inverse we introduce the maps ih : G −→ G × G

and

jg : G −→ G × G

x −→ (x, h)

x −→ (g, x)

which are locally analytic by Example 8.5.4). We have pr1 ◦ ih = idG

and

pr2 ◦ ih = constant map with value h

and hence T (pr1 ) ◦ T (ih ) = T (idG ) = idT (G) and T (pr2 ) ◦ T (ih ) = T (constant map) = 0. This means that the composed map T (ih )

T (pr )×T (pr )

1 2 −−−−− → Tg (G) × Th (G) Tg (G) −−−−→ T(g,h) (G × G) −−−−−

sends t to (t, 0). Analogously the composed map (17)

T (jg)

T (pr )×T (pr )

1 2 Th (G) −−−−→ T(g,h) (G × G) −−−−− −−−−− → Tg (G) × Th (G)

13

Definitions and Foundations

91

sends t to (0, t). We conclude that Tg (ih ) + Th (jg) : Tg (G) × Th (G) −→ T(g,h) (G × G) (t1 , t2 ) −→ Tg (ih )(t1 ) + Th (jg)(t2 ) is the inverse of T(g,h) (pr1 ) × T(g,h) (pr2 ). Lemma 13.4. The diagram T (m)

T(g,h) (G × G)

Tgh (G)

∼ = T (pr1 )×T (pr2 )

Tg (rh )+Th (lg )

Tg (G) × Th (G) is commutative for any g, h ∈ G. Proof. We compute T(g,h) (m) ◦ (T(g,h) (pr1 ) × T(g,h) (pr2 ))−1 = T(g,h) (m) ◦ (Tg (ih ) + Th (jg )) = Tg (m ◦ ih ) + Th (m ◦ jg ) = Tg (rh ) + Th (lg ).

Corollary 13.5. The diagram T (m)

T(e,e) (G × G)

Te (G)

∼ = +

T (pr1 )×T (pr2 )

Te (G) × Te (G) is commutative. Proposition 13.6. The map ι = ιG : G −→ G g −→ g −1 is a locally analytic isomorphism (of manifolds).

92

III

Lie Groups

Proof. Because of ι2 = idG it suffices to show that the map ι is locally analytic. To do so we use the bijective locally analytic map μ : G × G −→ G × G (x, y) −→ (xy, y). We claim that the tangent map T(g,h) (μ), for any g, h ∈ G, is bijective. By Lemma 13.4 the diagram T(g,h) (G × G)

T(g,h) (μ)

T(gh,h) (G × G)

T (pr1 )×T (pr2 ) ∼ =

∼ = T (pr1 )×T (pr2 )

Tg (G) × Th (G)

Tgh (G) × Th (G)

in which the lower horizontal arrow is given by (t1 , t2 ) −→ (Tg (rh )(t1 ) + Th (lg )(t2 ), t2 ) is commutative. Suppose that (t1 , t2 ) lies in the kernel of this latter map. Then t2 = 0 and hence 0 = Tg (rh )(t1 ) + Th (lg )(t2 ) = Tg (rh )(t1 ). The analog of Cor. 13.2 for the right multiplication implies that t1 = 0. We see that this lower horizontal map and therefore T(g,h) (μ) are injective. But all vector spaces in the diagram have the same finite dimension. Our claim that T(g,h) (μ) is bijective follows. We now may apply the criterion for local invertibility in Prop. 9.3 and we conclude that the inverse μ−1 is locally analytic as well. It remains to note that ι is the composite je

μ−1

pr

1 G −−→ G × G −−−→ G × G −−−→ G.

Corollary 13.7. For any g ∈ G the diagram Te (G)

Te (lg )

−1

Te (G)

Tg (G) Tg (ι)

Te (rg−1 )

Tg−1 (G)

is commutative; in particular, the map Te (ι) coincides with the multiplication by −1.

13

Definitions and Foundations

93

Proof. The composed map lg

rg

ι

ι

G −−→ G −→ G −−→ G −→ G is the identity. Hence the diagram Te (lg )

Te (G)

Tg (G) Tg (ι)

Te (ι)

Te (G)

Te (rg−1 )

Tg−1 (G)

is commutative. This reduces us to showing the special case in our assertion. We consider the diagram Te (G) t→(0,t)

T (je ) T (pr1 )×T (pr2 )

T(e,e) (G × G)

Te (G) × Te (G)

T(e,e) (μ−1 )=T(e,e) (μ)−1

(t1 ,t2 )→(t1 −t2 ,t2 ) T (pr1 )×T (pr2 )

T(e,e) (G × G)

Te (G) × Te (G) pr1

T (pr1 )

Te (G)

.

In the proof of Prop. 13.6 we have seen that the map μ−1 (x, y) = (xy −1 , y) is locally analytic and that the central square in the above diagram is commutative. The top triangle is commutative by (17). The commutativity of the bottom triangle is trivial. It remains to observe that passing from top to bottom along the left, resp. right, hand side is equal to Te (ι), resp. to the multiplication by −1. Corollary 13.8. For every n ∈ Z the map fn : G −→ G g −→ g n is locally analytic, and Te (fn ) coincides with the multiplication by n.

94

III

Lie Groups

Proof. Case 1: For n = 0 the map f0 is the constant map with value e and Te (f0 ) = 0. Case 2: Let n ≥ 1. We may view fn as the composite diag

mult

G G −−−→ G × · · · × G −−−→ g −→ (g, . . . , g) (g1 , . . . , gn ) −→ g1 . . . gn . Both maps are locally analytic, the left diagonal map by Example 8.5.4) and the right multiplication map by assumption. Hence in the diagram Te (G × · · · × G) Te (diag)

Te (mult) Q

Te (G)

i

Te (G)

T (pri )

(t1 ,...,tn )→t1 +···+tn

diag

Te (G) × · · · × Te (G) the top, resp. bottom, composed map is equal to Te (fn ), resp. the multiplication by n. But this diagram is commutative, the left triangle for trivial reasons and the right triangle as a consequence of Cor. 13.5. Case 3: Let n ≤ −1. Since fn = f−n ◦ ι we obtain, using the previous case and Cor. 13.7, that Te (fn ) = Te (f−n ) ◦ Te (ι) = (−n · id) ◦ (− id) = n · id .

Cor. 13.2 already indicates that the tangent space Te (G) in the unit element of G plays a distinguished role. We want to investigate this in greater detail. Proposition 13.9. The maps 

rT : Te (G) × G −−→ T (G)

and



lT : G × Te (G) −−→ T (G)

(t, g) −→ Te (rg )(t)

(g, t) −→ Te (lg )(t)

are locally analytic isomorphisms (of manifolds); the diagram Te (G) × G

rT

pr2

T (G) pG

G is commutative.

lT

pr1

G × Te (G)

13

Definitions and Foundations

95

Proof. By symmetry it suffices to discuss the map rT . We choose a chart c = (U, ϕ, K n ) for G around e. By Lemma 9.1.i. the map ∼ =

θc : K n −−→ Te (G) v −→ [c, v] is a K-linear isomorphism. We equip Te (G) with the unique structure of a manifold such that θc becomes a locally analytic isomorphism of manifolds. By Lemma 9.1.ii. this structure does not depend on the choice of the chart c. Of course, we then view Te (G) × G as the product manifold of Te (G) and ⊆

G. The inclusion map Te (G) −−→ T (G) is locally analytic since it can be viewed as the composite of the locally analytic maps θ−1

v→(e,v)

⊆

c Te (G) −−c−→ K n −−−−−−→ U × K n −−→ p−1 −→ T (G). G (U ) −

τ

We recall that τc ((g, v)) = [c, v] ∈ Tg (G) is locally analytic by the construction of T (G) as a manifold. Let ξ0 : G −→ T (G) g −→ 0 ∈ Tg (G) denote the “zero vector field”, i. e., the zero vector in the vector space Γ(G, T (G)). Using Lemma 13.4 we see that the composed locally analytic map ⊆× id

id ×ξ0

Te (G) × G −−−−→ T (G) × G −−−−→ T (G) × T (G) (T (pr )×T (pr ))−1

T (m)

−−−−−1−−−−−2−−−→ T (G × G) −−−−→ T (G) sends (t, g) to Te (rg )(t) + Tg (le )(0) = Te (rg )(t) and hence coincides with rT . This shows that the map rT is locally analytic. It is easy to check that the map T (G) −→ Te (G) × G t −→ (TpG (t) (rpG (t)−1 )(t), pG (t)) is inverse to rT . Its second component pG is locally analytic by Lemma 9.8. It therefore remains to prove that the map f : T (G) −→ Te (G) t −→ TpG (t) (rpG (t)−1 )(t)

96

III

Lie Groups

is locally analytic. Using again Lemma 13.4 we compute that the composed locally analytic map id ×p

id ×ξ0

id ×ι

T (G) −−−−G→ T (G) × G −−−−→ T (G) × G −−−−→ T (G) × T (G) (T (pr )×T (pr ))−1

T (m)

−−−−−1−−−−−2−−−→ T (G × G) −−−−→ T (G) sends t to TpG (t) (rpG (t)−1 )(t). It follows that the left vertical composite in the commutative diagram T (G)

=

f

Te (G)

f =

Te (G)

f =

Te (G)

t→(e,θc−1 (t))

⊆

T (G)

T (G)

τc

U × Kn

(g,v)→θc (v)

is locally analytic. Since τc is an open embedding we conclude that the right vertical composite is locally analytic. With the lower oblique arrow therefore also the upper oblique arrow f is locally analytic. Corollary 13.10. The maps ∼ =

∼ =

Γ(G, T (G)) ←−− C an (G, Te (G)) −−→ Γ(G, T (G)) l T ξf (g) := l ((g, f (g))) ←− f −→ ξfr (g) := rT ((f (g), g)) are isomorphisms of K-vector spaces. Proof. The maps ξ −→ pr2 ◦(lT )−1 ◦ξ and ξ −→ pr1 ◦(rT )−1 ◦ξ, respectively, are inverses. In C an (G, Te (G)) we have, for any t ∈ Te (G), the constant map constt (g) := t. We put l r ξtl (g) := ξconst (g) = Te (lg )(t) and ξtr (g) := ξconst (g) = Te (rg )(t). t t

Definition. A vector field ξ ∈ Γ(G, T (G)) is called left invariant, resp. right invariant, if ξ(g) = Te (lg )(ξ(e)), resp. ξ(g) = Te (rg )(ξ(e)), holds true for any g ∈ G.

13

Definitions and Foundations

97

Corollary 13.11. The maps ∼ =

Te (G) −−→ {ξ ∈ Γ(G, T (G)) : ξ is left invariant} t −→ ξtl and ∼ =

Te (G) −−→ {ξ ∈ Γ(G, T (G)) : ξ is right invariant} t −→ ξtr are K-linear isomorphisms. Proof. The map ξ −→ ξ(e) is the inverse in both cases. Let E be a K-Banach space. With any vector field ξ on G we had associated the K-linear map Dξ : C an (G, E) −→ C an (G, E) f −→ Dξ (f ) = df ◦ ξ. If ξ is left or right invariant what consequence does this have for the map Dξ ? We observe that as a consequence of Lemma 13.1 we have a left K-linear action by “left translation” G × C an (G, E) −→ C an (G, E) (h, f ) −→

h

f (g) := f (h−1 g)

of the group G on the vector space C an (G, E) as well as a right K-linear action by “right translation” C an (G, E) × G −→ C an (G, E) (f, h) −→ f h (g) := f (gh−1 ). Lemma 13.12. If ξ ∈ Γ(G, T (G)) is right, resp. left, invariant then we have Dξ (f h ) = Dξ (f )h ,

resp.

Dξ ( h f ) =

h

Dξ (f ),

for any f ∈ C an (G, E) and h ∈ G. In the case E = K the converse holds true as well.

98

III

Lie Groups

Proof. By symmetry we only consider the “right” case. First we suppose that ξ is right invariant, i. e., ξ(g) = Te (rg )(ξ(e)). It follows that T (rh−1 ) ◦ ξ(g) = Tg (rh−1 ) ◦ Te (rg )(ξ(e)) = Te (rgh−1 )(ξ(e)) = ξ(gh−1 ). We now compute Dξ (f h )(g) = df h ◦ ξ(g) = d(f ◦ rh−1 ) ◦ ξ(g) = df ◦ T (rh−1 ) ◦ ξ(g) = df ◦ ξ(gh−1 ) = Dξ (f )(gh−1 ) = Dξ (f )h (g) where for the second line we have used Lemma 9.12. If vice versa Dξ satisfies the asserted identity (for some E) then we have df ◦ T (rh−1 ) ◦ ξ(g) = Dξ (f h )(g) = Dξ (f )h (g) = Dξ (f )(gh−1 ) = df ◦ ξ(gh−1 ) for any f and any g, h. We rewrite this as df ◦ T (rh−1 ) ◦ ξ(gh) = df ◦ ξ(g). With ξ also

ξ h (g) := T (rh ) ◦ ξ(gh−1 )

is a vector field on G. Hence we obtain the identity Dξh = Dξ

for any h ∈ G.

Later on (Cor. 18.8) we will see that G is paracompact. In the case E = K the map ξ → Dξ therefore is injective by Prop. 9.16. It follows that ξ h = ξ, i. e., that T (rh ) ◦ ξ(gh−1 ) = ξ(g) holds true for any g, h ∈ G. In particular, for g = h we obtain T (rg )(ξ(e)) = ξ(g)

for any g ∈ G

which means that ξ is right invariant. The Lie product of vector fields is characterized by the identity Dξ ◦ Dη − Dη ◦ Dξ = D[ξ,η] (which even holds for general E by Prop. 9.20 and Cor. 18.8).

13

Definitions and Foundations

99

Corollary 13.13. If the vector fields ξ and η on G both are left or right invariant then so, too, is the vector field [ξ, η]. Proof. With Dξ and Dη obviously also D[ξ,η] satisfies the identity asserted in Lemma 13.12. Using Lemma 13.11 and Cor. 13.13 we see that for any s, t ∈ Te (G) there are uniquely determined tangent vectors [s, t]l and [s, t]r in Te (G) such that l ξ[s,t] = [ξsl , ξtl ] l

Then

r and ξ[s,t] = [ξsr , ξtr ]. r t→ξ l

t (Γ(G, T (G)), [ , ]) (Te (G), [ , ]l ) −−−−→

and

t→ξ r

t (Γ(G, T (G)), [ , ]) (Te (G), [ , ]r ) −−−−→

are injective maps of Lie algebras. Is there a relation between the two Lie products [ , ]l and [ , ]r on Te (G)? For any ξ ∈ Γ(G, T (G)) also ι

ξ(g) := Tg−1 (ι) ◦ ξ(g −1 )

is a vector field on G. This provides us with an involutory K-linear automorphism ι . : Γ(G, T (G)) −→ Γ(G, T (G)). Remark 13.14. The diagram Te (G)

ξ.l

Γ(G, T (G))

−1

Te (G)

ι.

ξ.r

Γ(G, T (G))

is commutative. Proof. Using Cor. 13.7 we compute ι

(ξtl )(g) = T (ι) ◦ ξtl (g −1 ) = T (ι) ◦ T (lg−1 )(t) = −T (rg )(t) = −ξtr (g).

Lemma 13.15. Any vector fields ξ and η on G satisfy [ι ξ, ι η] = ι [ξ, η].

100

III

Lie Groups

Proof. Using Lemma 9.12 we compute Dι ξ (f )(g) = df ◦ ι ξ(g) = df ◦ T (ι) · ξ(g −1 ) = d(f ◦ ι) ◦ ξ(g −1 ) = Dξ (f ◦ ι)(g −1 ). This amounts to the identity Dι ξ (f ) ◦ ι = Dξ (f ◦ ι). We continue computing (Dι ξ ◦ Dι η )(f )(g) = Dξ (Dι η (f ) ◦ ι)(g −1 ) = Dξ (Dη (f ◦ ι))(g −1 ) = (Dξ ◦ Dη )(f ◦ ι)(g −1 ) and consequently D[ι ξ,ι η] (f )(g) = [Dι ξ , Dι η ](f )(g) = [Dξ , Dη ](f ◦ ι)(g −1 ) = D[ξ,η] (f ◦ ι)(g −1 ) = Dι [ξ,η] (f )(g).

Corollary 13.16. We have [s, t]r = −[s, t]l

for any s, t ∈ Te (G).

Proof. Using Remark 13.14 and Lemma 13.15 we compute r r r = [ξsr , ξtr ] = [−ξsr , −ξtr ] = [ξ−s , ξ−t ] ξ[s,t] r

= [ι ξsl , ι ξtl ] = ι [ξsl , ξtl ] = ι ξ l[s,t]l r = ξ−[s,t] . l

From now on we simplify the notation by setting [s, t] := [s, t]r

and Dt := Dξtr

for any s, t ∈ Te (G). We then have the identity D[s,t] = Ds ◦ Dt − Dt ◦ Ds on C an (G, E). Definition. Lie(G) := (Te (G), [ , ]) is called the Lie algebra of G.

14

The Universal Enveloping Algebra

101

We obviously have dimK Lie(G) = dim G. The guiding question for the rest of this book is how much information the Lie algebra Lie(G) retains about the Lie group G. The answer requires several purely algebraic concepts which we discuss in the next few sections. Definition. Let G1 and G2 be two Lie groups over K; a homomorphism of Lie groups f : G1 −→ G2 is a locally analytic map which also is a group homomorphism. Definition. If (g1 , [ , ]1 ) and (g2 , [ , ]2 ) are two Lie algebras over K then a homomorphism (of Lie algebras) σ : g1 −→ g2 is a K-linear map which satisfies [σ(x), σ(y)]2 = σ([x, y]1 ) for any x, y ∈ g1 . We write HomK ((g1 , [ , ]1 ), (g2 , [ , ]2 )) for the set of all homomorphisms of Lie algebras σ : g1 −→ g2 . Exercise. For any homomorphism of Lie groups f : G1 −→ G2 the map Lie(f ) := Te (f ) : Lie(G1 ) −→ Lie(G2 ) is a homomorphism of Lie algebras.

14

The Universal Enveloping Algebra

In this section K is allowed to be a completely arbitrary field. Exercise. i. Let A be an associative K-algebra with a unit element. Then (A, [ , ]A ) with [x, y]A := xy − yx is a Lie algebra over K. In the case of a matrix algebra A = Mn×n (K) the corresponding Lie algebra is denoted by gln (K). ii. If the field K is nonarchimedean then we have gln (K) = Lie(GLn (K)). How general are the Lie algebras in this exercise? Obviously (A, [ , ]A ) may have Lie subalgebras which do not correspond to associative subalgebras. We want to show that any Lie algebra in fact arises as a subalgebra of an associative algebra. A K-linear map σ : g −→ A from a Lie algebra g into an associative algebra A, of course, will be called a homomorphism if it satisfies σ([x, y]) = σ(x)σ(y) − σ(y)σ(x)

for any x, y ∈ g.

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Lie Groups

At this point we need to recall the following general construction from multilinear algebra. Let E be any K-vector space. Then T (E) := ⊕n≥0 E ⊗n

where

E ⊗n := E ⊗K · · · ⊗K E

(n factors)

is an associative K-algebra with unit (note that E ⊗0 = K). The multiplication is given by the linear extension of the rule (v1 ⊗ · · · ⊗ vn )(w1 ⊗ · · · ⊗ wm ) := v1 ⊗ · · · ⊗ vn ⊗ w1 ⊗ · · · ⊗ wm . This algebra T (E) is called the tensor algebra of the vector space E. It has the following universal property. Any K-linear map σ : E −→ A into any associative K-algebra with unit A extends in a unique way to a homomorphism of K-algebras with unit σ ˜ : T (E) −→ A. In fact, this extension satisfies σ ˜ (v1 ⊗ · · · ⊗ vn ) = σ(v1 ) · . . . · σ(vn ). Let g be a Lie algebra over K. Viewed as a K-vector space we may form the tensor algebra T (g). In T (g) we consider the two sided ideal J(g) generated by all elements of the form x ⊗ y − y ⊗ x − [x, y]

for x, y ∈ g.

Note that x ⊗ y − y ⊗ x ∈ g⊗2 whereas [x, y] ∈ g⊗1 . Then U (g) := T (g)/J(g) is an associative K-algebra with unit and ε : g −→ U (g) x −→ x + J(g) is a homomorphism. Definition. U (g) is called the universal enveloping algebra of the Lie algebra g. This construction has the following universal property. Let σ : g −→ A be any homomorphism into any associative K-algebra with unit A. It extends uniquely to a homomorphism σ ˜ : T (g) −→ A of K-algebras with unit. Because of σ ˜ (x ⊗ y − y ⊗ x − [x, y]) = σ(x)σ(y) − σ(y)σ(x) − σ([x, y]) = 0

14

The Universal Enveloping Algebra

103

we have J(g) ⊆ ker(˜ σ ). Hence there is a uniquely determined homomorphism of K-algebras with unit σ ¯ : U (g) −→ A with σ ¯ ◦ ε = σ, i. e., the diagram σ

g

A

⊆

ε

σ ˜

T (g)

σ ¯

pr

U (g) is commutative. The tensor algebra T (E) has the increasing filtration T0 (E) ⊆ T1 (E) ⊆ · · · ⊆ Tm (E) ⊆ · · · defined by

Tm (E) := ⊕0≤n≤m E ⊗n .

The Tm (E) do not form ideals in T (E). But they satisfy Tl (E) · Tm (E) ⊆ Tl+m (E)

for any l, m ≥ 0.

Correspondingly we obtain an increasing filtration U0 (g) ⊆ U1 (g) ⊆ · · · ⊆ Um (g) ⊆ · · · in U (g) defined by Um (g) := Tm (g) + J(g)/J(g) and which satisfies (18)

Ul (g) · Um (g) ⊆ Ul+m (g)

for any l, m ≥ 0.

For example, we have U0 (g) = K · 1 and U1 (g) = K · 1 + ε(g). We define gr• U (g) := ⊕m≥0 grm U (g)

with

grm U (g) := Um (g)/Um−1 (g)

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(and the convention that U−1 (g) := {0}). Because of (18) the K-bilinear maps grl U (g) × grm U (g) −→ grl+m U (g) (y + Ul−1 (g), z + Um−1 (g)) −→ yz + Ul+m−1 (g) are well defined. Together they make gr• U (g) into an associative K-algebra with unit. Theorem 14.1. (Poincar´e-Birkhoff-Witt) The algebra gr• U (g) is isomorphic to a polynomial ring over K in possibly infinitely many variables Xi and, in particular, is commutative. More precisely, let {xi }i∈I be a K-basis of g; then ∼ =

K[{Xi }i∈I ] −−→ gr• U (g) Xi −→ ε(xi ) + U0 (g) ∈ gr1 U (g) is an isomorphism of K-algebras with unit. Proof. Compare [B-LL] Chap. I §2.7 or [Hum] §17.3. Corollary 14.2. The map ε : g −→ U (g) is injective. Because of this fact the map ε usually is viewed as an inclusion and is omitted from the notation. We see that g indeed is a Lie subalgebra of an associative algebra. Corollary 14.3. Let d := dimK g < ∞; if x1 , . . . , xd is an (ordered ) Kbasis of g then {xi1 · . . . · xim : m ≥ 0, 1 ≤ i1 ≤ · · · ≤ im ≤ d} is a K-basis of U (g). Proof. The Theorem 14.1 of Poincar´e-Birkhoff-Witt implies that, for any m ≥ 0, the set {xi1 · . . . · xim + Um−1 (g) : 1 ≤ i1 ≤ · · · ≤ im ≤ d} is a K-basis of Um (g)/Um−1 (g) (recall the convention that the empty product, in the case m = 0, is equal to the unit element). This last corollary obviously remains true, by choosing a total ordering of a K-basis of g, even if g is not finite dimensional. Let τ : g1 −→ g2 be a homomorphism of Lie algebras. Applying the universal property gives a homomorphism of K-algebras with unit U (τ ) : U (g1 ) −→ U (g2 )

14

The Universal Enveloping Algebra

105

such that the diagram τ

g1

g2

⊆

⊆

U (g1 )

U (τ )

U (g2 )

is commutative. We want to apply this in two specific situations. First let g1 and g2 two Lie algebras. Obviously, g1 × g2 again is a Lie algebra with respect to the componentwise Lie product. There are the corresponding monomorphisms of Lie algebras i1 : g1 −→ g1 × g2

and

i2 : g2 −→ g1 × g2

x −→ (x, 0)

y −→ (0, y).

Lemma 14.4. The map ∼ =

U (g1 ) ⊗K U (g2 ) −−→ U (g1 × g2 ) a ⊗ b −→ U (i1 )(a) · U (i2 )(b) is an isomorphism of K-algebras with unit. Proof. Obviously, we have the K-bilinear map U (g1 ) × U (g2 ) −→ U (g1 × g2 ) (a, b) −→ U (i1 )(a) · U (i2 )(b). By the universal property of the tensor product it induces the map in the assertion as a K-linear map. The latter is bijective by a straightforward application of Cor. 14.3. Since we have [i1 (x), i2 (y)] = [(x, 0), (0, y)] = ([x, 0], [0, y]) = (0, 0) for any x ∈ g1 and any y ∈ g2 it follows easily that U (i1 )(a) and U (i2 )(b), for any a ∈ U (g1 ) and any b ∈ U (g2 ), commute with one another. This implies that the asserted map is a homomorphism and hence an isomorphism of K-algebras with unit. We point out that under the isomorphism in the above lemma the elements x ⊗ 1 + 1 ⊗ y ←→ (x, y)

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correspond to each other. Secondly, for any Lie algebra g the diagonal map Δ : g −→ g × g x −→ (x, x) is a homomorphism of Lie algebras. We obtain the commutative diagram g

Δ

⊆

U (g)

g×g ⊆

U (Δ)

U (g × g)

∼ =

U (g) ⊗K U (g).

Definition. The composed map U (g) −→ U (g) ⊗K U (g) in the lower line of the above diagram is denoted (by abuse of notation) again by Δ and is called the diagonal (or comultiplication) of the algebra U (g). We note that for x ∈ g ⊆ U (g) we have Δ(x) = x ⊗ 1 + 1 ⊗ x. Proposition 14.5. If the field K has characteristic zero then we have g = {a ∈ U (g) : Δ(a) = a ⊗ 1 + 1 ⊗ a}. Proof. Compare [B-LL] Chap. II §1.5 Cor.

15

The Concept of Free Algebras

In this section K again is an arbitrary field. We will discuss the following problem. Let A be a specific class (or category) of K-algebras. We have in mind the following list of examples: – ComK := all commutative and associative K-algebras with unit; – AssK := all associative K-algebras with unit; – LieK := all Lie algebras over K; – AlgK := all K-algebras, i. e., all K-vector spaces A equipped with a K-bilinear “multiplication” map A × A −→ A.

15

The Concept of Free Algebras

107

We suppose given a finite set X = {X1 , . . . , Xd }, and we ask for an algebra AX in the class A together with a map X −→ AX which have the following universal property: For any map X −→ A from the set X into any algebra A in the class A there is a unique homomorphism AX −→ A of algebras in A such that the diagram A

X

AX is commutative. If it exists AX is called the free A-algebra on X. The case ComK : The polynomial ring AX := K[X1 , . . . , Xd ] over K in the variables X1 , . . . , Xd has the requested universal property. The case AssK : As we have recalled in Sect. 14 the tensor algebra AX := AsX := T (K d ) of the standard K-vector space K d together with the map X −→ K d ⊆ T (K d ) Xi −→ i-th standard basis vector ei satisfies the requested universal property. It sometimes is useful to view AsX as the ring of all “noncommutative” polynomials  a(i1 ,...,im ) Xi1 · . . . · Xim P (X1 , . . . , Xd ) = (i1 ,...,im )

with coefficients a(i1 ,...,im ) ∈ K where the sum runs over finitely many tuples (i1 , . . . , im ) with entries from the set {1, . . . , d} (including possibly the empty tuple). The multiplication is determined by the rule that the variables commute with the coefficients but not with each other. The algebra AsX in (n) a natural way is graded by AsX := K d ⊗K · · · ⊗K K d (n factors) which means that (n)

AsX = ⊕n≥0 AsX

(l)

(m)

(l+m)

with AsX · AsX ⊆ AsX

for any l, m ≥ 0.

The case AlgK : Here we have to preserve the information about the order in which the multiplications in a “monomial” Xi1 · . . . · Xim are performed (and we have to omit the unit element). This can be done in the following way. We inductively define sets X(n) for n ≥ 1 by X(1) := X and X(n) := disjoint union of all X(p) × X(q) for p + q = n,

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and we put MX := disjoint union of all X(n). ⊆

The obvious inclusion maps X(m) × X(n) −−→ X(m + n) combine into a “multiplication” map μX : MX × MX −→ MX . We now form the K-algebra AX := the K-vector space on the basis MX in which the multiplication is given by the linear extension of the map μX . There are the obvious inclusions X ⊆ MX ⊆ AX . Let γ : X −→ A be any map into any K-algebra A. We inductively extend γ to a map γ : MX −→ A by γ : X(n) ⊇ X(p) × X(q) −→ A (x, y) −→ γ(x)γ(y). This extension by construction is multiplicative in the sense that the diagram M X × MX

μX

MX γ

γ×γ ·

A×A

A

is commutative. Hence it further extends by linearity to a homomorphism of K-algebras γ˜ : AX −→ A. We stress that the algebra AX is graded by (n)

AX := the K-vector space on the basis X(n), i. e., we have (n)

AX = ⊕n∈N AX

(l)

(m)

(l+m)

with AX · AX ⊆ AX

for any l, m ≥ 1.

The case LieK : In AX we consider the two sided ideal JX which is generated by all expressions of the form aa

and (ab)c + (bc)a + (ca)b

for a, b, c ∈ AX .

15

The Concept of Free Algebras

109

Then LX := AX /JX

with [a + JX , b + JX ] := ab + JX

is a Lie algebra over K. Let γ : X −→ g be any map into any Lie algebra g over K. As discussed above it extends to a homomorphism of K-algebras γ˜ : AX −→ g. We obviously have JX ⊆ ker(˜ γ ). Hence there is a uniquely determined homomorphism of Lie algebras γ¯ : LX −→ g such that the diagram γ

X

g

⊆

γ ˜

AX

γ ¯

pr

LX is commutative. Exercise.

(n)

i. We have JX = ⊕n∈N JX ∩ AX and hence (n)

LX = ⊕n∈N LX (n)

with (n)

(l)

(m)

(l+m)

[LX , LX ] ⊆ LX

for any l, m ≥ 1

(n)

if we define LX := AX /JX ∩AX (i. e., the Lie algebra LX is graded ). (1)

ii. The set X is (more precisely, maps bijectively onto a) K-basis of LX . (2)

iii. The set {[Xi , Xj ] : i < j} is a K-basis of LX . ⊆

The inclusion map X −−→ AsX extends uniquely to a homomorphism of Lie algebras φ : LX −→ (AsX , [ , ]AsX ). By the universal property of the universal enveloping algebra this map φ further extends uniquely to a homomorphism of associative K-algebras with unit Φ : U (LX ) −→ AsX . Proposition 15.1. The map Φ is an isomorphism of algebras.

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Proof. The composed map X −→ LX −→ U (LX ) extends uniquely to a homomorphism of associative K-algebras with unit Ψ : AsX −→ U (LX ). We have the commutative diagram X ⊆

⊆

LX φ

AsX

Ψ

U (LX )

Φ

AsX

.

It therefore follows from the unicity in the universal property of AsX that Φ ◦ Ψ = id. We also have the commutative diagram X

LX

⊆

LX

φ

U (LX )

AsX

Φ

U (LX )

Ψ

.

The universal property of LX then implies that even the diagram LX

U (LX )

Φ

AsX

Ψ

U (LX )

is commutative. Using the universal property of U (.) we conclude that Ψ ◦ Φ = id as well. Corollary 15.2. The map φ : LX −→ AsX is injective. Proof. Combine Cor. 14.2 and Prop. 15.1.

16

The Campbell-Hausdorff Formula

111

Exercise. i. If we view AsX as the ring of noncommutative polynomials P (X1 , . . . , Xd ) over K in the variables X1 , . . . , Xd then φ(LX ) is the smallest K-vector subspace of AsX which contains the variables X1 , . . . , Xd and is closed under the operation (P, Q) −→ P · Q − Q · P . (n)

(n)

ii. φ(LX ) ⊆ AsX for any n ∈ N, i. e., the homomorphism φ is graded.

16

The Campbell-Hausdorff Formula

Again K is an arbitrary field and X = {X1 , . . . , Xd } is a fixed finite set. We recall that the free associative K-algebra with unit AsX on X is graded: (n)

(l)

AsX = ⊕n≥0 AsX

(m)

(l+m)

and AsX · AsX ⊆ AsX

Therefore

 X := As



for any l, m ≥ 0.

(n)

AsX

n≥0

with the multiplication  (an )n · (bn )n :=

n  i=0

 ai bn−i n

also is an associative K-algebra with unit (containing AsX as a subalgebra). It is called the Magnus algebra on X. Similarly as for AsX it is useful to  X as the ring of all “noncommutative” formal power series over K view As  X we have the two sided maximal ideal in the variables X1 , . . . , Xd . In As  X : a0 = 0}. ˆ X := {(an )n ∈ As m Lemma 16.1.

 × i. As X = {(an )n ∈ AsX : a0 = 0}.

× ˆ X is a subgroup of As ii. 1 + m X. Proof. i. The map  X −→ K As (an )n −→ a0 is a homomorphism of K-algebras with unit. The group of multiplicative  × therefore must be contained in the complement of the kernel of units As X

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 X be an element such that a0 = 0. this map. Vice versa let a = (an )n ∈ As We have a = a0 · (1 − u)

where

−1 u := (0, −a−1 0 · a1 , . . . , −a0 · an+1 , . . .).

  (n) Since um ∈ {0} × · · · × {0} × n≥m AsX the sum m≥0 um is well defined

m we then obtain ab = ba = 1.  X . For b := a−1 ·  in As 0 m≥0 u ˆ X is the kernel of the homoii. This is obvious from i. Note that 1 + m morphism of groups × × As X −→ K

(an )n −→ a0 .

In the last proof we have used a special case of the following general  (n) principle. For each m ≥ 0 let u(m) ∈ {0} × · · · × {0} × n≥m AsX be some   X is well defined. In particular, for element. Then the sum m≥0 u(m) ∈ As ˆ X we have the well defined homomorphism of K-algebras with any u ∈ m unit X εu : K[[T ]] −→ As F (T ) −→ F (u). Proposition 16.2. If the field K has characteristic zero then the maps ˆX ˆ X −→ 1 + m exp : m  un u −→ n!

and

n≥0

ˆ X −→ m ˆX log : 1 + m  un (−1)n+1 1 + u −→ n n≥1

are well defined and inverse to each other. Proof. Because of exp(u) = εu (exp(T )) and

log(1 + u) = εu (log(1 + T ))

the maps in the assertion are well defined. Applying εu to the identities exp(log(1 + T )) = 1 + T

and

log(exp(T )) = T

in the ring Q[[T ]] shows that they are inverse to each other.

16

The Campbell-Hausdorff Formula

113

ˆ X commute with each other (multiplicatively) then we Exercise. If a, b ∈ m have exp(a + b) = exp(a) · exp(b). Using Cor. 15.2 and the subsequent Exercise we may view LX as the Lie subalgebra of AsX “generated” by the elements X1 , . . . , Xd . Moreover, we have (1) (2) LX ⊕ LX ⊕ ··· LX = ∩ ∩ ∩ (1) (2) AsX = K⊕ AsX ⊕ AsX ⊕ · · · We now define ˆ X := L



(n) X. ˆ X ⊆ As LX ⊆ m

n≥1

X. ˆ X is a Lie subalgebra of As Lemma 16.3. L ˆ X . We have to Proof. Let a = (an )n and b = (bn )n be any two elements of L ˆ show that ab − ba ∈ LX holds true. For any m ≥ 1 we put a(m) := (0, a1 , . . . , am , 0, . . .), v (m) := (0, . . . , 0, am+1 , am+2 , . . .), b(m) := (0, b1 , . . . , bm , 0, . . .), u(m) := (0, . . . , 0, bm+1 , bm+2 , . . .). Then a(m) , b(m) ∈ LX and hence a(m) b(m) − b(m) a(m) ∈ LX . Moreover ab − ba = (a(m) + v (m) )(b(m) + u(m) ) − (b(m) + u(m) )(a(m) + v (m) ) = a(m) b(m) − b(m) a(m) + (0, . . . , 0, cm+1 , . . .). It follows that for n ≤ m we have (n)

n-th component of ab − ba = n-th component of a(m) b(m) − b(m) a(m) ∈ LX . ˆX . Since m was arbitrary we conclude that ab − ba ∈ L Since U (LX ) = AsX by Prop. 15.1 we may view the comultiplication Δ of U (LX ) as a homomorphism of K-algebras with unit Δ : AsX −→ AsX ⊗K AsK . It satisfies Δ(Xi ) = Xi ⊗ 1 + 1 ⊗ Xi for any 1 ≤ i ≤ d. Since the X1 , . . . , Xd (1) form a K-basis of AsX it follows that (1)

(1)

(0)

(0)

(1)

Δ(AsX ) ⊆ AsX ⊗K AsX + AsX ⊗K AsX

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and then inductively that (n)

Δ(AsX ) ⊆



(m)

(l)

AsX ⊗K AsX

l+m=n

for any n ≥ 0. This makes it possible to extend Δ to the homomorphism of K-algebras with unit  (n)  (l) (m)  X := ˆ : As X = X⊗ ˆ K As AsX ⊗K AsX Δ AsX −→ As n≥0

l,m≥0

= (an )n −→



 

(l)

(m)

AsX ⊗K AsX

n≥0 l+m=n

Δ(an ) n .

Lemma 16.4. If the field K has characteristic zero then we have ˆ X = {a ∈ As  X : Δ(a) ˆ L = a ⊗ 1 + 1 ⊗ a}.  X be any element. We have Δ(a) ˆ Proof. Let a = (an )n ∈ As = a⊗1+1⊗a if and only if Δ(an ) = an ⊗ 1 + 1 ⊗ an for any n ≥ 0. By Prop. 14.5 the latter (n) (n) is equivalent to an ∈ LX ∩ AsX = LX for any n ≥ 0 which exactly is the ˆX . condition that a ∈ L Theorem 16.5. (Campbell-Hausdorff ) Suppose that K has characteristic zero; then the map ∼ ˆ X −− ˆ ˆ X : Δ(b) exp : L → {b ∈ 1 + m = b ⊗ b}

is a well defined bijection; moreover, the right hand side is a subgroup of ˆ X. 1+m ˆ being Proof. The second part of the assertion follows immediately from Δ ˆ ˆ X and ˆ X the map exp is defined on L a ring homomorphism. Since LX ⊆ m is injective by Prop. 16.2. For the subsequent computations we observe that ˆ implies that the componentwise construction of the ring homomorphism Δ ˆ ˆ ˆ Δ commutes with the maps exp and log. First let a ∈ LX . Then Δ(a) = a ⊗ 1 + 1 ⊗ a, and we compute ˆ ˆ Δ(exp(a)) = exp(Δ(a)) = exp(a ⊗ 1 + 1 ⊗ a) = exp(a ⊗ 1) · exp(1 ⊗ a) = (exp(a) ⊗ 1) · (1 ⊗ exp(a)) = exp(a) ⊗ exp(a).

16

The Campbell-Hausdorff Formula

115

This shows that exp(a) indeed lies in the target of the asserted map. Vice ˆ ˆ X such that Δ(b) versa let b ∈ 1 + m = b ⊗ b. By Prop. 16.2 we may define ˆ X so that b = exp(a). We compute a := log(b) ∈ m ˆ ˆ ˆ Δ(a) = Δ(log(b)) = log(Δ(b)) = log(b ⊗ b) = log((b ⊗ 1) · (1 ⊗ b)) = log(b ⊗ 1) + log(1 ⊗ b) = log(b) ⊗ 1 + 1 ⊗ log(b) = a ⊗ 1 + 1 ⊗ a. ˆ X . We see that the asserted map is Hence Lemma 16.4 implies that a ∈ L surjective. ˆX Corollary 16.6. Suppose that the field K has characteristic zero; then L equipped with the multiplication a  b := log(exp(a) · exp(b)) is a group whose neutral element is the zero vector 0 and such that −a is the inverse of a. Proof. Since exp(0) = 1 the neutral element for  must be the zero vector 0. Furthermore, since a and −a commute with respect to the usual multi X we have exp(a) · exp(−a) = exp(−a) · exp(a) = exp(0) = 1. plication in As Hence a  (−a) = (−a)  a = log(1) = 0. Definition. For the field K = Q and the two-element set {Y, Z} we call  {Y,Z} ˆ {Y,Z} ⊆ As H(Y, Z) := Y  Z ∈ L the Hausdorff series (in Y and Z). As alluded to earlier we should view H(Y, Z) as a noncommutative formal power series in the variables Y, Z with coefficients in the field Q. We have exp(Y ) · exp(Z) = 1 + W

with W =

 Y r Zs · r! s!

r+s≥1

and hence

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Wm m m≥1  m  (−1)m+1  Y r Zs = · m r! s!

H(Y, Z) =

(−1)m+1

m≥1

 

=

r+s≥1

n 

n≥1 r+s=n m=1

(−1)m+1 m

m  Y ri



r1 +···+rm =r i=1 s1 +···+sm =s r1 +s1 ≥1,...,rm +sm ≥1

ri !

·

Z si . si !

 Here and in the following the product sign m i=1 always has to be understood in such a way that the corresponding multiplications are carried out in the order of the enumeration i = 1, . . . , m. It is convenient to use the abbreviations Hr,s

r+s  (−1)m+1 := m m=1



m  Y ri

r1 +···+rm =r i=1 s1 +···+sm =s r1 +s1 ≥1,...,rm +sm ≥1

and Hn :=



ri !

·

Z si si !

Hr,s .

r+s=n

We note that Hr,s is a sum of noncommutative monomials of degree r in Y and s in Z. As a sum of noncommutative monomials of total degree n the (n) element Hn lies in As{Y,Z} . We have  Hn or, more formally, H = (Hn )n . H= n≥1 (n)

From the theory we know that Hn ∈ L{Y,Z} for each n ≥ 1 but this is not visible from the above explicit formula. Examples.

1) H1,0 = Y , H0,1 = Z, and H1 = Y + Z.

2) Hr,0 = H0,r = 0 for any r ≥ 2 (observe, for example, that Hr,0 is the term of degree r in log(exp(Y )) = Y ). 3) H2 = H2,0 + H1,1 + H0,2 = H1,1 = Y Z − 12 (Y Z + ZY ) = 12 [Y, Z]. If g is any Lie algebra over (any) K then the K-linear map ad(z) : g −→ g x −→ [z, x],

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The Campbell-Hausdorff Formula

117

for any z ∈ g, is a derivation in the sense that ad(z)([x, y]) = [ad(z)(x), y] + [x, ad(z)(y)]

for any x, y ∈ g

holds true. This is just a reformulation of the Jacobi identity in g. Proposition 16.7. (Dynkin’s formula) For r + s ≥ 1 we have Hr,s =

1  (H  + Hr,s ) r + s r,s

 defined as with Hr,s

 (−1)m−1 m≥1

m



   m−1  ad(Y )ri ad(Z)si ad(Y )rm ◦ (Z) ◦ ri ! si ! rm !

r1 +···+rm =r s1 +···+sm−1 =s−1 r1 +s1 ≥1,...,rm−1 +sm−1 ≥1

i=1

and  Hr,s

 (−1)m−1 := m m≥1



 m−1  ad(Y )ri

r1 +···+rm−1 =r−1 s1 +···+sm−1 =s r1 +s1 ≥1,...,rm−1 +sm−1 ≥1

i=1

ri !

 ad(Z)si ◦ (Y ). si !

Proof. Compare [B-LL] Chap. II §6.4. We note that the above defining sums (n) are finite and make it visible that Hn lies in L{Y,Z} . Remark 16.8. Suppose that K has characteristic zero; then we have a  b = H(a, b)

ˆX . for any a, b ∈ L

Proof. The above explicit computations including Dynkin’s formula were completely formal and therefore are valid for any a, b (instead of Y, Z). The X expression H(a, b), of course, has to be calculated componentwise in As using the observation before Prop. 16.2. The exploitation of these “universal” considerations is based upon the following technique. For any finite dimensional K-vector space V let Map(V × V ; V ) := K-vector space of all maps f : V × V −→ V. We pick a K-basis e1 , . . . , ed of V .

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Definition. A map f : V × V −→ V is called polynomial (of degree (r, s)) if there are (homogeneous) polynomials Pi (X1 , . . . , Xd , Y1 , . . . , Yd ) over K (of degree r in X1 , . . . , Xd and degree s in Y1 , . . . , Yd ), for 1 ≤ i ≤ d, such that   d d d    ai ei , bi ei = Pi (a1 , . . . , ad , b1 , . . . , bd )ei for any ai , bi ∈ K. f i=1

i=1

i=1

In Map (V × V ; V ) we have the vector subspace Pol(V × V ; V ) of all polynomial maps. It decomposes into Pol(V × V ; V ) = ⊕n≥0 Poln (V × V ; V ) = ⊕n≥0 ⊕r+s=n Polr,s (V × V ; V ) where Polr,s (V ×V ; V ) denotes the subspace of all polynomial maps of degree (r, s) and Poln (V × V ; V ) := ⊕r+s=n Polr,s (V × V ; V ) is the subspace of all polynomial maps of total degree n. Lemma 16.9. Given any f ∈ Polr,s (V × V ; V ) and gi ∈ Polli ,mi (V × V ; V ) for i = 1, 2 the map (v, w) −→ f (g1 (v, w), g2 (v, w)) lies in Polrl1 +sl2 ,rm1 +sm2 (V × V ; V ). Proof. Straightforward. Corollary 16.10. The property of a map f : V × V −→ V of being polynomial (of a certain degree) does not depend on the choice of the K-basis of V . Proof. View the change of bases as a polynomial map and apply Lemma 16.9. Suppose now that the vector space V is a Lie algebra g of finite dimension d := dimK g. Then also the vector space Map(g × g; g) is a Lie algebra with respect to the Lie product [f, g](x, y) := [f (x, y), g(x, y)]. Corollary 16.11. Pol(g × g; g) is a Lie subalgebra of Map(g × g; g); more precisely, we have [Polr,s (g × g; g), Polr ,s (g × g; g)] ⊆ Polr+r ,s+s (g × g; g).

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The Campbell-Hausdorff Formula

119

Proof. Apply Lemma 16.9 to the map f := [, ] lying in Pol1,1 (g × g; g). We identify the two-element set {Y, Z} with the subset of Pol(g × g; g) consisting of the two projection maps pri : g × g −→ g by sending Y to pr1 and Z to pr2 . By the universal property of free Lie algebras this extends uniquely to a homomorphism of graded Lie algebras θ : L{Y,Z} −→ Pol(g × g; g). It satisfies (19)

θ([Y, a])(y, z) = [y, θ(a)(y, z)] and θ([Z, a])(y, z) = [z, θ(a)(y, z)]

for any a ∈ L{Y,Z} . We define Pow(g × g; g) :=



Poln (g × g; g)

n≥0

as a K-vector space. The elements of Pow(g × g; g) can be viewed (if K is infinite, and after the choice of a K-basis of g) as d-tuples of usual formal power series in the variables Y1 , . . . , Yd , Z1 , . . . , Zd with coefficients in K. As a consequence of Cor. 16.11 the Lie product on Pol(g × g; g) extends by    [(fn )n , (gn )n ] := [fl , gm ] l+m=n

n

to a Lie product on Pow(g × g; g). Being graded θ extends to the K-linear map ˆ {Y,Z} −→ Pow(g × g; g) θˆ : L (fn )n −→ (θ(fn ))n . Using the trick in the proof of Lemma 16.3 we obtain for any m ∈ N, with the notations in this proof, that ˆ θ([a, b]) ≡ θ([a(m) , b(m) ]) ≡ [θ(a(m) ), θ(b(m) )]  ˆ ˆ Poln (g × g; g) ≡ [θ(a), θ(b)] mod n>m

ˆ {Y,Z} . Since m is arbitrary this means that θˆ also is a homofor any a, b ∈ L morphism of Lie algebras. From now on we assume for the rest of this section that the field K has characteristic zero. We put

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ˆ ˜ := H g := θ(H) ∈ Pow(g × g; g). H More precisely, we have  ˜ = ˜ r,s H H

˜ r,s := θ(Hr,s ) ∈ Polr,s (g × g; g). with H

r+s≥1

Using (19) Dynkin’s formula in Prop. 16.7 implies that ˜ r,s = H

1  ˜ + H ˜ r,s (H ) r + s r,s

with  ˜ r,s : g × g −→ g H

 (−1)m−1  (y, z) −→ m ··· m≥1

 m−1  ad(y)ri i=1

ad(z)si ◦ si !

ri !



 ad(y)rm ◦ (z) rm !

and ˜  : g × g −→ g H r,s  (−1)m−1  (y, z) −→ m ··· m≥1

 m−1  ad(y)ri i=1

ri !

 ad(z)si (y). ◦ si !

˜ 1,0 = pr1 , H ˜ 0,1 = pr2 , and Examples 16.12. H ˜ 1,1 : g × g −→ g H 1 (y, z) −→ [y, z]. 2 ˜ First of all we recall from Cor. 16.6 What else do we know about H? and Remark 16.8 that we have (20)

H(a, H(b, c)) = H(H(a, b), c), H(a, 0) = H(0, a) = a, and H(a, −a) = 0

ˆ {Y,Z} . In order to use this we reinterpret the evaluation of for any a, b, c ∈ L H in a and b in the following way. ˆ {Y,Z} be any two elements. By the universal property of free Let a, b ∈ L Lie algebras there is a unique homomorphism of Lie algebras ˆ {Y,Z}

a,b : L{Y,Z} −→ L

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The Campbell-Hausdorff Formula

121

mapping Y to a and Z to b. By construction it satisfies  (n) (m) L{Y,Z}

a,b (L{Y,Z} ) ⊆ {0} × · · · × {0} × n≥m

for any m ≥ 1 and therefore extends, by the observation before Prop. 16.2, to the K-linear map ˆ {Y,Z} −→ L ˆ {Y,Z}

ˆa,b : L  (cn )n −→

a,b (cn ). n≥1

The same reasoning as for θˆ shows that ˆa,b in fact is a homomorphism of Lie algebras. On the other hand of course, a,b is the restriction of a corresponding unique homomorphism of associative K-algebras with unit  {Y,Z} .

a,b : As{Y,Z} −→ As Viewing an element in As{Y,Z} as a noncommutative polynomial G(Y, Z) it is clear that

a,b (G) = G(a, b) holds true. It follows that  

a,b (Hn ) = Hn (a, b) = H(a, b).

ˆa,b (H) = n≥1

n≥1

There is an analogous construction for the Lie algebra Pow(g × g; g). Quite generally, given any g1 , g2 ∈ Map(V × V ; V ) there is the homomorphism (of Lie algebras in case V = g) Map(V × V ; V ) −→ Map(V × V ; V ) f −→ f (g1 , g2 )(v, w) := f (g1 (v, w), g2 (v, w)). If g1 , g2 ∈ Pol(V × V ; V ) then Lemma 16.9 says that it restricts to Pol(V × V ; V ) −→ Pol(V × V ; V ) and satisfies f (g1 , g2 ) ∈ Polrn1 +sn2 (V × V ; V ) if f ∈ Polr,s (V × V ; V ) and gi ∈ Polni (V × V ; V ).

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III

Hence for g1 , g2 in Pow0 (g × g; g) := {0} ×



Lie Groups

Poln (g × g; g)

n≥1

we obtain, by the usual componentwise procedure, a homomorphism of Lie algebras Pow(g × g; g) −→ Pow(g × g; g) f −→ f (g1 , g2 ). Indeed, this is just a reformulation of the fact that a formal power series without constant term can be inserted into any formal power series. We note that pri (g1 , g2 ) = gi . ˆ ˆ lie ˆ {Y,Z} be any two elements. Then θ(a), As before let now a, b ∈ L θ(b)

in Pow0 (g × g; g), and we have the commutative diagram ˆa,b

ˆ {Y,Z} L (21)

ˆ ˆ Y,Z→θ(a), θ(b)

Y,Z→pr1 ,pr2 θˆ

Pow(g × g; g)

ˆ {Y,Z} L

ˆ ˆ f →f (θ(a), θ(b))

θˆ

Pow(g × g; g).

Lemma 16.13. Suppose that K has characteristic zero; for any y ∈ g and r + s ≥ 1 we have: ˜ r,s (y, −y) = 0; i. H  ˜ r,s (y, 0) = y ii. H 0  ˜ r,s (0, y) = y iii. H 0

if r = 1, s = 0, otherwise; if r = 0, s = 1, otherwise.

Proof. We apply the commutative diagram (21) to the Hausdorff series H ∈ ˆ {Y,Z} and various choices for the elements a and b. For a := Y and b := −Y L we have θ(a) = pr1 and θ(b) = − pr1 and we obtain from (21) that H

H(Y, −Y )

˜ H

˜ 1 , − pr1 ). H(pr

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The Campbell-Hausdorff Formula

123

Since H(Y, −Y ) = 0 by (20) the assertion i. follows. For a := Y and b := 0 we similarly obtain H

H(Y, 0)

˜ H

˜ 1 , 0). H(pr

˜ 1 , 0) = pr1 which is Again by (20) we have H(Y, 0) = Y and hence H(pr the assertion ii. The last assertion iii. comes symmetrically from the choice a := 0 and b := Y . The discussion leading to the commutative diagram (21) can easily be generalized to the three-element set {U, Y, Z} and the Lie algebra Pow(g × g × g; g) of d-tuples of formal power series over K in 3d variables. We leave the details to the reader. This leads to the homomorphism of Lie algebras ˆ {U,Y,Z} −→ Pow(g × g × g; g) θˆ : L which sends U, Y, and Z to pr1 , pr2 , and pr3 , respectively. For any choice ˆ {U,Y,Z} we obtain, analogously to (21), the commutative of elements a, b ∈ L diagram ˆ {Y,Z} L (22)

Y,Z→a,b

θˆ

Pow(g × g; g)

ˆ {U,Y,Z} L θˆ

ˆ ˆ f →f (θ(a), θ(b))

Pow(g × g × g; g).

Lemma 16.14. Suppose that K has characteristic zero; we then have ˜ 1 , H(pr ˜ 2 , pr3 )) = H( ˜ H(pr ˜ 1 , pr2 ), pr3 ). H(pr ˆ {Y,Z} and the two choices Proof. Apply (22) to the Hausdorff series H ∈ L a := U, b := H(Y, Z) and a := H(U, Y ), b := Z, respectively, and use the first part of (20).

124

17

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Lie Groups

The Convergence of the Hausdorff Series

We fix a Lie algebra g of finite dimension d over a field K of characteristic zero. We also pick a K-basis e1 , . . . , ed of g. k ∈ K, for 1 ≤ i, j, k ≤ d, defined by the Definition. The elements γij equations d  k [ei , ej ] = γij ek k=1

are called the structure constants of g with respect to the basis {ei }1≤i≤d . If we define the Lie product [ , ] on K d by      1 d (23) [(v1 , . . . , vd ), (w1 , . . . , wd )] := γij vi wj , . . . , γij vi wj i,j

i,j

then the isomorphism g ∼ = K d becomes an isomorphism of Lie algebras. Using this same isomorphism we also may view the element ˜ = H g ∈ Pow0 (g × g; g) H as a d-tuple H(Y , Z) := H g (Y , Z) = (H(1) (Y , Z), . . . , H(d) (Y , Z)) of formal power series H(i) (Y , Z) over K in the variables Y = (Y1 , . . . , Yd ) and Z = (Z1 , . . . , Zd ). The Examples 16.12 tell us that H(i) (Y , Z) = Yi + Zi +

1 i γjk Yj Zk + · · · . 2 j,k

Lemma 17.1.

i. H(Y , 0) = Y , H(0, Z) = Z.

ii. H(Y , −Y ) = 0. iii. H(U , H(Y , Z)) = H(H(U , Y ), Z). Proof. This is a restatement of Lemma 16.13 and Lemma 16.14. From now on let (K, | |) be a nonarchimedean field of characteristic zero. Via the linear isomorphism g ∼ = K d we may view g as a manifold over K (but which structure does not depend on the choice of the basis).

17

The Convergence of the Hausdorff Series

125

Let us suppose at this point that there is an ε > 0 such that (24)

H(Y , Z) ∈ Fε (K d × K d ; K d )

and Hε ≤ ε

(where K d is equipped with the usual maximum norm). We then consider the open submanifold Gε := Bε (0) ⊆ K d ∼ = g. Obviously Gε × Gε −→ Gε (g, h) −→ gh := H(g, h) is a well defined locally analytic map. Prop. 5.4 and Lemma 17.1 together imply that we have g1 0 = 0g1 = g1 , g1 (−g1 ) = 0, and g1 (g2 g3 ) = (g1 g2 )g3 for any g1 , g2 , g3 ∈ Gε . This proves the first half of the following fact provided (24) holds true. Proposition 17.2. Gε is a d-dimensional Lie group over K whose neutral element is the zero vector 0 and such that −g is the inverse of g ∈ Gε ; moreover, if g, h ∈ Gε satisfy [g, h] = 0 then gh = g + h. Proof. The second half of the assertion is immediate from the fact that, ˜ r,s for (r, s) = (1, 0), (0, 1) are iterated by Dynkin’s formula, all maps H commutators and therefore vanish on (g, h). If two ε ≥ ε > 0 satisfy (24) then Gε of course is an open subgroup Gε . Definition. {Gε }ε is called the Campbell-Hausdorff Lie group germ of the Lie algebra g. What is the Lie algebra of Gε (still assuming (24))? We have the “global” chart c := (Gε , ⊆, K d ) for the manifold Gε and correspondingly the locally analytic isomorphism 

τc : Gε × K d −−→ T (Gε ) (g, v) −→ [c, v] ∈ Tg (Gε )

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as well as the linear isomorphism ∼ =

C an (Gε , K d ) −−→ Γ(Gε , T (Gε )) f −→ ξf (g) = τc (g, f (g)) = [c, f (g)] ∈ Tg (Gε ). From Remark 9.21.i. we know that the Lie product of vector fields corresponds on the left hand side to the Lie product [f1 , f2 ](g) = Dg f1 (f2 (g)) − Dg f2 (f1 (g)). On the other hand the Lie product on Lie(Gε ) = T0 (Gε ) is induced via the inclusion T0 (Gε ) −→ Γ(Gε , T (Gε )) t −→ ξt (g) = T0 (rg )(t) by the Lie product of vector fields. By the construction of the tangent map T0 (rg ) in Sect. 9 we have the commutative diagram Kd

D0 rg

Kd

∼ =

v→[c,v]

T0 (Gε )

∼ = v→[c,v]

Tg (Gε ).

T0 (rg )

We define fv (g) := D0 (rg )(v) and compute ξ[c,v] (g) = T0 (rg )([c, v]) = [c, D0 rg (v)] = ξf v (g) which means that the diagram Kd v→[c,v]

v→fv

∼ = f →ξf

∼ =

T0 (Gε )

C an (Gε , K d )

t→ξt

Γ(Gε , T (Gε ))

is commutative. Let [ , ] denote the Lie product on K d which under the left perpendicular arrow corresponds to the Lie product of the Lie algebra Lie(Gε ) = T0 (Gε ). The commutativity of the diagram then says that we have f[v,w] = [fv , fw ] for any v, w ∈ K d .

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The Convergence of the Hausdorff Series

127

Observing that fv (0) = D0 idGε (v) = v we deduce that (25)

[v, w] = [fv , fw ](0) = D0 fv (fw (0)) − D0 fw (fv (0)) = D0 fv (w) − D0 fw (v)

for any v, w ∈ K d . We summarize that we have identified both Lie algebras, g and Lie(Gε ), with K d thereby obtaining the two Lie products [ , ] and [ , ] on K d . Proposition 17.3. Lie(Gε ) ∼ = g as Lie algebras. Proof. By the above discussion it suffice to show that [ , ] = [ , ] holds true. To further compute the Lie product [ , ] we start from the identity rg (h) = H(h, g). Since, by Lemma 17.1.i., H does not contain monomials of degree (0, s) in (Y , Z) with s ≥ 2 we may write H(i) (Y , Z) = Zi +

d 

P(i,j) (Z)Yj + terms of degree ≥ 2 in Y .

j=1

Using Prop. 5.6 we deduce that  

∂H(i) (Y , g) D0 rg = = P(i,j) (g) i,j  ∂Yj Y =0 i,j and hence that



fv (g) = D0 (rg )(v) =

d  j=1

vj P(1,j) (g), . . . ,

d 

 vj P(d,j) (g)

j=1

for any v = (v1 , . . . , vd ) ∈ K d . To derive the function fv in 0 we must derive the P(i,j) (Z) in Z and subsequently set Z = 0. By the Examples 16.12 we may write 1 i γjk Zk + terms of degree ≥ 2 in Z 2 d

P(i,j) (Z) = δij +

k=1

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Lie Groups

where δij denotes the Kronecker symbol. It follows that ∂P(i,j) (Z) 1 = γi  ∂Zk Z=0 2 jk 

and hence that D 0 fv =

1 i γjk vj 2 d

j=1

 i,k

and  D0 fv (w) =

d d 1  1 1  d γjk vj wk , . . . , γjk vj wj 2 2 j,k=1



j,k=1

1 = [v, w] 2 for any v = (v1 , . . . , vd ), w = (w1 , . . . , wd ) ∈ K d . We conclude that 1 1 [v, w] = D0 fv (w) − D0 fw (v) = [v, w] − [w, v] = [v, w] . 2 2

Having seen the interesting consequences of a possible convergence of the Hausdorff series we now must address the main question of this section whether any ε satisfying (24) exists. Using the isomorphism g ∼ = K d any element f ∈ Pol(g × g; g) can be viewed as a d-tuple f of polynomials in the variables Y and Z and hence, in particular, as an element f ∈ Fε (K d × K d ; K d ) for any ε > 0. Since the ˜ r,s are homogeneous of total degree r + s we have polynomials in Hr,s := H Hr,s ε = Hr,s 1 εr+s . Suppose that there is a 0 < ε0 ≤ 1 such that (26)

−(r+s−1)

Hr,s 1 ≤ ε0

for any r + s ≥ 1.

It follows that for any 0 < ε < ε0 we have Hr,s ε = Hr,s 1 εr+s ≤ Hr,s 1 εr+s−1 ε≤ε 0 and

for any r + s ≥ 1

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The Convergence of the Hausdorff Series

129

lim Hr,s ε ≤ ε · lim Hr,s 1 εr+s−1

r+s→∞

r+s→∞

=ε·



lim Hr,s 1 εr+s−1 0 r+s→∞ 

≤ ε · lim

r+s→∞

ε ε0

r+s−1

ε ε0

r+s−1

= 0. As H=



Hr,s

r+s≥1

we conclude that (26) implies (24) for any 0 < ε < ε0 . The coefficients of the Hausdorff series H are explicitly known and their absolute values therefore can easily be estimated. But in order to translate this knowledge into an estimate for the norms Hr,s 1 we need a particularly well behaved basis of the K-vector space L{Y,Z} .  The free K-algebra A{Y,Z} by construction has the K-basis M{Y,Z} = n≥1 {Y, Z}(n). For any x ∈ M{Y,Z} we let ex denote its image in the factor algebra L{Y,Z} . These ex obviously generate L{Y,Z} as a K-vector space. Hence there exist subsets B ⊆ M{Y,Z} such that {ex }x∈B is a K-basis of L{Y,Z} . In the following we have to make a particularly clever choice of such a subset B. But first we note that also the free associative K-algebra with unit As{Y,Z} has an obvious K-basis which is the set Mon{Y,Z} of all noncommutative monomials in Y and Z. All of this is valid over an arbitrary field K. Since our K is nonarchimedean we may introduce the oK -submodules  As≤1 := oK μ {Y,Z} μ∈Mon{Y,Z}

of As{Y,Z} and

≤1 L≤1 {Y,Z} := L{Y,Z} ∩ As{Y,Z}

of L{Y,Z} . Proposition 17.4. that we have

i. (K arbitrary) There is a subset B ⊆ M{Y,Z} such

– {ex }x∈B is a K-basis of L{Y,Z} , – {Y, Z} ⊆ B, and – for any x ∈ B \ {Y, Z} there are x , x ∈ B with x = x x and, in particular, ex = [ex , ex ].

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ii. (K nonarchimedean) There is a subset B ⊆ M{Y,Z} as in i. and such that  L≤1 oK ex . {Y,Z} = x∈B

Proof. Apply [B-LL] Chap. II §2.10 Prop. 11, §2.11 Thm. 1, and §3.1 Remark 1) over K and oK , respectively. We now define the constant ε0 by  1 if K is p-adic for some p, |p| p−1 ε−1 1 ε0 := −1 otherwise ε1 where k |). ε1 := max(1, max |γij i,j,k

We note that 0 < ε0 ≤ 1. The constant ε1 has the property that [f, g]1 ≤ ε1 f 1 g1

for any f, g ∈ Pol(g × g; g).

Lemma 17.5. Let {ex }x∈B be any K-basis of L{Y,Z} as in Prop. 17.4.i.; we then have θ(ex )1 ≤ εn−1 1

for any x ∈ B(n) := B ∩ {Y, Z}(n).

Proof. We proceed by induction with respect to n. For x = Y we have θ(eY ) = pr1 and hence θ(eY ) = (Y1 , . . . , Yd ) so that θ(eY )1 = 1 = ε01 . The case x = Z is analogous. Any x ∈ B(n) with n ≥ 2 can be written as x = x x

with x ∈ B(l), x ∈ B(m), and l + m = n.

Since l, m < n we may apply the induction hypothesis to x and x and obtain θ(ex )1 = θ([ex , ex ])1 = [θ(ex ), θ(ex )]1 ≤ ε1 θ(ex )1 θ(ex )1 m−1 ≤ ε1 εl−1 = εn−1 . 1 ε1 1

Proposition 17.6. For any 0 < ε < ε0 we have H ∈ Fε (K d × K d ; K d ) and Hε ≤ ε.

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The Convergence of the Hausdorff Series

131

Proof. As discussed after (26) it suffices to show that −(r+s−1)

Hr,s 1 ≤ ε0

for any r + s ≥ 1.

We fix n := r + s ≥ 1. We also pick a basis {ex }x∈B as in Prop. 17.4.i. and write  Hr,s = cx ex . x∈B (n)

Since Hr,s ∈ L{Y,Z} we in fact have Hr,s =



where B(n) = B ∩ {Y, Z}(n).

cx ex

x∈B(n)

Lemma 17.5 then implies that Hr,s 1 ≤ max |cx |θ(ex )1 ≤ εn−1 max |cx |. 1 x∈B(n)

x∈B(n)

In order to estimate the |cx | we have to distinguish cases. But we emphasize that this is a question solely about the Hausdorff series (and not the Lie algebra g) and therefore, in principle, can be treated over the field Q. Case 1: K is not p-adic for any p. Since Q ⊆ K we may choose the basis already over the field Q. Then all coefficients cx lie in Q. By Exercise 2.1.i. −(n−1) we have |cx | = 0 or 1 and hence Hr,s 1 ≤ εn−1 = ε0 . 1 Case 2: K is p-adic for some p. By Exercise 2.1.ii. we have Qp ⊆ K and −

|a| = |a|p

log |p| log p

for any a ∈ Qp .

Hence we may assume without loss of generality that (K, | |) = (Qp , | |p ), and we choose B as in Prop. 17.4.ii. We want to show that n−1

max |cx |p ≤ p p−1 .

x∈B(n)

Since the left hand side is an integral power of p this amounts to showing that for any x ∈ B(n) pl cx ∈ Zp := oQp where l is the unique integer such that l≤

n−1 < l + 1. p−1

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By our particular choice of the set B this is equivalent to pl Hr,s ∈ L≤1 {Y,Z}

and hence to

pl Hr,s ∈ As≤1 {Y,Z} .

The explicit form of the coefficients of Hr,s then reduces us to showing that  m    m     pl  1     ri !si ! ≥ p−l   ≤ 1, or equivalently, m   m ri !si !  i=1

i=1

p

p

whenever 1 ≤ m ≤ n, r1 + · · · + rm = r, s1 + · · · + sm = s, and ri + si ≥ 1. But Lemma 2.2 implies   m    1   ri !si ! ≥ p− p−1 ((m−1)+(r1 +s1 −1)+···+(rm +sm −1)) m   i=1

p

= p− p−1 . n−1

Since the left hand side is an integral power of p it indeed must be ≥ p−l . (For an alternative argument compare the proof of the later Lemma 32.4.)

18

Formal Group Laws

Let K be any field of characteristic zero. We fix a natural number d, and let R := K[[Y1 , . . . , Yd , Z1 , . . . , Zd ]] denote the ring of formal power series over K in the variables Y = (Y1 , . . . , Yd ) and Z = (Z1 , . . . , Zd ). Definition. A formal group law (of dimension d over K) is a d-tuple F = (F1 , . . . , Fd ) of power series Fi ∈ R such that we have: (i) F (Y , 0) = Y and F (0, Z) = Z, (ii) F (U , F (Y , Z)) = F (F (U , Y ), Z). We observe that the condition (i) implies that (27)

Fi (Y , Z) = Yi + Zi + terms of degree ≥ 1 both in Y and Z.

Hence the two sides in the condition (ii) are well defined. Examples.

1) Fi (Y , Z) = Yi + Zi .

2) F (Y, Z) = Y + Z + Y Z (for d = 1).

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Formal Group Laws

133

3) (Lemma 17.1) F := H g for a finite dimensional Lie algebra g over K (and some choice of K-basis of g). The last example has a converse. Let F be any formal group law. We have  Fi (Y , Z) = Yi + Zi + cijk Yj Zk + terms of degree ≥ 3. j,k

We define a bilinear map bF : K d × K d −→ K d by     bF ((v1 , . . . , vd ), (w1 , . . . , wd )) := c1jk vj wk , . . . , cdjk vj wk , j,k

j,k

and we put [v, w]F := bF (v, w) − bF (w, v)

for v, w ∈ K d .

Lemma 18.1. [ , ]F satisfies the Jacobi identity. Proof. Compare [Haz] §14.1 or [Se2] Part II, Chap. IV §7 formula 6). We see that [ , ]F is a Lie product on K d . In the case of the formal group law H g it follows from the Examples 16.12 that [ , ]H g coincides (up to the isomorphism g ∼ = K d ) with the Lie product on g. Next we discuss a seemingly very different construction of a formal group law from a finite dimensional Lie algebra g over K by using the universal enveloping algebra U (g). We have the following list of K-linear maps: • (multiplication) m = mg : U (g) ⊗K U (g) −→ U (g), • (unit) e = eg : K −→ U (g) sending a to a · 1, U (Δ) • (comultiplication) Δ = Δg : U (g) −−−−→ U (g × g) ∼ = U (g) ⊗K U (g), pr

• (counit) c = cg : U (g) = T (g)/J(g) −−→ g⊗0 = K. Of course, the maps m and e satisfy the axioms for a (noncommutative) associative K-algebra with unit, and Δ and c are homomorphisms of K-algebras with unit. In addition, the maps Δ and c have the following properties: (28) (29)

(counit property) (coassociativity)

(c ⊗ id) ◦ Δ = id = (id ⊗ c) ◦ Δ; (id ⊗ Δ) ◦ Δ = (Δ ⊗ id) ◦ Δ;

134

(30)

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(cocommutativity)

Lie Groups

the diagram U (g)

Δ

U (g) ⊗K U (g)

Δ

x⊗y→y⊗x

U (g) ⊗K U (g)

is commutative. They easily follow, by applying the universal property of U (g), from the corresponding properties of the diagonal map Δ : g −→ g × g. We now consider the K-linear dual U (g)∗ := HomK (U (g), K) together with the K-linear map Δ∗

can

[U (g) ⊗K U (g)]∗ −−−→ U (g)∗ . μ : U (g)∗ ⊗K U (g)∗ −−−→ −→ [x ⊗ y → l1 (x) · l2 (y)] l 1 ⊗ l2 Proposition 18.2. (U (g)∗ , μ, c) is a commutative and associative K-algebra with unit. Proof. The required axioms are exactly dual to the properties (28)–(30). In order to determine the algebra U (g)∗ explicitly we pick an (ordered) K-basis e1 , . . . , ed of g. We know from Cor. 14.3 of the Poincar´e-BirkhoffWitt theorem that the eα :=

eαd eα1 1 · ... · d α1 ! αd !

for α = (α1 , . . . , αd ) ∈ Nd0

form a K-basis of U (g). Proposition 18.3. The map ∼ =

U (g)∗ −−→ K[[U1 , . . . , Ud ]]  (eα )U α  −→ F (U ) := α∈Nd0

is an isomorphism of K-algebras with unit onto the ring of formal power series over K in the variables U = {U1 , . . . , Ud }.

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Proof. The fact that {eα }α is a K-basis of U (g) immediately implies that the asserted map is a K-linear isomorphism. The unit element c of U (g)∗ is the projection map onto Ke0 ∼ = K which is mapped to Fc = 1. For the multiplicativity we first recall that m m Δ(em k ) = Δ(ek ) = (ek ⊗ 1 + 1 ⊗ ek ) m    m (eik ⊗ 1)(1 ⊗ em−i = ) k i i=0 m    m i e ⊗ em−i = k i k i=0

for any 1 ≤ k ≤ d and any m ≥ 0. By induction one deduces that  eβ ⊗ eγ for any α ∈ Nd0 (31) Δ(eα ) = β+γ=α

holds true. We now compute   μ(1 , 2 )(eα )U α = (1 ⊗ 2 )(Δ(eα ))U α Fμ( 1 , 2 ) (U ) = α

=

 

α

1 (eβ )2 (eγ )U β+γ

α β+γ=α

 =





1 (eβ )U β



 2 (eγ )U γ

γ

β

= F 1 (U )F 2 (U ).

By dualizing the multiplication map m U (g × g) ∼ = U (g) ⊗K U (g) −−→ U (g)

we obtain a K-linear map m∗

U (g)∗ −−−→ U (g × g)∗ . Applying Prop. 18.3 to both sides (with (e1 , 0), . . . , (ed , 0), (0, e1 ), . . . , (0, ed ) as an ordered K-basis for g × g) we may view the latter as a K-linear map m∗

K[[U1 , . . . , Ud ]] −−−→ K[[Y1 , . . . , Yd , Z1 , . . . , Zd ]] = R.

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We define F(i) := m∗ (Ui ) ∈ R

and F g := (F(1) , . . . , F(d) ).

At this point we have to recall a few basic facts about formal power series rings. First of all, the formal power series ring K[[U1 , . . . , Ur ]] has a unique maximal ideal mU which is the ideal generated by U1 , . . . , Ur . This is an immediate consequence of the fact that any formal power series F over K with F (0) = 0 is invertible. Definition. i. A commutative ring with unit is called local if it has a unique maximal ideal. ii. A homomorphism of local rings is called local if it maps the maximal ideal into the maximal ideal. Let K[[U1 , . . . , Ur ]] and K[[V1 , . . . , Vs ]] be two formal power series rings. For any F = (F1 , . . . , Fr ) ∈ mV × · · · × mV the map εF : K[[U1 , . . . , Ur ]] −→ K[[V1 , . . . , Vs ]] G −→ G(F ) := G(F1 , . . . , Fr ) is a well defined local homomorphism of local rings. We have εF (Ui ) = Fi . Lemma 18.4. Let ε : K[[U1 , . . . , Ur ]] −→ K[[V1 , . . . , Vs ]] be any homomorphism of K-algebras with unit which is local ; we then have ε = εF

with

Fi := ε(Ui ).

Proof. Since ε is local we have Fi ∈ mV so that εF is well defined. Both ε and εF are homomorphisms of K-algebras with unit. Hence the identities ε(Ui ) = εF (Ui ) imply that ε(G) = εF (G)

for any polynomial G ∈ K[U1 , . . . , Ur ].

We now write an arbitrary formal power series G ∈ K[[U1 , . . . , Ur ]] as  G= Gn n≥0

where Gn is a homogeneous polynomial of degree n. In particular, Gn lies in mnU . Since ε is local we obtain ε(Gn ) ∈ mnV . Therefore the element  n≥0

ε(Gn ) ∈ K[[V1 , . . . , Vs ]]

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137

is well defined. We have ε(G) −





ε(Gn ) = ε(G0 ) + · · · + ε(Gk ) + ε

n≥0

 =ε



k+1 k≥0 mV

 Gn

−

 −

Gn





ε(Gn )

n≥0

n>k

n>k k+1 mV

∈ for any k ≥ 0. Since





ε(Gn )

n>k

= {0} we conclude that

ε(G) =



ε(Gn ).

n≥0

The same reasoning, of course, applies to εF . Hence   ε(G) = ε(Gn ) = εF (Gn ) = εF (G). n≥0

Proposition 18.5.

n≥0

i. m∗ is a local homomorphism.

ii. m∗ = εF g . iii. F g is a formal group law. iv. [ , ]F g coincides (up to the isomorphism g ∼ = K d given by the basis e1 , . . . , ed ) with the Lie product on g. Proof. i. The multiplicativity of m∗ amounts to the commutativity of the outer square in the diagram U (g)∗ ⊗K U (g)∗

m∗ ⊗m∗

can

[U (g) ⊗K U (g)]∗

can (m⊗m)∗

[U (g × g) ⊗K U (g × g)]∗ Δ∗g×g

Δ∗g

U (g)∗

U (g × g)∗ ⊗K U (g × g)∗

m∗

U (g × g)∗ .

The upper square is commutative for trivial reasons. The lower square is the dual of the diagram

138

III Δg×g

U (g × g)

Lie Groups

U (g × g) ⊗K U (g × g) ∼ =⊗∼ =

∼ =

U (g) ⊗K U (g)

[U (g) ⊗K U (g)] ⊗K [U (g) ⊗K U (g)]

m

m⊗m Δg

U (g)

U (g) ⊗K U (g).

We check its commutativity on the K-basis eα,β :=

(ed , 0)αd (0, e1 )β1 (0, ed )βd (e1 , 0)α1 · ... · · · ... · , α1 ! αd ! β1 ! βd !

for α = (α1 , . . . , αd ), β = (β1 , . . . , βd ) ∈ Nd0 , of U (g × g). Under the algebra isomorphism U (g × g) ∼ = U (g) ⊗K U (g) an element (x, y) ∈ g × g corresponds to x ⊗ 1 + 1 ⊗ y. It follows that eα,β corresponds to eα ⊗ eβ . Using the formula (31) we obtain 

Δg×g

eα,β

eγ,λ ⊗ eδ,μ

(γ,λ)+(δ,μ)=(α,β)



eα ⊗ eβ

(eγ ⊗ eλ ) ⊗ (eδ ⊗ eμ )

γ+δ=α λ+μ=β m

eα eβ

m⊗m Δg

 γ+δ=α

eγ ⊗ eδ

 λ+μ=β

eλ ⊗ eμ =



eγ eλ ⊗ eδ eμ .

γ+δ=α λ+μ=β

This establishes the multiplicativity of m∗ . The unit element in U (g)∗ , resp. in U (g×g)∗ , is the linear form cg , resp. cg×g , which is the projection map onto Ke0 ∼ = K, resp. onto Ke0,0 ∼ = K. Since cg is a homomorphism of algebras we have  1 if (α, β) = (0, 0), ∗ m (cg )(eα,β ) = cg (eα eβ ) = cg (eα )cg (eβ ) = 0 otherwise. It follows that m∗ (cg ) = cg×g . The dual of the map e : K −→ U (g) is the linear form  −→ (e0 ) on U (g)∗ , resp. the linear form F −→ F (0) on K[[U1 , . . . , Ud ]]. We therefore

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see that under the isomorphism K[[U1 , . . . , Ud ]] ∼ = U (g)∗ in Prop. 18.3 the maximal ideal mU is the orthogonal complement of the subspace im(e) = K · 1 ⊆ U (g). But the multiplication map m respects unit elements, i. e., m◦eg×g = eg . Hence m∗ respects the corresponding orthogonal complements which means that m∗ is local. ii. This is immediate from i. and Lemma 18.4. iii. The associativity of the multiplication in U (g) can be expressed as the identity m ◦ (id ⊗ m) = m ◦ (m ⊗ id). Dually we obtain (id ⊗ m∗ ) ◦ m∗ = (m∗ ⊗ id) ◦ m∗ . By ii. the evaluation on U gives the identity F g (U , F g (Y , Z)) = F g (F g (U , Y ), Z). The commutative diagram U (g)

U (g) ⊗K U (g) ∼ = U (g × g)

e⊗id =

m

U (g) gives rise on the dual side to the commutative diagram K[[Y1 , . . . , Yd , Z1 , . . . , Zd ]]

Yi →0,Zi →Ui

K[[U1 , . . . , Ud ]]

=

m∗

K[[U1 , . . . , Ud ]]. This means that F g (0, Z) = Z. Analogously we obtain F g (Y , 0) = Y . iv. We write  cijk Yj Zk + terms of degree ≥ 3. m∗ (Ui ) = F(i) (Y , Z) = Yi + Zi + j,k

The linear form i :

 α∈Nd0

U (g) −→ K cα eα −→ ci

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(where as before i = (. . . , 0, 1, 0, . . .) with 1 in the i-th place) satisfies F i = Ui . Hence Fm∗ ( i ) = m∗ F i = m∗ (Ui ) = F(i) and therefore cijk = m∗ (i )(ej,0 e0,k ) = i (ej ek ). It follows that [ej , ek ] =

=

d 

i ([ej , ek ])ei =

i=1 d 

d 

i (ej ek − ek ej )ei

i=1

(cijk − cikj )ei .

i=1

Hence the structure constants cijk − cikj of the Lie product [ , ]F g coincide with the structure constants of g. Example. For commutative g we have F g = (Y1 + Z1 , . . . , Yd + Zd ). As a last construction we will associate a formal group law with any Lie group. Let K be a nonarchimedean field of characteristic zero, and let G be a d-dimensional Lie group over K with Lie algebra g := Lie(G). We pick a chart c = (U, ϕ, K d ) for G around the unit element e ∈ G such that ϕ(e) = 0. (The latter, of course, can always be achieved by translating ϕ by ϕ(e).) Since the multiplication mG is continuous we find an open neighbourhood V ⊆ U of e such that V · V ⊆ U . Then also (V, ϕ|V, K d ) is a chart of G around e, by Remark 8.1, and the map ϕ◦mG ◦(ϕ−1 ×ϕ−1 )

ϕ(V ) × ϕ(V ) −−−−−−−−−−−−→ ϕ(U ) is locally analytic. Hence there exists, for sufficiently small ε > 0, a d-tuple of power series F G,c ∈ Fε (K d × K d ; K d ) such that F G,c (v, w) = ϕ(ϕ−1 (v)ϕ−1 (w))

for any v, w ∈ Bε (0).

By the identity theorem Cor. 5.8 the d-tuple F G,c (Y , Z) of formal power series does not depend on the choice of ε. Proposition 18.6.

i. F G,c is a formal group law. ∼ =

ii. [ , ]F G,c coincides, modulo the isomorphism θc−1 : g = Te (G) −−→ K d , with the Lie product on g.

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141

Proof. i. Because of ϕ(e) = 0 we have F G,c (v, 0) = v

and F G,c (0, w) = w

for any v, w ∈ Bε (0).

Again using the identity theorem Cor. 5.8 this translates into the identities of formal power series F G,c (Y , 0) = Y

and F G,c (0, Z) = Z.

In particular, the formation of F G,c (U , F G,c (Y , Z))

and F G,c (F G,c (U , Y ), Z)

is well defined. For sufficiently small δ > 0 these formations commute, by Prop. 5.4, with the evaluation in any points u, v, w ∈ Bδ (0). But by the associativity of the multiplication in G we have F G,c (u, F G,c (v, w)) = F G,c (F G,c (u, v), w)

for any u, v, w ∈ Bδ (0).

By a third application of Cor. 5.8 this translates into the identity of formal power series F G,c (U , F G,c (Y , Z)) = F G,c (F G,c (U , Y ), Z). ii. This is exactly the same computation as in the proof of Prop. 17.3. Lemma 18.7. The Lie group G has a family {H|a| }|a| of open subgroups indexed by the sufficiently big |a| ∈ |K| which forms a fundamental system of open neighbourhoods of e ∈ G and such that each H|a| is isomorphic, via ϕ, to (B 1 (0), F G,c ). |a|

Proof. With ε > 0 as above we put ε0 := F G,c ε , and we choose any 1 |a| ≥ max( 1ε , εε02 ) (so that, in particular, |a| ≤ ε). We claim that F G,c 

1 |a|

≤

1 |a|

holds true. Let F G,c (Y , Z) = Y + Z +

 |α|,|β|≥1

vα,β Y α Z β

with vα,β ∈ K d .

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We have vα,β  ≤ ε0 ε−|α|−|β| and hence   1 1 F G,c  1 = max , max vα,β  |α|+|β| |a| |a| |α|,|β|≥1 |a|     1 1 |α|+|β| ≤ max , max ε0 |a| |α|,|β|≥1 |a|ε   2  1 1 ≤ max , ε0 |a| |a|ε 1 ≤ . |a| By possibly enlarging the lower bound for |a| we can make exactly the same argument for the power series expansion of the map g −→ g −1 on G in a sufficiently small neighbourhood of ϕ(e) = 0 (cf. Prop. 13.6). The family H|a| := ϕ−1 (B 1 (0)) then has the required properties. |a|

Corollary 18.8. Every Lie group is paracompact. Proof. By Lemma 18.7 we find an open subgroup H ⊆ G which as a manifold is isomorphic to a ball Bε (0). Any coset gH, for g ∈ G, then is isomorphic, as a manifold, to Bε (0) as well. By Lemma 1.4 the ultrametric space Bε (0) and therefore any coset gH is strictly paracompact. As a disjoint union of cosets gH the Lie group G also is strictly paracompact. Remark 18.9. If Gε is the Campbell-Hausdorff Lie group germ of a Lie algebra g then we have H g = F Gε ,c for the chart c := (Gε , ⊆, K d ). In the present situation of a Lie group G and with the choice of the K-basis of g which corresponds to the standard basis of K d under the iso∼ = morphism θc−1 : g −−→ K d we now have the three formal group laws H g , F g , and F G,c whose Lie products [ , ]H g = [ , ]F g = [ , ]F G,c coincide and coincide with the Lie product on g (Example 16.12, Lemma 17.1, Prop. 18.5, Prop. 18.6). In order to compare formal group laws we need the following concept.

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Definition. Let F and F  be formal group laws over K of dimension d and d , respectively. A formal homomorphism Φ : F −→ F  is a d -tuple Φ = (Φ1 , . . . , Φd ) of formal power series Φi ∈ K[[U1 , . . . , Ud ]] such that Φi (0) = 0 and Φ(F (Y , Z)) = F  (Φ(Y ), Φ(Z)). The formal group laws F and F  are called isomorphic if d = d and if there are formal homomorphisms Φ : F −→ F  and Φ : F  −→ F such that Φ(Φ (U )) = U = Φ (Φ(U )). We write Hom(F , F  ) for the set of all formal homomorphisms Φ : F −→ F , and we consider the linear map 



Hom(F , F  ) −→ HomK (K d , K d )   ∂Φi (U ) . Φ −→ σΦ :=  ∂Uj U =0 i,j The chain rule implies that σΦ ◦Φ = σΦ ◦ σΦ Φ

Φ

holds true for any two formal homomorphisms F −−→ F  −−→ F  . In a first step we study the particular situation where F is a formal group law over K of dimension d and where F g is the formal group law which we have constructed above for the Lie algebra g := (K d , [ , ]F ) (and its (ordered) standard basis e1 , . . . , ed ). We want to show that these two formal group laws F and F g are isomorphic. This will be achieved by comparing the universal enveloping algebra U (g) to an algebra U (F ) which is constructed directly from F . Given a local K-algebra R with maximal ideal m we let R∗ := HomK (R, K) denote its K-linear dual and we introduce the K-vector subspace Rd := { ∈ R∗ : |mn = 0 for some n ≥ 0}. Any local homomorphism of K-algebras α : R0 −→ R1 induces a K-linear map αd : R1d −→ R0d  −→  ◦ α. We apply this to the local homomorphism εF : K[[U1 , . . . , Ud ]] −→ K[[Y1 , . . . , Yd , Z1 , . . . , Zd ]] = R

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and obtain the K-linear map εdF : Rd −→ U (F ) := K[[U1 , . . . , Ud ]]d . Exercise. The inclusion of K-algebras K[[U1 , . . . , Ud ]] ⊗K K[[U1 , . . . , Ud ]] −→ R Ui ⊗ 1 −→ Yi 1 ⊗ Ui −→ Zi ∼ =

induces a K-linear isomorphism Rd − → U (F ) ⊗K U (F ). We therefore may introduce the composite map εd

F m : U (F ) ⊗K U (F ) ∼ = Rd −−→ U (F ).

Moreover, we let e ∈ U (F ) denote the linear form which sends a formal power series to its constant coefficient. Proposition 18.10. (U (F ), m, e) is an associative K-algebra with unit. Proof. See [Haz] (36.1.5) or [Se2] Part II, Chap. V §6 Lemma 1. Let ψ : g −→ U (F ) denote the K-linear map which maps the standard basis vector ei to the linear form sending a formal power series to the coefficient of the monomial Ui . Lemma 18.11. ψ([x, y]F ) = ψ(x)ψ(y) − ψ(y)ψ(x) for any x, y ∈ g. Proof. See [Haz] Lemma 36.2.3 or [Se2] Part II, Chap. V §6 Thm. 1. By the universal property of the universal enveloping algebra the map ψ therefore extends uniquely to a homomorphism of K-algebras with unit ψ : U (g) −→ U (F ). ∼ =

Theorem 18.12. The map ψ : U (g) −−→ U (F ) is an isomorphism. Proof. See [Haz] Thm. 37.4.7 or [Se2] Part II, Chap. V §6 Thm. 2. We deduce from this theorem the commutative diagram of local homomorphisms K[[U ]] εF

R = Rd∗

U (F )∗

∼ = ψ∗

m∗

(U (F ) ⊗K U (F ))∗

U (g)∗

∼ =

K[[U ]]

m∗ (ψ⊗ψ)∗

(U (g) ⊗K U (g))∗

εF g ∼ =

R.

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Proposition 18.13. If g := (K d , [ , ]F ) then there is an isomorphism of ∼ =

formal group laws Φ : F g −−→ F such that σΦ = idK d . Proof. According to Lemma 18.4 the composed isomorphism in the upper row of the above diagram is of the form εΦ for some d-tuple Φ = (Φ1 , . . . , Φd ) of formal power series Φi ∈ K[[U1 , . . . , Ud ]] with Φi (0) = 0. The commutativity of the diagram then amounts to the equation Φ(F g (Y , Z)) = F (Φ(Y ), Φ(Z)). This says that Φ : F g −→ F is an isomorphism of formal group laws. We compute  1 if i = j, ∂Φi (U ) ∗ = ψ (Ui )(ej ) = ψ(ej )(Ui ) =  ∂Uj U =0 0 if i = j. Hence σΦ = idK d . Theorem 18.14. For any two formal group laws F and F  over K of dimension d and d , respectively, the map ∼ =



Hom(F , F  ) −−→ HomK ((K d , [ , ]F ), (K d , [ , ]F  )) Φ −→ σΦ is well defined and bijective; in particular, the formal group laws F and F  are isomorphic if and only if the corresponding Lie products [ , ]F and [ , ]F  are isomorphic. Proof. Well defined : Let d 

Fi (Y , Z) = Yi + Zi +

cijk Yj Zk + terms of degree ≥ 3

j,k=1 

Fi (Y

d 

, Z) = Yi + Zi +

dijk Yj Zk + terms of degree ≥ 3

j,k=1

Φi (U ) =

d 

=1

ai U +

d 

bimn Um Un + terms of degree ≥ 3.

m,n=1

The terms of degree ≤ 2 in Φi (F (Y , Z)) are

146

III



 ai

Y + Z +



=

ai (Y +Z )+





bimn (Ym + Zm )(Yn + Zn )

m,n

bimn (Ym Yn +Zm Zn )+



m,n

whereas in

Fi (Φ(Y



 bijk +bikj +

j,k



 ai c jk

Y j Zk

), Φ(Z)) they are



ai (Y +Z )+



bimn (Ym Yn +Zm Zn )+

m,n

=

+

j,k





 c jk Yj Zk

Lie Groups

ai (Y + Z ) +

 dimn

m,n



bimn (Ym Yn + Zm Zn ) +



m,n

j,k





 am j Yj

j



 ank Zk

k



 n dimn am j ak Yj Zk .

m,n

The equation Φi (F (Y , Z)) = Fi (Φ(Y ), Φ(Z)) therefore implies   n bijk + bikj + ai c jk = dimn am j ak . m,n

We finally compute



[σΦ (ej ), σΦ (ek )]F  =



am j em ,



m

=

 m,n

=

 i

=

n am j ak



ank en

= F

n

  (dimn − dinm )ei = i





i







n am j ak [em , en ]F 

m,n



(dimn

−

n dinm )am j ak

ei

m,n

    ai (c jk − c kj ) ei = (c jk − c kj ) ai ei 

 (c jk − c kj )σΦ (e ) = σΦ







i

(c jk − c kj )e

= σΦ ([ej , ek ]F ) which shows that σΦ is a homomorphism of Lie algebras. Injectivity: Differentiating the equation Φ(F (Y , Z)) = F  (Φ(Y ), Φ(Z)) gives the matrix equation     ∂Fi (Y , Z) ∂Φi (U ) ·  ∂Uj U =F (Y ,Z) i,j ∂Yj i,j      ∂Fi (Y , Z) ∂Φi (U ) = · .   ∂Yj ∂Uj U =Y i,j (Y ,Z)=(Φ(Y ),Φ(Z)) i,j

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Setting Y = 0 and remembering that F (0, Z) = Z and Φ(0) = 0 we obtain 

∂Φi (U )  ∂Uj U =Z



 · i,j

∂Fi (Y , Z)  ∂Yj Y =0

 i,j



=

∂Fi (Y , Z)  ∂Yj (Y ,Z)=(0,Φ(Z))

 · σΦ . i,j

Assuming that σΦ = 0 we then have     ∂Fi (Y , Z) ∂Φi (U ) · = 0.   ∂Uj U =Z i,j ∂Yj Y =0 i,j (Y ,Z)  Equation (27) shows that ( ∂Fi∂Y  j

the determinant of the matrix

)i,j is the identity matrix. Hence Y =Z=0 (Y ,Z)  ( ∂Fi∂Y Y =0 )i,j is a formal power series with j

nonzero constant term, i. e., is a unit the ring K[[Z1 , . . . , Zd ]]. It follows that (Y ,Z)  the matrix ( ∂Fi∂Y )i,j is invertible and consequently that  j Y =0



∂Φi (U ) ∂Uj

 = 0. i,j

Since Φ(0) = 0 (and K has characteristic zero) this implies that Φ = 0.  Surjectivity: We abbreviate g := (K d , [ , ]F ) and g := (K d , [ , ]F  ). As a consequence of Prop. 18.13 we have a commutative diagram of the form Hom(F , F  )

∼ =

Φ→σΦ



Hom(F g , F g ) Φ→σΦ

Hom(g, g ). It therefore suffices to establish the surjectivity of the right oblique arrow. Let σ : g −→ g be any homomorphism of Lie algebras. It extends uniquely to a homomorphism of K-algebras U (σ) : U (g) −→ U (g ) such that (U (σ) ⊗ U (σ)) ◦ Δg = Δg ◦ U (σ), cg ◦ U (σ) = cg , and eg = U (σ) ◦ eg . Hence U (σ) dualizes to a local homomorphism U (σ)∗ : U (g )∗ ∼ = K[[V1 , . . . , Vd ]] −→ U (g)∗ ∼ = K[[U1 , . . . , Ud ]].

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As such it is, by Lemma 18.4, of the form U (σ)∗ = εΦ for a unique d -tuple Φ of formal power series in K[[U1 , . . . , Ud ]]. The identity U (σ × σ)∗ ◦ m∗g = m∗g ◦ U (σ)∗ 

implies that Φ : F g −→ F g , in fact, is a formal homomorphism. If i ∈ U (g )∗ are the linear forms corresponding to the variables Vi then the restrictions 1 |g , . . . , d |g form the K-basis dual to the standard basis e1 , . . . , ed of g . The formal power series Φi (U ) then corresponds to the linear form U (σ)∗ (i ) = i ◦ U (σ) ∈ U (g)∗ . It follows that  i ◦ U (σ)(eα )U α Φi (U ) = α∈Nd0

and hence that

∂Φi (U ) = i (σ(ej )).  ∂Uj U =0

We conclude that σΦ is the matrix of σ with respect to the standard basis. Corollary 18.15. The three formal group laws H g , F g , and F G,c are mutually isomorphic. Proposition 18.16. Let G1 and G2 be two Lie groups over K and let ci = (Ui , ϕi , K di ), for i = 1, 2, be a chart for Gi around the unit element ei ∈ Gi such that ϕi (ei ) = 0; for any formal homomorphism Φ : F G1 ,c1 −→ F G2 ,c2 there is an ε > 0 such that Φ ∈ Fε (K d1 ; K d2 ). Proof. In a first step we consider the special case that G1 = (K, +) is the additive group of the field K and the chart is c1 = (K, id, K). The F G1 ,c1 = Y + Z. We abbreviate d := d2 and F := F G2 ,c2 . The formal homomorphism Φ is a d-tuple of formal power series in one variable U which satisfies Φ(0) = 0

and

Φ(Y + Z) = F (Φ(Y ), Φ(Z)).

Deriving the last identity with respect to Z and then setting Z equal to zero leads to ∂F (Φ(U ), 0) · Φ (0). Φ (U ) = ∂Z

18

Formal Group Laws

149

We define G(Y ) :=

∂F (Y , 0) · Φ (0) ∂Z

and obtain the system of differential equations Φ (U ) = G(Φ(U ))

with

Φ(0) = 0.

We write 

G(Y ) =

Y α (Mα · Φ (0))

with Mα ∈ Md×d (K)

α∈Nd0

and Φ(U ) =

 n≥1

Un

wn n!

with wn = (wn,1 , . . . , wn,d ) ∈ K d .

Our system of differential equations now reads α1 αd       w w w m,1 m,d n+1 Un Um ·. . .· Um (Mα ·Φ (0)). = n! m! m! d

n≥0

α∈N0

m≥1

m≥1

By comparing coefficients we obtain the equations     wmi,j ,i · n!  (Mα · Φ (0)) wn+1 = m ! i,j d m +···+m =n α∈N0

1,1

d,αd

i,j

for n ≥ 0, where the second summation runs over all |α|-tuples (m1,1 , . . . , m1,α1 , m2,1 , . . . , m2,α2 , . . . , md,1 , . . . , md,αd ) n! is an integer of integers ≥ 1 whose sum is equal to n. Since each m1,1 !·...·m d,αd ! it follows that      d |wmi,j ,i | ·Mα ·Φ (0) : α ∈ N0 , m1,1 +· · ·+md,αd = n wn+1  ≤ max

 ≤ max

i,j







wmi,j  ·Mα ·Φ (0) : α ∈ Nd0 , m1,1 +· · ·+md,αd = n .

i,j

In the proof of Lemma 18.7 we have seen that F 

1 |a|

≤

1 |a|

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holds true for any sufficiently big |a| ∈ |K|. This implies the existence of some |a| ≥ 1 such that Mα  ≤ |a||α|

for any α ∈ Nd0 .

We claim that wn+1  ≤ |a|n · Φ (0)n+1

for any n ≥ 0.

The case n = 0 is obvious from w1 = Φ (0). We now proceed by induction with respect to n. Since 1 ≤ mi,j < n the induction hypothesis gives wmi,j  ≤ |a|mi,j −1 · Φ (0)mi,j . We deduce



wmi,j  ≤ |a|n−|α| · Φ (0)n

i,j

and therefore wn+1  ≤ max |a|n−|α| · Φ (0)n · |a||α| · Φ (0) = |a|n · Φ (0)n+1 . α

Using Exercise 2.1.i. and Lemma 2.2 we conclude that there are appropriate ε0 , ε1 > 0 such that 

wn  ≤ ε0 εn1 n!

for any n ≥ 1.

It follows that Φ ∈ Fε (K; K d ) for any 0 < ε < ε−1 1 . We now consider the general case, and we fix a K-basis x1 , . . . , xd1 of g1 (where gi := Lie(Gi )). For any x ∈ g1 we may apply Thm. 18.14 to the homomorphism of Lie algebras K −→ g1 a −→ ax and obtain a unique formal homomorphism Φx : F(K,+),id −→ F G1 ,c1 such that

(x). Φx (0) = σΦx (1) = θc−1 1

We introduce the homomorphism of Lie algebras σ := θc2 ◦ σΦ ◦ θc−1 : g1 −→ g2 . 1

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Formal Group Laws

151

The unicity part of Thm. 18.14 implies in addition that we must have (32)

Φσ(x) (U ) = Φ(Φx (U )).

By the special case which we have treated already we find an ε > 0 such that Φxi ∈ Fε (K; K d1 ) and Φσ(xi ) ∈ Fε (K; K d2 )

for any 1 ≤ i ≤ d1 .

Hence, for sufficiently small ε > 0, the maps G1 −1 f1 ((a1 ,...,ad1 )):=ϕ−1 1 (Φx (a1 ))·...·ϕ1 (Φx 1

d1

(ad1 ))

K d1 ⊇ Bε (0) −1 f2 ((a1 ,...,ad1 )):=ϕ−1 2 (Φσ(x ) (a1 ))·...·ϕ2 (Φσ(x 1

d1 )

(ad1 ))

G2 are well defined and locally analytic. Using Cor. 13.5 we see that the tangent map at 0 of the upper map is equal to (a1 , . . . , ad1 ) −→ a1 x1 + · · · + ad1 xd1 which is a bijection. Hence by Prop. 9.3 the upper map can be inverted as a locally analytic map in a sufficiently small open neighbourhood V1 ⊆ U1 of e1 ∈ G1 . Because of (32) the resulting composed locally analytic map f2 ◦ f1−1 : V1 −→ Bε (0) −→ G2 has Φ as its power series expansion (with respect to the charts ϕ1 |V1 and ϕ2 ) around ϕ1 (e1 ) = 0. Proposition 18.17. Let G1 and G2 be Lie groups over K, and let σ : Lie(G1 ) −→ Lie(G2 ) be a homomorphism of Lie algebras; we then have: i. There exist open subgroups Hi ⊆ Gi as well as a homomorphism of Lie groups f : H1 −→ H2 such that Lie(f ) = σ; ii. if (H1 , H2 , f  ) is another triple as in i. then f |H = f  |H on some open subgroup H ⊆ H1 ∩ H1 . Proof. i. We choose, for i = 1, 2, a chart ci = (Ui , ϕi , K di ) for Gi around ei satisfying ϕi (e) = 0. By Lemma 18.7 there is an open subgroup Hi ⊆ Gi such that ϕi restricts to an isomorphism of Lie groups ∼ =

Hi −−→ (B

1 |ai |

(0), F Gi ,ci )

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for some sufficiently big |ai | ∈ K. On the other hand, according to Thm. 18.14, there is a formal homomorphism Φ : F G1 ,c1 −→ F G2 ,c2 such that the diagram σ

Lie(G1 )

Lie(G2 )

∼ = θc1

∼ = θc2

(K d1 , [ , ]F G

1 ,c1

)

σΦ

(K d2 , [ , ]F G

2 ,c2

)

is commutative (note that the perpendicular arrows are isomorphisms of Lie algebras by Prop. 18.6.ii.). After possibly enlarging |a1 | we have Φ ∈ F 1 (K d1 ; K d2 ), by Prop. 18.16, with Φ 1 ≤ |a12 | . Hence Φ defines a |a1 |

|a1 |

homomorphism of Lie groups f˜ : (B

1 |a1 |

(0), F G1 ,c2 ) −→ (B

1 |a2 |

(0), F G2 ,c2 )

x −→ Φ(x) ˜ such that Lie(f˜) = σΦ . We put f := ϕ−1 2 ◦ f ◦ ϕ1 . Using Lemma 9.2 we see that = σ. Lie(f ) = Lie(ϕ2 )−1 ◦ Lie(f˜) ◦ Lie(ϕ1 ) = θc2 ◦ σΦ ◦ θc−1 1 ii. By shrinking the involved open subgroups we may assume: – f and f  both are homomorphisms from H1 to H2 ; – there are charts c1 and c2 with domains of definition H1 and H2 , respectively, with respect to which f and f  are given by tuples Φ and Φ , respectively, of convergent power series; – with respect to ci the multiplication in Hi is given by the formal group law F Gi ,ci . The identity theorem Cor. 5.8 then implies that Φ, Φ : F G1 ,c1 −→ F G2 ,c2 both are formal homomorphisms. But σΦ = θc−1 ◦ Lie(f ) ◦ θc1 = θc−1 ◦ σ ◦ θc1 = θc−1 ◦ Lie(f  ) ◦ θc1 = σΦ . 2 2 2 Hence Thm. 18.14 implies that Φ = Φ and a fortiori that f = f  .

18

Formal Group Laws

153

Corollary 18.18. Let G1 and G2 be Lie groups over K; if Lie(G1 ) ∼ = Lie(G2 ) then there are open subgroups Hi ⊆ Gi such that H1 ∼ = H2 . ∼ =

Proof. We fix an isomorphism σ : Lie(G1 ) −−→ Lie(G2 ). By applying Prop. 18.17 to σ as well as σ −1 we find open subgroups H1 ⊆ H1 ⊆ G1 and H2 ⊆ H2 ⊆ G2 together with homomorphisms of Lie groups g

f

H2 −→ H1 −→ H2 such that f ◦ g = idH2 , f (H1 ) ⊆ H2 , and g ◦ (f |H1 ) = idH1 . Hence both maps f

g

H1 −→ H2 −→ H1 are injective with their composite being the identity of H1 . It follows that for any open subset V ⊆ H1 we have f (V ) = g −1 (V ). This shows that f restricts to an isomorphism between H1 and the open subgroup H2 := g −1 (H1 ). Corollary 18.19. Let G be a Lie group over K, and let {Gε }ε>0 be the Campbell-Hausdorff Lie group germ of Lie(G); for any sufficiently small ε there is a homomorphism of Lie groups expG,ε : Gε −→ G such that: i. expG,ε (Gε ) is open in G; ii. expG,ε is an isomorphism of Lie groups onto its image; iii. Lie(expG,ε ) = idLie(G) . For any two such homomorphisms expG,ε1 and expG,ε2 there is a 0 < ε ≤ min(ε1 , ε2 ) such that expG,ε1 |Gε = expG,ε2 |Gε . Proof. We fix a sufficiently small ε > 0 and view G as an open submanifold of Lie(G). Then Lie(G ) = Lie(G) by Prop. 17.3. Hence the assertion follows from Prop. 18.17 and Cor. 18.18. Definition. expG,ε is called an exponential map for the Lie group G.

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Part B

The Algebraic Theory of p-Adic Lie Groups

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Chapter IV

Preliminaries Throughout this Part B we fix a complete discrete valuation ring O with field of fractions K. We also pick a prime element π in O. Then O = limm O/π m O ←− as a topological ring.

19

Completed Group Rings

Let G be any profinite group. We denote by N (G) the set of open normal subgroups in G. Then (33)

G=

lim ←−

G/N

N ∈N (G)

is the projective limit, as a topological group, of the finite groups G/N . By functoriality, the algebraic group rings O[G/N ] form, for varying N in N (G), a projective system of rings. Its projective limit Λ(G) := O[[G]] :=

lim ←−

O[G/N ]

N ∈N (G)

is called the completed group ring or the Iwasawa algebra of G over O. As an immediate consequence of (33) the natural map O[G] −→ Λ(G) is injective. We therefore will always view O[G] as a subring of Λ(G). Each O[G/N ] is finitely generated and free as an O-module and therefore is a complete topological O-algebra for the π-adic topology. We equip Λ(G) with the projective limit topology of these π-adic topologies and make it in this way into a complete topological O-algebra. By construction the subring O[G] is dense in Λ(G). If the residue field O/πO is finite then all rings O, O[G/N ], and hence Λ(G) are compact. For general O/πO we contemplate the identity Λ(G) =

lim ←−

lim (O/π m O)[G/N ] . ←−

N ∈N (G) m≥1

Each factor ring (O/π m O)[G/N ] obviously is (left and right) artinian. We see that Λ(G), as a left or right module over itself, is the projective limit of P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8 4, © Springer-Verlag Berlin Heidelberg 2011

157

158

IV

Preliminaries

modules of finite length. This means that Λ(G) is a pseudocompact topological ring. In the Appendix we will collect the basic properties of this notion which, for our purposes, retains sufficiently many features of compact rings. We also see that the two-sided ideals   pr Jm,N (G) := ker Λ(G) −−→ (O/π m O)[G/N ] , for m ≥ 1 and N ∈ N (G), form a fundamental system of open neighbourhoods of zero in Λ(G). The ideal m(G) := J1,G (G) satisfies Λ(G)/m(G) = O/πO and therefore is a maximal (left or right) ideal in Λ(G). The group G is contained as a subgroup in the group of units O[G]× of O[G] and a fortiori in the group of units Λ(G)× of Λ(G). Lemma 19.1. The inclusion map G → Λ(G) is a homeomorphism onto its image. Proof. The map in question is the projective limit of the inclusion maps G/N → O[G/N ] for N ∈ N (G). Hence it is continuous and injective. Since G is compact and Λ(G) is Hausdorff it then necessarily is a homeomorphism onto its image. Proposition 19.2. (Universal property) Let M be any complete (and Hausdorff ) topological O-module in which the open O-submodules form a fundamental system of neighbourhoods of zero, and let f : G −→ M be any continuous map; then there is a unique continuous O-module homomorphism fΛ : Λ(G) −→ M such that fΛ |G = f . Proof. It is clear that f  extends uniquely to an O-module homomorphism  fO : O[G] −→ M by fO ( g cg g) := g cg f (g). Since O[G] is dense in Λ(G) the continuous extension fΛ of fO to Λ(G) is unique if it exists. To establish the existence let (Mi )i∈I be the family of all open O-submodules in M . We first consider an arbitrary but fixed index i ∈ I. By the continuity of f we find, for any g ∈ G, an Ng ∈ N (G) such that f (gNg ) ⊆ f (g) + Mi . Because of the compactness of G finitely many cosets g1 Ng1 , . . . , gs Ngs cover G. We put Ni := Ng1 ∩ · · · ∩ Ngs ∈ N (G). Then f (gNi ) ⊆ f (g) + Mi

for any g ∈ G.

On the other hand one easily checks that the kernel of the projection map O[G] −→ O[G/Ni ] is generated, as an O-module, by the elements gh − g for g ∈ G and h ∈ Ni . This shows the existence of the commutative diagram of

19

Completed Group Rings

159

O-module homomorphisms O[G]

fO

M

pr

pr

O[G/Ni ]

fi

M/Mi .

Since G/Ni is finite the map fi is continuous for the π-adic topology on O[G/Ni ] and the discrete topology on M/Mi . Hence we even have a commutative diagram O[G]

fO

M

pr

pr

(O/π mi O)[G/Ni ]

fi

M/Mi

for some sufficiently large mi ≥ 1. We now vary i ∈ I and obtain in the limit the continuous extension lim f

i ← − fΛ : Λ(G) −→ lim O[G]/(O/π mi O)[G/Ni ] −− −−→ lim M/Mi = M ←− ←−

i∈I

i∈I

of fO . Corollary 19.3. Let A be any complete (and Hausdorff ) topological Oalgebra in which the open O-submodules form a fundamental system of neighbourhoods of zero, and let f : G −→ A× be any group homomorphism such f

⊆

that the composed map G − → A× − → A is continuous; then there is a unique continuous unital O-algebra homomorphism fΛ : Λ(G) −→ A such that fΛ |G = f . Proof. Applying Prop. 19.2 to M := A we obtain the O-module homomorphisms fO and fΛ . The assumption that f is a group homomorphism immediately implies that fO is a unital ring homomorphism and by continuity fΛ then has to be a ring homomorphism as well. Corollary 19.4. Let f : G −→ H be a continuous homomorphism of profinite groups; we then have: i. f extends uniquely to a continuous unital O-algebra homomorphism Λ(f ) : Λ(G) −→ Λ(H);

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IV

Preliminaries

ii. if f is injective then Λ(f ) is a homeomorphism onto its image; iii. if f is surjective then Λ(f ) is a topological quotient map; iv. if f is the inclusion map of an open subgroup G in H then Λ(H) is finitely generated free of rank [H : G] as a left or right Λ(G)-module (via Λ(f )). Proof. i. This is immediate from applying Cor. 19.3 to A := Λ(H). ii. By compactness f is a homeomorphism onto its image. Hence {G∩N : N ∈ N (H)} is cofinal in N (G). This implies that Λ(f ) is the projective limit of the maps O[G/G ∩ N ] −→ O[H/N ] induced by f which clearly are homeomorphisms onto their images (for the π-adic topologies). iii. We may view Λ(H) via Λ(f ) as a pseudocompact (left) Λ(G)-module so that Λ(f ) becomes a continuous homomorphism of pseudocompact Λ(G)modules. In this situation we quite generally know, by Thm. 22.3.ii., that im(Λ(f )) is closed in Λ(H) and that on im(Λ(f )) the subspace topology induced by Λ(H) coincides with the quotient topology induced by Λ(f ). But im(Λ(f )) obviously contains O[H] which is dense in Λ(H). Hence im(Λ(f )) = Λ(H) and its topology is the quotient topology induced by Λ(f ). iv. The set N of all open normal subgroups in H which are contained in G is cofinal in both N (H) and N (G). Hence Λ(f ) is the projective limit of the inclusion maps O[G/N ] → O[H/N ] for N ∈ N . If h1 , . . . , hs ∈ H is any set of representatives for the right or left cosets of G in H then each O[H/N ] is free as a left or right O[G/N ]-module with basis h1 , . . . , hs . It follows that h1 , . . . , hs also is a basis of Λ(H) as a left or right Λ(G)-module. Next we establish a few straightforward facts about the ideal structure of Λ(G). Let   pr JN (G) := ker Λ(G) −−→ O[G/N ] for any N ∈ N (G). Proposition 19.5. Suppose that the subgroup N ∈ N (G) is topologically generated by the finitely many elements h1 , . . . , hr ; we then have: i. JN (G) is generated, as a left or as a right ideal, by the finitely many elements h1 − 1, . . . , hr − 1; ii. Jm,N (G), as a left or as a right ideal, is generated by the elements π m , h1 − 1, . . . , hr − 1.

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Completed Group Rings

161

Proof. The second assertion is a trivial consequence of the first one. We will prove only the left version of i. the argument for the right version being completely analogous. Suppose first that G is finite in which case the hi generate algebraically. We have used already the simple fact that then JN (G) as a left ideal in Λ(G) = O[G] is generated by the elements h − 1 for h ∈ N . Using the identities h h − 1 = h (h − 1) + (h − 1)

and

h−1 − 1 = −h−1 (h − 1)

for any h, h ∈ N we conclude that JN (G) in fact is generated by h1 − 1, . . . , hr − 1. In general we have JN (G) =

JN/N  (G/N  ).

lim ←−

N  ∈N (G),N  ⊆N

Since each finite group N/N  is algebraically generated by the elements h1 N  , . . . , hr N  it follows that the maps O[G/N  ]r −→ JN/N  (G/N  ) r  ¯ r ) −→ ¯ i (hi − 1) ¯1, . . . , λ λ (λ i=1

are surjective. Passing to the projective limit with respect to N  gives the map Λ(G)r −→ JN (G) r  (λ1 , . . . , λr ) −→ λi (hi − 1) i=1

which, by Thm. 22.3.iv., remains surjective. We recall that, quite generally for any unital ring R, the Jacobson radical Jac(R) is defined to be the intersection of all maximal left ideals (cf. [Lam] §4). It is a two-sided ideal which equivalently can be characterized as the intersection of all maximal right ideals, and it contains any left or right nil ideal. We put J(G) := Jac(Λ(G)). Obviously, J(G) ⊆ m(G). We also recall that a nonzero unital ring R is called local if the subset of nonunits R \ R× is additively closed; in this case one has R \ R× = Jac(R) (cf. [Lam] §19). Lemma 19.6. Suppose that char(O/πO) = p > 0 and that G is a finite p-group; then, for any m ≥ 1, the ring (O/π m O)[G] is local and its maximal ideal is nilpotent.

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IV

Preliminaries

Proof. Let m denote the kernel of the augmentation map (O/π m O)[G] −→ O/πO   cg g −→ cg mod π. g

g

This obviously is a maximal (left) ideal and therefore contains the Jacobson radical. We first consider the case m = 1 and show, by induction with respect 2 to the order |G| of G, that then m|G| = 0. Let Z ⊆ G be the center of G which is nontrivial. By the induction hypothesis we have m|G/Z| ⊆ ker(O/πO[G] −→ O/πO[G/Z]). 2

This right hand kernel is generated, as an ideal, by the central elements h−1 for h ∈ Z. Since the order of Z is a power of the characteristic of O/πO we obtain (h − 1)|Z| = h|Z| − 1 = 1 − 1 = 0. Hence m|G| ⊆ ker(O/πO[G] −→ O/πO[G/Z])|Z| = 0. 2

2

For general m we deduce that m|G| ⊆ π(O/π m O)[G] and hence iteratively 2 that m|G| ·m = 0. It follows that m is contained in the Jacobson radical. Therefore m is the unique maximal left ideal which means that the ring (O/π m O)[G] is local. 2

Proposition 19.7. Suppose that char(O/πO) = p > 0 and that G is a pro-p-group; we then have: i. Λ(G) is a local ring with residue field O/πO; ii. m(G) = J(G); iii. if G is topologically finitely generated then the J(G)i , for i ≥ 1, are finitely generated as left or right ideals and form a fundamental system of open neighbourhoods of zero (i. e., the topology of Λ(G) is the J(G)adic one). Proof. It is a direct consequence of Lemma 19.6 that each ideal Jm,N (G) contains a suitable power of the ideal m(G). This means that the m(G)-adic topology is finer than the pseudocompact topology. The lemma also says that any element λ ∈ Λ(G) \ m(G) is a unit modulo each ideal Jm,N (G). Hence λ is a unit in Λ(G). This shows that Λ(G) is local, that m(G) = J(G), and that the J(G)-adic topology is finer than the pseudocompact one. We now suppose that G is topologically generated by the finitely many elements

20

The Example of the Group Zdp

163

g1 , . . . , gr . We then know from Prop. 19.5.ii. that m(G) = J(G) is generated by π, g1 − 1, . . . , gr − 1 as a left as well as a right ideal. This implies, by induction with respect to i, that J(G)i as a left as well as a right ideal is generated by the finitely many elements {π  (gj1 − 1) · · · (gjm − 1) : , m ≥ 0,  + m = i, 1 ≤ j1 , . . . , jm ≤ r}. As a finitely generated left ideal J(G)i is closed in Λ(G) by Cor. 22.4. Moreover, each subquotient J(G)i /J(G)i+1 is a finite dimensional vector space over Λ(G)/J(G) = O/πO. It follows inductively that Λ(G)/J(G)i with the quotient topology is a pseudocompact Λ(G)-module of finite length (cf. Thm. 22.3.i.). The topology of any such module necessarily is discrete. We conclude that J(G)i is open in Λ(G). Lemma 19.8. Suppose that the ring Λ(G) is noetherian; then its topology is the J(G)-adic one, and any left or right ideal is closed in Λ(G). Proof. By Prop. 22.5 and Cor. 22.4 this is a general fact about noetherian pseudocompact rings.

20

The Example of the Group Zdp

Let G = Zdp , for some d ≥ 1, be the d-fold direct product of the additive group of p-adic integers Zp with itself. Its completed group ring can be computed explicitly by comparing it with the ring O[[X1 , . . . , Xd ]] of formal power series in the variables X1 , . . . , Xd over O. The latter is a local ring with maximal ideal m generated by π, X1 , . . . , Xd . The m-adic topology makes it into a pseudocompact ring. Let gi := (. . . , 0, 1, 0, . . .) with the entry 1 in the i-th place, for 1 ≤ i ≤ d, denote the standard topological generators of the group Zdp . Proposition 20.1. Suppose that char(O/πO) = p > 0; there is a unique ∼ = topological isomorphism of pseudocompact O-algebras O[[X1 , . . . , Xd ]] − → d Λ(Zp ) which sends Xi to gi − 1. Proof. We obviously have the unique O-algebra homomorphism f0 : O[X1 , . . . , Xd ] −→ Λ(Zdp ) Xi −→ gi − 1 where the left hand side is the polynomial ring over O in the variables X1 , . . . , Xd . From Prop. 19.7 we know that Λ(Zdp ) is a local ring with maximal ideal m(Zdp ) generated by π, g1 − 1, . . . , gd − 1, and that the pseudocompact topology of Λ(Zdp ) is the m(Zdp )-adic one. Since clearly f0 (m ∩

164

IV

Preliminaries

O[X1 , . . . , Xd ]) ⊆ m(Zp ) it follows that f0 extends uniquely to a continuous ring homomorphism f : O[[X1 , . . . , Xd ]] −→ Λ(Zdp ). We observe that O[[X1 , . . . , Xd ]]/πO[[X1 , . . . , Xd ]] = O/πO[[X1 , . . . , Xd ]] and O[[X1 , . . . , Xd ]] = lim O[[X1 , . . . , Xd ]]/π n O[[X1 , . . . , Xd ]] ←− n

as well as, by Thm. 22.3.iv. and Lemma 22.1, that Λ(Zdp )/πΛ(Zdp ) = O/πO[[Zdp ]] and Λ(Zdp ) = lim Λ(Zdp )/π n Λ(Zdp ). ←− n

Hence the bijectivity of f can be shown modulo π n for any n ≥ 1. Since O[[X1 , . . . , Xd ]] and Λ(Zdp ), both by construction, are O-torsion free this reduces, by an inductive argument, to the bijectivity of f modulo π. But f mod π is the projective limit of the corresponding ring homomorphisms n

n

O/πO[[X1 , . . . , Xd ]]/ X1p , . . . , Xdp  −→ O/πO[Zdp /pn Zdp ] n

n

which are well defined because of the identities 0 = gip − 1 = (gi − 1)p in O/πO[Zdp /pn Zdp ]. These are easily seen to be isomorphisms.

21

Continuous Distributions

The completed group ring Λ(G) of a profinite group G also has an important functional analytic description. We let C ∞ (G), resp. C(G), denote the Omodule of all O-valued locally constant, resp. continuous, functions on G. Clearly C ∞ (G) is a submodule of C(G). If G is finite we have the natural O-module isomorphism ∼ =

O[G] − → HomO (C(G), O) which sends a group element g to the linear form which evaluates a function in g. By passing to the projective limit we then obtain for a general G a

22

Appendix: Pseudocompact Rings

165

corresponding O-module isomorphism Λ(G) ∼ =

(34)

lim ←−

HomO (C(G/N ), O)

N ∈N (G)

= HomO (lim C(G/N ), O) −→ N

= HomO (C ∞ (G), O). We always equip C(G) with the π-adic topology. Because of C(G) = lim{functions G −→ O/π m O} = lim C(G)/π m C(G) ←− ←− m

m

the submodule C ∞ (G) is dense in C(G). Moreover, C ∞ (G) ∩ π m C(G) = π m C ∞ (G) which implies that the π-adic topology on C ∞ (G) is the subspace topology induced by the π-adic topology on C(G). Lemma 21.1. Λ(G) = HomO (C(G), O). Proof. Because of (34) it suffices to see that any O-linear form on C ∞ (G) extends uniquely to an O-linear form on C(G). But this follows from the discussion before the lemma once one observes that any O-linear form is automatically continuous for the π-adic topology. In this sense we may view Λ(G) as the space of O-valued continuous distributions on G.

22

Appendix: Pseudocompact Rings

A left pseudocompact ring R is a unital topological ring which is Hausdorff and complete and which has a fundamental system (Li )i∈I of open neighbourhoods of zero consisting of left ideals Li ⊆ R such that R/Li is of finite length as an R-module. A pseudocompact left R-module M is a unital topological R-module which is Hausdorff and complete and which has a fundamental system (Mi )i∈I of open neighbourhoods of zero consisting of submodules Mi ⊆ M such that the R-module M/Mi is of finite length. In the following let R be a left pseudocompact ring. The two technical key facts about pseudocompact R-modules are the following.

166

IV

Preliminaries

Lemma 22.1. Let M be a pseudocompact R-module and (Mi )i∈I be a decreasingly filtered family of closed submodules; then the natural map M −→ limi M/Mi is surjective. ←− Proof. [Gab] IV.3 Prop. 10. Lemma 22.2. Let M be a pseudocompact R-module, N a closed submodule, and (Mi )i∈I a decreasingly filtered family of closed submodules; then   Mi = (N + Mi ). N+ i

i

Proof. [Gab] IV.3 Prop. 11. Theorem 22.3. i. Let M be a pseudocompact R-module and N ⊆ M be a closed submodule; then N , with the subspace topology, and M/N , with the quotient topology, are pseudocompact R-modules. ii. Let α : M −→ N be a continuous homomorphism of pseudocompact R-modules; we then have: a) im(α) is closed in N ; b) on im(α) the subspace topology induced by N coincides with the quotient topology induced by α. iii. The R-module projective limit of any projective system of pseudocompact R-modules is pseudocompact for the projective limit topology. iv. The formation of filtered projective limits of pseudocompact R-modules is exact. Proof. [Gab] IV.3 Thm. 3. Corollary 22.4. Any finitely generated submodule N in a pseudocompact R-module M is closed. Proof. Write N as the image of some R-module homomorphism α : Rm −→ M . Since α automatically is continuous we may apply Thm. 22.3.ii. Suppose that R is left noetherian, and let M be an abstract finitely generated left R-module. We may write M as a quotient Rm  M of some finitely generated free R-module. The latter carries the product topology and M then the corresponding quotient topology.

22

Appendix: Pseudocompact Rings

167

Exercise. This quotient topology on M is independent of the choice of the presentation Rm  M . Since the kernel of Rm  M is finitely generated, by our assumption on R, it is closed in Rm by Cor. 22.4. Hence Thm. 22.3.i. implies that M is pseudocompact in the quotient topology. In fact, this is the unique pseudocompact topology on M . Proposition 22.5. If R is left noetherian then the pseudocompact topology on R is the Jac(R)-adic topology. Proof. According to [Gab] IV.3 Prop. 13.b the ring R/ Jac(R) is a product of endomorphism rings of vector spaces over division rings. But R/ Jac(R) by assumption is left noetherian. Hence it must be left artinian. Each quotient Jac(R)i / Jac(R)i+1 is a finitely generated left module over the left artinian ring R/ Jac(R) and hence is of finite length. It follows inductively that each R/ Jac(R)i is a left R-module of finite length. On the other hand Jac(R)i is closed in R by Cor. 22.4. Hence, by Thm. 22.3.i., R/ Jac(R)i is a pseudocompact R-module of finite length. Its topology therefore must be the discrete one which means that Jac(R)i is open.  Again by [Gab] IV.3 Prop. 13.b we have i Jac(R)i = 0. Let  L ⊆ Ri be any open left ideal. Then Lemma 22.2 implies that L = L + i Jac(R) =  i ). Since R/L is of finite length it follows that Jac(R)i ⊆ L (L + Jac(R) i for any sufficiently big i. There are obvious “right” versions of everything above. A unital topological ring R which is left as well as right pseudocompact will be called pseudocompact.

Chapter V

p-Valued Pro-p-Groups From now on we fix once and for all a prime number p.

23

p-Valuations

Let G be any abstract group. A p-valuation ω on G is a real valued function ω : G \ {1} −→ (0, ∞) which, with the convention that ω(1) = ∞, satisfies (a) ω(g) >

1 p−1 ,

(b) ω(g −1 h) ≥ min(ω(g), ω(h)), (c) ω([g, h]) ≥ ω(g) + ω(h), (d) ω(g p ) = ω(g) + 1 for any g, h ∈ G. Here the commutator is normalized to be [g, h] = ghg −1 h−1 ; as usual, for any two subsets A, B ⊆ G we will write [A, B] for the subgroup of G generated by the set of commutators {[g, h] : g ∈ A, h ∈ B}. The reason for the somewhat mysterious axiom (a) will only become clear later. Applying (b) with h = 1 gives ω(g −1 ) ≥ ω(g) and hence by symmetry ω(g −1 ) = ω(g)

for any g ∈ G.

From (b) and (c) we deduce ω(ghg −1 ) = ω([g, h]h) ≥ min(ω([g, h]), ω(h)) ≥ min(ω(g) + ω(h), ω(h)) = ω(h) and therefore, by symmetry, (35)

ω(ghg −1 ) = ω(h)

for any g, h ∈ G.

Finally, if ω(g) > ω(h) then ω(h) = ω(g −1 (gh)) ≥ min(ω(g), ω(gh)) ≥ min(ω(g), ω(h)) = ω(h) P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8 5, © Springer-Verlag Berlin Heidelberg 2011

169

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p-Valued Pro-p-Groups

and hence ω(gh) = ω(h). It follows that (36)

ω(gh) = min(ω(g), ω(h)) if

ω(g) = ω(h).

For any real number ν > 0 we put Gν := {g ∈ G : ω(g) ≥ ν} and Gν+ := {g ∈ G : ω(g) > ν}. These are subgroups by (b), and they are normal by (35). Remark 23.1. Gν /Gν+ , for any ν > 0, is a central subgroup of G/Gν+ . Proof. By (c) we have ω(ghg −1 h−1 ) > ω(h) for any g, h ∈ G. This means that ghg −1 h−1 ∈ Gω(h)+ and hence that ghGω(h)+ = hgGω(h)+ . The subgroups Gν form a decreasing exhaustive and separated filtration of G with the additional properties Gν =



Gν 

and [Gν , Gν  ] ⊆ Gν+ν  .

ν  <ν

There is a unique (Hausdorff) topological group structure on G for which the Gν form a fundamental system of open neighbourhoods of the identity element. It will be called the topology defined by ω. Example 23.2. Let E be a finite extension of the field Qp of p-adic numbers, oE its ring of integers, πE ∈ oE a prime element, and v its additive valuation normalized by v(p) = 1. We fix an n ∈ N, and we consider the algebra Mn×n (E) of n × n-matrices over E. For any nonzero matrix A = (aij ) we put w(A) := min v(aij ), i,j

and w(0) := ∞. This function w on Mn×n (E) has the following straightforward properties: (i) w(A + B) ≥ min(w(A), w(B)); (ii) w(A + B) = min(w(A), w(B)) if w(A) = w(B); (iii) w(AB) ≥ w(A) + w(B); (iv) w(A) = 0 for any A ∈ GLn (oE ).

23

p-Valuations

Now let

171

 G := g ∈ GLn (E) : w(g − 1) >

1 p−1



and ω(g) := w(g − 1)

for g ∈ G.

Then G is an open neighbourhood of 1 in GLn (E). By its defining condition any g ∈ G satisfies g ∈ Mn×n (oE ) and g ≡ 1 mod πE , hence det g ∈ o× E and therefore g ∈ GLn (oE ). We now compute: 1) For g ∈ G we have ω(g −1 ) = w(g −1 − 1) = w(−g −1 (g − 1)) ≥ w(g − 1) = ω(g). In particular, also g −1 lies in G, and then by symmetry ω(g −1 ) = ω(g). Using (iii) it moreover follows that G is a subgroup of GLn (oE ). 2) For g, h ∈ G we have ω(g −1 h) = w(g −1 h − 1) = w((g −1 − 1)(h − 1) + (g −1 − 1) + (h − 1)) ≥ min(w(g −1 − 1), w(h − 1)) = min(ω(g −1 ), ω(h)) = min(ω(g), ω(h)). 3) For g, h ∈ G we also have ω([g, h]) = w([g, h] − 1) = w(((g − 1)(h − 1) − (h − 1)(g − 1))g −1 h−1 ) ≥ w((g − 1)(h − 1) − (h − 1)(g − 1)) ≥ min(w((g − 1)(h − 1)), w((h − 1)(g − 1))) ≥ w(g − 1) + w(h − 1) = ω(g) + ω(h). 4) For g = 1 in G we have in Mn×n (E) the equation p    p p p (g − 1)j g − 1 = ((g − 1) + 1) − 1 = j j=1

= p(g − 1) + p(g − 1)

p−2  j=1

aj (g − 1)j + (g − 1)p

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 j with appropriate integers aj ∈ Z. Since w( p−2 j=1 aj (g − 1) ) > 0 and w((g − 1)p ) ≥ pw(g − 1) > 1 + w(g − 1) we obtain ω(g p ) = w(g p − 1) = w(p(g − 1)) = 1 + w(g − 1) = ω(g) + 1. Altogether these computations tell us that ω is a p-valuation on G. It is clear from the definition of G that ω defines the subspace topology on the open subgroup G of GLn (E). Example 23.3. Retaining the notations of the previous example we let Gn ⊆ GLn (Zp ), for any n ≥ 2, denote the open pro-p-subgroup of all matrices (aij ) such that v(aij ) ≥ 1 for any 1 ≤ j < i ≤ n and v(aii − 1) ≥ 1 for any 1 ≤ i ≤ n. If n ≥ p − 1 then Gn contains an element of order p (exercise!) and hence cannot carry a p-valuation. We therefore assume in the following that 2 ≤ n < p − 1. Then we may pick a rational number Gn we define

1 p−1

<α<

p−2 (p−1)(n−1) .

For g = (aij ) ∈

ωα (g) := min( min ((j − i)α + v(aij )), min v(aii − 1)). 1≤i≤n

1≤i=j≤n

In order to see that ωα indeed is a p-valuation on Gn we will embed (Gn , ωα ) into a group of the form discussed in the previous example. We pick a finite extension E of Qp which contains an element c of valuation v(c) = α. Let C ∈ GLn (E) denote the diagonal matrix with entries cn , cn−1 , . . . , c and consider the monomorphism of groups f : Gn −→ GLn (E) g −→ CgC −1 . For g = (aij ) we have f (g) = (cj−i aij ). This immediately implies ωα (g) = w(f (g) − 1)

for any g ∈ Gn .

Furthermore, by the choice of α, we have v(aii − 1) ≥ 1 >

1 p−1

(j − i)α + v(aij ) ≥ α + v(aij ) ≥ α >

for any 1 ≤ i ≤ n, 1 p−1

for any 1 ≤ i < j ≤ n,

23

p-Valuations

173

and (j − i)α + v(aij ) ≥ (1 − n)α + v(aij ) > (1 − n)

1 p−2 +1 = (p − 1)(n − 1) p−1 for any 1 ≤ j < i ≤ n

if (aij ) ∈ Gn . It follows that f maps Gn into the subgroup of GLn (E) on which w(. − 1) is a p-valuation by the previous example. It is clear that ωα defines the topology of Gn . For later use we recall the following commutator identities in G. Exercise. For any g, h, k ∈ G we have: (A) [gh, k] = g[h, k]g −1 [g, k], (B) [g, hk] = [g, h]h[g, k]h−1 , (C) [[g, h], hkh−1 ][[h, k], kgk −1 ][[k, g], ghg −1 ] = 1. We now form, for each ν > 0, the subquotient group grν G := Gν /Gν+ . It is commutative by (c) and therefore will be denoted additively. One of our important goals in the following is to investigate the graded abelian group gr G := grν G. ν>0

An element ξ ∈ gr G is called, as usual, homogeneous (of degree ν) if it lies in grν G. Furthermore, in this case any g ∈ Gν such that ξ = gGν+ is called a representative of ξ. We immediately observe that gr G has considerably more structure. First of all, by (d) we have pξ = 0 for any homogeneous element ξ ∈ gr G. Hence gr G in fact is an Fp -vector space. Lemma 23.4. For any ν, ν  > 0 the map grν G × grν  G −→ grν+ν  G (ξ, η) −→ [ξ, η] := [g, h]G(ν+ν  )+ , where g and h are representatives of ξ and η, respectively, is a well defined bi-additive map; we have [ξ, ξ] = 0 and [ξ, η] = −[η, ξ].

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Proof. First of all, by (c), we have [g, h] ∈ Gν+ν  so that [ξ, η] indeed is a homogeneous element of degree ν + ν  . Let now k ∈ Gν  be any element. Then [g, hk] = [g, h]h[g, k]h−1 by (B). Using Remark 23.1 we obtain [g, hk]G(ν+ν  )+ = [g, h]h[g, k]h−1 G(ν+ν  )+ = [g, h][g, k]G(ν+ν  )+ . If k ∈ Gν  + then [g, k] ∈ G(ν+ν  )+ and hence [g, hk]G(ν+ν  )+ = [g, h]G(ν+ν  )+ . This proves the independence of the choice of the representative as well as the additivity in the second component. For the first component the argument is completely analogous but using (A). It remains to note the trivial identities [g, g] = 1 and [g, h] = [h, g]−1 . By bilinear extension we therefore obtain a graded Fp -bilinear map [ , ] : gr G × gr G −→ gr G which satisfies [ξ, ξ] = 0

for any ξ ∈ gr G.

Lemma 23.5. Any ζ, ξ, η ∈ gr G satisfy the Jacobi identity [[ζ, ξ], η] + [[ξ, η], ζ] + [[η, ζ], ξ] = 0. Proof. We may assume that ζ, ξ, η are homogeneous of degrees ν, ν  , ν  with representatives g, h, and k, respectively. According to (C) we have [[ζ, ξ], hkh−1 Gν  + ] + [[ξ, η], kgk −1 Gν+ ] + [[η, ζ], ghg −1 Gν  + ] = 0. But as a consequence of Remark 23.1 the set of representatives of any given homogeneous element is invariant under conjugation. It follows that hkh−1 Gν  + = η, kgk−1 Gν+ = ζ, and ghg −1 Gν  + = ξ.

Altogether we see that gr G is a graded Lie algebra over Fp . Due to the last axiom (d) there is another piece of additional structure, though. Its construction requires less straightforward arguments which we will prepare for in the next section.

24

The Free Group on Two Generators

24

175

The Free Group on Two Generators

Let M denote the set of all noncommutative monomials in two variables X and Y , i. e., the free monoid generated by X and Y . If μ ∈ M is μ = μ1 · · · μd with each μi being equal to X or Y then deg μ := d is called the degree of μ. For any d ≥ 0 let M(d) be the free abelian group on the subset of M of all monomials of degree d. The Magnus algebra M in the variables X and Y is defined to be ring of all associative (but not commutative) formal power series in X and Y with coefficients in Z. Each element F ∈ M can uniquely be written as  cμ μ with cμ ∈ Z. F = μ∈M

Equivalently we may view M as the direct product M = d≥0 M(d) where  F corresponds to (F (d) )d with F (d) := deg μ=d cμ μ. If F = 0 then the nonnegative integer ord F := min{deg μ : cμ = 0} is called the order of F . By convention, ord 0 = ∞. Instead of ord(F − F  ) ≥ we often will write F ≡ F

mod deg .

If in the difference F −F  all monomials of degree < occur with a coefficient divisible by p we write F ≡ F

mod (p, deg ).

Using the geometric series we see that the elements 1 + X and 1 + Y are units in the ring M. More generally we have the following. Exercise 24.1. Let a ∈ Z and F, U ∈ M with ord U ≥ 1; we then have: i. F is invertible in M if and only if F (0) = ±1; ii. (1 + U )a ≡ 1 + aU (ord U ) mod deg((ord U ) + 1); iii. (1 + U )pa ≡ (1 + U p )a ≡ 1 mod (p, deg(p · ord U )). We let F denote the subgroup of the group of units M× generated by 1 + X and 1 + Y . Proposition 24.2. F is the free group generated by 1 + X and 1 + Y . Proof. [B-LL] Chap. II §5.3 Thm. 1.

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Next we put Fn := {F ∈ F : ord(F − 1) ≥ n} for any n ≥ 1. Theorem 24.3. F1 = F and Fn+1 = [Fn , F] for n ≥ 1. Proof. [B-LL] Chap. II §5.4 Thm. 2. Corollary 24.4. Fn /Fn+1 , for any n ≥ 1, is a finitely generated abelian group. For example, F/F1 is a free abelian group of rank two. In fact, this corollary can be made much more precise. First of all, by completely analogous arguments as for Lemma 23.4 and Lemma 23.5 one shows that the graded abelian group gr F := Fn /Fn+1 n≥1

is a graded Lie algebra over Z with the Lie bracket being the sum of the maps F /F+1 × Fn /Fn+1 −→ F+n /F+n+1 (EF+1 , F Fn+1 ) −→ EF E −1 F −1 F+n+1 . On the other hand the associative algebra M is a Lie algebra in the usual way for the Lie bracket (E, F ) −→ EF − F E. Let L ⊆ M be the Lie subalgebra generated by the variables X and Y . This L is known to be the free Lie algebra on the set {X, Y }, and it is naturally graded by L ∩ M(d) ; L= d≥1

in addition each abelian group L∩M(d) is a direct summand of M(d) ([B-LL] Chap. II §3.1 Thm. 1.a and Remark 3). Theorem 24.5. The unique homomorphism of graded Lie Z-algebras ∼ =

φ : L −−→ gr F sending X to (1 + X)F2 and Y to (1 + Y )F2 is an isomorphism. Its inverse is given by F Fn+1 −→ F (n) .

24

The Free Group on Two Generators

177

Proof. [B-LL] Chap. II §5.4 Thm. 3 (in particular part C) in the proof). For our purposes we need the following fact. n

Notation. If H is any group then H p denotes the subgroup generated the pn -th powers of all the elements of H. For any two subsets A, B ⊆ H let AB := {ab : a ∈ A, b ∈ B}. Proposition 24.6. For any pm ≥ ≥ m ≥ 1 we have {F ∈ Fm : F ≡ 1

mod (p, deg )} ⊆ Fpm F .

Proof. As a consequence of Cor. 24.4 we find a strictly increasing sequence of integers d0 = 0 < d1 = 2 < d2 < · · · < dn < · · · and a sequence E1 , . . . , Ei , . . . in F such that the Ei for dn−1 < i ≤ dn lie in Fn with their cosets modulo Fn+1 forming a Z-basis of Fn /Fn+1 . We then may write ad

+1

ad

−1 m−1 F = Edm−1 +1 · · · Ed−1 F1

with F1 ∈ F and adm−1 +1 , . . . , ad−1 ∈ Z. We claim that each integer adm−1 +1 up to ad−1 is divisible by p. Suppose we already have shown this to hold for all adm−1 +1 , . . . , adn with some m − 1 ≤ n < − 1. We then obtain from Ex. 24.1.iii. that ad

a

+1

dn m−1 Edm−1 +1 · · · Edn ≡ 1 mod (p, deg pm).

On the other hand we have ad

+1

ad

m−1 n+1 F ≡ Edm−1 +1 · · · Edn+1

mod deg(n + 2).

With the help of Ex. 24.1.ii., and observing that pm ≥ ≥ n + 2, we deduce a

ad

+1 n+1 · · · Edn+1 F ≡ Edndn+1

mod (p, deg(n + 2))

(n+1)

(n+1)

≡ (1 + adn +1 Edn +1 ) · · · (1 + adn+1 Edn+1 ) (n+1)

(n+1)

≡ 1 + adn +1 Edn +1 + · · · + adn+1 Edn+1

mod (p, deg(n + 2))

mod (p, deg(n + 2)).

Since ≥ n + 2 our assumption on F therefore implies (n+1)

(n+1)

adn +1 Edn +1 + · · · + adn+1 Edn+1 ∈ pM(n+1) . (n+1)

(n+1)

But according to Thm. 24.5 the Edn +1 , . . . , Edn+1 form a Z-basis of L ∩ M(n+1) . Since L ∩ M(n+1) is a direct summand of M(n+1) this forces the coefficients adn +1 , . . . , adn+1 to be divisible by p. Our claim follows by applying this argument iteratively.

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i. (1 + Y )−p (1 + X)−p ((1 + X)(1 + Y ))p ∈ Fp2 Fp .

Corollary 24.7.

ii. [1 + X, 1 + Y ]−p [(1 + X)p , 1 + Y ] ∈ Fp3 Fp+1 (here [ , ] denotes the commutator in the group F). Proof. i. It suffices to show that F := (1 + Y )−p (1 + X)−p ((1 + X)(1 + Y ))p . satisfies F ≡ 1 mod deg 2

and

F ≡ 1 mod (p, deg p).

But these congruences are immediate from Ex. 24.1.ii. and iii., respectively. ii. Here it suffices to show that F := [1 + X, 1 + Y ]−p [(1 + X)p , 1 + Y ] satisfies F ≡ 1 mod deg 3

and

F ≡ 1 mod (p, deg(p + 1)).

One easily sees that [1 + X, 1 + Y ] ≡ 1 + XY − Y X

mod deg 3.

By Ex. 24.1.ii. and iii., respectively, this implies [1 + X, 1 + Y ]−p ≡ 1 − p(XY − Y X) mod deg 3, [1 + X, 1 + Y ]−p ≡ 1 mod (p, deg 2p). On the other hand, by direct calculation we obtain [(1 + X)p , 1 + Y ] ≡ 1 + p(XY − Y X)

mod deg 3,

[(1 + X) , 1 + Y ] ≡ [1 + X , 1 + Y ] ≡ 1 mod (p, deg(p + 1)). p

25

p

The Operator P

Again ω is a fixed p-valuation on the abstract group G. The last axiom (d) says that ω(g p ) = ω(g) + 1 for any g ∈ G.

25

The Operator P

179

Proposition 25.1. For any two elements g, h ∈ G we have: i. ω(h−p g −p (gh)p ) > max(ω(g), ω(h)) + 1; ii. ω(g −p hp ) = ω(g −1 h) + n for any n ≥ 1. n

n

Proof. i. We may assume that the group G is generated by the two elements g and h (otherwise replace G by the subgroup generated by g and h and ω by its restriction this subgroup). There is a unique epimorphism of groups ψ : F −→ G which sends 1 + X to g and 1 + Y to h. Cor. 24.7.i. implies that h−p g −p (gh)p lies in ψ(F2 )p ψ(Fp ). Since ψ(F2 ) is the commutator subgroup of G it follows from the commutator identities (A) and (B) in Sect. 23 that it is the smallest normal subgroup of G which contains the commutators [g, h], [g −1 , h], [g, h−1 ], and [g −1 , h−1 ] (in fact, [g, h] would suffice because of relations like [g −1 , h] = g −1 [g, h]−1 g). The axioms (b) – (d) for ω then imply that ω has values ≥ ω(g) + ω(h) > max(ω(g), ω(h)) on ψ(F2 ) and > max(ω(g), ω(h)) + 1 on ψ(F2 )p . Similarly ψ(Fp ) is the smallest normal subgroup of G which contains all iterated commutators with p entries from the set {g, g −1 , h, h−1 }. So axioms (b) and (c) imply that the values of ω on ψ(Fp ) are ≥ (p−1) min(ω(g), ω(h))+max(ω(g), ω(h)). Using now for the first time the axiom (a) we see that this latter number is > max(ω(g), ω(h)) + 1 as well. ii. By induction it suffices to consider the case n = 1. We may assume that ω(g) ≤ ω(h). Applying the inequality in i. to the pair g and g −1 h and using the axioms (b) and (d) gives ω((g −1 h)−p g −p hp ) > max(ω(g), ω(g −1 h)) + 1 = ω(g −1 h) + 1 = ω((g −1 h)p ). Hence (36) implies ω(g −p hp ) = min(ω((g −1 h)p ), ω((g −1 h)−p g −p hp )) = ω((g −1 h)p ) = ω(g −1 h) + 1.

Let ν > 0 and g, h ∈ G such that ω(h) ≥ ν = ω(g). As a consequence of Prop. 25.1.i. we have (gh)p G(ν+1)+ = g p hp G(ν+1)+ . If ω(h) > ν then ω(hp ) > ν + 1 and hence (gh)p G(ν+1)+ = g p G(ν+1)+ .

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This shows that the map grν G −→ grν+1 G gGν+ −→ g p G(ν+1)+ is well defined and Fp -linear. For varying ν the direct sum of these maps then is an Fp -linear map of degree one P : gr G −→ gr G. We therefore may and always will view gr G as a graded module over the polynomial ring Fp [P ] in one variable over Fp . Remark 25.2. The Fp [P ]-module gr G is torsionfree. Proof. Let q(P ) be a nonzero polynomial over Fp of degree d with highest  term aP d and let ξ = ν ξν ∈ gr G be a finite nonzero sum with homogeneous components ξν such that q(P )(ξ) = 0. By the axiom (d) the operator P on gr G is injective and of degree one. The latter property implies that P d (aξν0 ) = aP d (ξν0 ) = 0 where ν0 := max{ν : ξν = 0}. But this contradicts the injectivity of P . Proposition 25.3. ω([g, h]−p [g p , h]) > ω(g) + ω(h) + 1 for any g, h ∈ G. Proof. The proof is exactly analogous to the proof of Prop. 25.1.i. by using Cor. 24.7.ii. Observe that, with the notations in the proof of Prop. 25.1.i. the values of ω on ψ(F3 )p are ≥ 2 min(ω(g), ω(h)) + max(ω(g), ω(h)) + 1 > ω(g) + ω(h) + 1 and on ψ(Fp+1 ) are ≥ p min(ω(g), ω(h)) + max(ω(g), ω(h)) = (p − 1) min(ω(g), ω(h)) + ω(g) + ω(h) > 1 + ω(g) + ω(h) where the last inequality uses axiom (a). Prop. 25.3 implies that for g, h ∈ G \ {1} we have [g p , h]G(ω(g)+ω(h)+1)+ = [g, h]p G(ω(g)+ω(h)+1)+ . This means that the Lie bracket on gr G is bilinear for the Fp [P ]-module structure. In other words, gr G is a Lie algebra over the ring Fp [P ]. This completes the discussion of the formal structure of gr G.

26

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Let G be a profinite group and ω be a p-valuation on G which we assume to define the topology of G. Then the Gν are open subgroups, hence the G/Gν are finite, and G = limν G/Gν . The axiom (d) implies that each G/Gν is a p←− group. We see that in the presence of a “defining” p-valuation G necessarily is a pro-p-group. Lemma 26.1. ω(G \ {1}) is a discrete subset of the interval (0, ∞). Proof. This is an immediate consequence of the finiteness of the G/Gν . In the last section we have seen that grG is a graded Lie algebra over the polynomial ring Fp [P ] and is torsionfree as an Fp [P ]-module. Definition. The pair (G, ω) is called of finite rank if gr G is finitely generated as an Fp [P ]-module. Later on we will show that the property of being of finite rank in fact does not depend on the choice of the p-valuation. From that point onwards we will simply speak of G being of finite rank. For the rest of this section we suppose (G, ω) to be of finite rank. By the elementary divisor theorem a finitely generated torsionfree module over the principal ideal domain Fp [P ] is free. We call rank(G, ω) := rankFp [P ] gr G the rank of the pair (G, ω). Exercise 26.2. For any closed subgroup H ⊆ G we have rank(H, ω|H) ≤ rank(G, ω); if H is open in G then rank(H, ω|H) = rank(G, ω). For any 0 < ν ≤ 1 we let (gr G)(ν) := grν G ⊕ grν+1 G ⊕ · · · ⊕ grν+n G ⊕ · · · Since the operator P is of degree one the obvious decomposition gr G = (gr G)(ν) 0<ν≤1

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is a decomposition of Fp [P ]-modules. Because of the finite generation it follows that at most finitely many of the (gr G)(ν) can be nonzero. This, in 1 particular, shows that there are finitely many real numbers ν1 , . . . , νs > p−1 such that  νj + N0 . ω(G \ {1}) = 1≤j≤s

Before we continue to explore the structure of gr G we need to make a simply observation on p-adic powers in the pro-p-group G. Let g ∈ G be any element. We then have the group homomorphism c : Z −→ G m −→ g m . Since G/N , for any N ∈ N (G), is a p-group we obtain c−1 (N ) = paN Z for some aN ≥ 0. It follows that c extends uniquely to a continuous group homomorphism c

c : Zp −→ lim Z/paN Z −−−−→ lim G/N = G ←− ←− N

N

which we always will write as g x := c(x). More generally, for any finitely many elements g1 , . . . , gr ∈ G, we have the continuous map (37)

Zrp −→ G (x1 , . . . , xr ) −→ g1x1 · . . . · grxr

but which, in general, depends on the order of the gi and therefore is not a group homomorphism. Nevertheless we introduce the following notion, where v denotes the usual p-adic valuation on Qp (i. e., v(pm x) = m for any m ∈ Z and x ∈ Z× p , and v(0) = ∞). Definition. The sequence of elements (g1 , . . . , gr ) in G is called an ordered basis of (G, ω) if the map (37) is a bijection (and hence, by compactness, a homeomorphism) and ω(g1x1 · · · grxr ) = min (ω(gi ) + v(xi )) 1≤i≤r

for any x1 , . . . , xr ∈ Zp .

Notation. For any element g = 1 in G we put σ(g) := gGω(g)+ ∈ gr G. ¯ ∈ F× Remark 26.3. Let g = 1 in G and x = 0 in Zp ; further let a p denote −v(x) x; we then have the image of p ω(g x ) = ω(g) + v(x)

and

σ(g x ) = (¯ aP v(x) )(σ(g)).

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Proof. This is straightforward from the definitions and axiom (d) once we convince ourselves that ω(g x ) = ω(g) holds whenever x ∈ Z× p is a p-adic unit. Since Gω(g) is open and hence closed in G we certainly have ω(g x ) ≥ ω(g). −1 Hence ω(g) = ω((g x )x ) ≥ ω(g x ) ≥ ω(g). Lemma 26.4. For any sequence (g1 , . . . , gr ) in G \ {1} the following assertions are equivalent: i. The elements σ(g1 ), . . . , σ(gr ) ∈ gr G are linearly independent over Fp [P ]; ii. ω(g1x1 · · · grxr ) = min1≤i≤r (ω(gi ) + v(xi )) for any x1 , . . . , xr ∈ Zp ; iii. ω(g −1 h) = min1≤i≤r (ω(gi ) + v(xi − yi )) for g := g1x1 · · · grxr , h := g1y1 · · · gryr , and any x1 , . . . , xr , y1 , . . . , yr ∈ Zp . Proof. i. =⇒ ii. By axiom (b) and Remark 26.3 we have ω(g1x1 · · · grxr ) ≥ ν := min1≤i≤r (ω(gi ) + v(xi )). Let us therefore suppose that g1x1 · · · grxr ∈ Gν+ . If {i1 < · · · < is } = {1 ≤ i ≤ r : ω(gixi ) = ω(gi ) + v(xi ) = ν} xi x then already gi1 1 · · · gisis ∈ Gν+ since Gν+ is normal in G. It follows that xi x σ(gi1 1 ) + · · · + σ(gisis ) = 0 in grν G. By Remark 26.3 the latter is in fact a nontrivial linear relation in gr G of the form (¯ ai1 P v(xi1 ) )(σ(gi1 )) + · · · + (¯ ais P v(xis ) )(σ(gis )) = 0 which contradicts i. ii. =⇒ i. We consider any relation q1 (P )(σ(g1 )) + · · · + qr (P )(σ(gr )) = 0  ¯ij P j in Fp [P ] where the coefficients in gr G with polynomials qi (P ) = j a a ¯ij are the images in Fp of integers aij ∈ Z. Let νi := ω(gi ). Inserting the definitions we may rewrite this relation more explicitly as r   i=1

a pj

gi ij G(νi +j)+ = 0.

j

If we collect, for any ν, the terms in grν G then we obtain r i=1

ai,ν−νi pν−νi

gi

∈ Gν+

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where we set ai,ν−νi := 0 if ν − νi is not a nonnegative integer. But by ii. the value of ω on this latter product has to be equal to min (ω(gi ) + ν − νi + v(ai,ν−νi )) = ν + min v(ai,ν−νi )

1≤i≤r

1≤i≤r

which therefore is strictly bigger than ν. It follows that a ¯i,ν−νi = 0 for any i and then, since ν was arbitrary, that all a ¯ij = 0. Hence q1 (P ) = · · · = qr (P ) = 0. iii. =⇒ ii. This implication is trivial. ii. =⇒ iii. Let ν := min1≤i≤r (ω(gi ) + v(xi − yi )). Since Gν /Gν+ is central in G/Gν+ by Remark 23.1 we compute iteratively g −1 hGν+ = gr−yr · · · g1−y1 g1x1 · · · grxr Gν+ = gr−yr · · · g2−y2 g1x1 −y1 g2x2 · · · grxr Gν+ = g1x1 −y1 (gr−yr · · · g3−y3 g2x2 −y2 g3x3 · · · grxr )Gν+ .. . = g1x1 −y1 · · · grxr −yr Gν+ . It follows that ω(g −1 h) = ω((g1x1 −y1 · · · grxr −yr )k) with ω(k) > ν and ω(g1x1 −y1 · · · grxr −yr ) = ν by ii. Hence (36) implies that ω(g −1 h) = ν. Proposition 26.5. If (G, ω) is of finite rank then for any sequence of elements (g1 , . . . , gr ) in G \ {1} the following assertions are equivalent: i. (g1 , . . . , gr ) is an ordered basis of (G, ω); ii. σ(g1 ), . . . , σ(gr ) is a basis of the Fp [P ]-module gr G. Proof. First we assume i. to hold true. Then σ(g1 ), . . . , σ(gr ) are linearly independent by Lemma 26.4. To see that they generate gr G as an Fp [P ]module let g = g1x1 · · · grxr be any element = 1 in G, and let {i1 < · · · < is } = {1 ≤ i ≤ r : ω(gixi ) = ω(g)}. Then xi

x

xi

x

σ(g) = σ(gi1 1 · · · gisis ) = σ(gi1 1 ) + · · · + σ(gisis ) ais P v(xis ) )(σ(gis )) = (¯ ai1 P v(xi1 ) )(σ(gi1 )) + · · · + (¯ by Remark 26.3.

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We now assume, vice versa, that ii. holds true. Because of Lemma 26.4 it only remains to be seen that any g ∈ G can be written in the form g = g1x1 · · · grxr . In other words, if X ⊆ G denotes the image of the continuous map Zrp −→ G

c:

(x1 , . . . , xr ) −→ g1x1 · . . . · grxr then we have to show that X = G. As the continuous image of a compact space, X is closed in G. Hence it suffices to establish that, given any g = 1 in G and any ν > 0, there exists a g  ∈ X such that gGν = g  Gν . Step 1: By ii. we may write σ(g) = q1 (P )(σ(g1 ))+· · ·+qr (P )(σ(gr )) with appropriate polynomials qi (P ) ∈ Fp [P ]. By comparing degrees we see that, more precisely, there is a sequence 1 ≤ i1 < · · · < is ≤ r and p-adic units a1 , . . . , as ∈ Z× ¯1 , . . . , a ¯s ∈ F× p with images a p such that all ω(g) − ω(gij ) are nonnegative integers and we have σ(g) = (¯ a1 P ω(g)−ω(gi1 ) )(σ(gi1 )) + · · · + (¯ as P ω(g)−ω(gis ) )(σ(gis )) xi

x

= σ(gi1 1 · · · gisis )

with xij := aj pω(g)−ω(gij ) . xi

x

In this way we have found an element g (1) := gi1 1 · · · gisis ∈ X such that gGω(g)+ = g (1) Gω(g)+ . But note that in addition we have (38)

xi

x

ω(g) = ω(g (1) ) = ω(gi1 1 ) = · · · = ω(gisis ).

Step 2: We write g = g (1) h with ω(h) > ω(g). If h = 1 we have g ∈ X. Otherwise we apply the first step to h and obtain the element h(1) ∈ X such that gGω(h)+ = g (1) hGω(h)+ = g (1) h(1) Gω(h)+ . Using the additional property (38) for h(1) and the centrality of Gω(h) /Gω(h)+ in G/Gω(h)+ , by Remark 23.1, we conclude that g (1) h(1) Gω(h)+ = g (2) Gω(h)+

for some g (2) ∈ X.

Hence g = g (2) k with ω(k) > ω(h). Again, if k = 1 we have g ∈ X, and otherwise we apply the first step to k. Proceeding inductively in this way we either arrive at g ∈ X after finitely many steps or we construct an infinite sequence g (1) , . . . , g (m) , . . . in X as well as a strictly increasing sequence of real numbers ω(g) =: ν1 < · · · < νm < · · · such that gGνm + = g (m) Gνm +

for any m ≥ 1.

It remains to note that, since the set of values of ω is discrete by Lemma 26.1, we have νm ≥ ν for any sufficiently big m.

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We point out that the implication from i. to ii. in the above proposition does not require the assumption that (G, ω) is of finite rank. It therefore shows that the existence of an ordered basis implies the finite rank property. Proposition 26.6. Any (G, ω) of finite rank has an ordered basis (of length equal to rank(G, ω)). Proof. The graded Fp [P ]-module gr G is free of some finite rank r. In view of Prop. 26.5 all that remains to be shown is that we find a basis of gr G consisting of homogeneous elements. For this it suffices to consider any of the direct summands M := (gr G)(ν). Let ν1 := min{ν + : ≥ 0, grν+ G = 0}. Then M = Mν1 ⊕ Mν1 +1 ⊕ · · · with Mν1 + := grν1 + G. We define M (1) ⊆ M to be the Fp [P ]-submodule generated by Mν1 which, in fact, is equal to M (1) = Mν1 ⊕ P (Mν1 ) ⊕ P 2 (Mν1 ) ⊕ · · · Since M is torsionfree this shows that any Fp -basis of Mν1 is an Fp [P ]-basis of M (1) . It also shows that on the quotient module M/M (1) the multiplication by P again is injective. Hence, by the argument in the proof of Remark 25.2, the module M/M (1) remains torsionfree. We therefore may repeat this process by defining M (2) ⊆ M to be the submodule such that M (2) /M (1) is generated by its homogeneous part of smallest degree ν2 > ν1 . In this way we construct an increasing sequence of submodules M (1) ⊆ M (2) ⊆ · · · ⊆ M (s) = M (which necessarily is finite by the finite generation of M ) with the property that each step has an Fp [P ]-basis consisting of homogeneous elements. By lifting all this basis elements to homogeneous elements of M we obtain such a basis for M . We point out that the method of proof of Prop. 26.6 in fact shows that gr G always is a free Fp [P ]-module and that it has a basis consisting of homogeneous elements; the only difference is that the increasing sequence M (i) of finitely generated free submodules in the above proof is exhaustive but possibly not finite. Corollary 26.7. If (G, ω) is of finite rank then G is topologically finitely generated. Corollary 26.8. The members of an ordered basis of (G, ω) cannot be p-th powers in G.

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Proof. Let (g1 , . . . , gr ) be any ordered basis and suppose that gi0 = hp with h ∈ G for some 1 ≤ i0 ≤ r. By Prop. 26.5 we have σ(h) = q1 (P )(σ(g1 )) + · · · + qr (P )(σ(gr )) with appropriate polynomials qi (P ) ∈ Fp [P ]. Hence σ(gi0 ) = P (σ(h)) = (P q1 (P ))(σ(g1 )) + · · · + (P qr (P ))(σ(gr )). This contradicts the fact that, by Prop. 26.5, the σ(g1 ), . . . , σ(gr ) are linearly independent over Fp [P ]. There is an important additional property a p-valuation can have. Note p that ω(g p ) > p−1 for any g ∈ G. Definition. (G, ω) is called saturated if any g ∈ G such that ω(g) > a p-th power in G.

p p−1

is

n

Remark 26.9. If (G, ω) is saturated then {g p : g ∈ G} = G(n+ 1 )+ is a p−1 subgroup for any n ≥ 0. Lemma 26.10. (G, ω) is saturated if and only if P (gr G) = ⊕ν>

p p−1

grν G.

Proof. The necessity of the condition is obvious. Vice versa let g ∈ G be any p . By assumption we have element with ν0 := ω(g) > p−1 g = g1p k1

with ν1 := ω(k1 ) > ν0

as well as k1 ∈ hp1 Gν1 + . Hence g ∈ g1p hp1 Gν1 + . Note that ω(g1p ) = ν0 < ν1 = ω(hp1 ). Prop. 25.1.i. therefore implies g1p hp1 Gν1 + = (g1 h1 )p Gν1 + . We see that g = g2p k2

with g2 := g1 h1 and ν2 := ω(k2 ) > ν1 .

We now repeat this reasoning for k2 and continue inductively. In this way we obtain a sequence g1 , . . . , gm , . . . in G as well as a strictly increasing sequence of real numbers ω(g) = ν0 < · · · < νm < · · · such that p Gνm−1 + gGνm−1 + = gm

for any m ≥ 1.

Using Prop. 25.1.ii. we deduce −p −1 p ω(gm+1 gm ) + 1 = ω(gm+1 gm ) > νm−1

for any m ≥ 1.

Since the sequence (νm )m is unbounded by Lemma 26.1 it follows that the sequence (gm )m converges to some h ∈ G and that hp = g.

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Proposition 26.11. Suppose that (G, ω) is of finite rank with ordered basis (g1 , . . . , gr ); then it is saturated if and only if 1 p < ω(gi ) ≤ p−1 p−1

for any 1 ≤ i ≤ r.

Proof. By Cor. 26.8 the gi are not p-th powers. Hence, if (G, ω) is saturated, p they must satisfy ω(gi ) ≤ p−1 . Let us now assume, vice versa, that the latter inequalities are satisfied. Then the Fp [P ]-basis σ(g1 ), . . . , σ(gr ) of gr G (cf. p Prop. 26.5) lies in ⊕ν≤ p grν G. Therefore each grν G for ν > p−1 must lie p−1 in the image of the operator P . It follows that the assumption of Lemma 26.10 is satisfied, and (G, ω) consequently is saturated. Corollary 26.12. Suppose that (G, ω) is of finite rank and saturated and let (g1 , . . . , gr ) be an ordered basis; for any n ≥ 0 we then have: n

i. Gp = {g1x1 · · · grxr : v(x1 ), . . . , v(xr ) ≥ n}; ii. two elements g1x1 · · · grxr and g1y1 · · · gryr lie in the same coset modulo n Gp if and only if v(x1 − y1 ), . . . , v(xr − yr ) ≥ n; n

iii. [G : Gp ] = pn rank(G,ω) . n

Proof. By Remark 26.9 we have g1x1 · · · grxr ∈ Gp if and only if v(xi ) > 1 1 − ω(gi ) for any 1 ≤ i ≤ r. But −1 ≤ p−1 − ω(gi ) < 0 according to n + p−1 Prop. 26.11. Hence, since v(xi ) is a nonnegative integer, the above inequalities are equivalent to v(xi ) ≥ n. This shows the first assertion. The second one follows by exactly the same argument based on Lemma 26.4.iii. The last assertion is an immediate consequence of the second one. We now turn to the question of variation of the p-valuation on the pro-pgroup G. But we do keep our initial p-valuation which defines the topology   of G and is of finite rank. To avoid confusion we write grων G and grω G for the associated graded Lie algebra when it is formed with respect to any other p-valuation ω  on G. There is one obvious possibility of modifying ω. Let C be any real number which satisfies

 1 0 < C < min ω(g) − . g=1 p−1 We note that by Lemma 26.1 this minimum exists and is strictly bigger than 1 p−1 so that such constants C do exist. It is straightforward to see that ωC (g) := ω(g) − C

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is another p-valuation on G which defines the topology of G. But, in fact, the axiom (c) holds in the stronger form (39)

ωC ([g, h]) > ωC (g) + ωC (h)

for any g, h ∈ G. This, of course, means that the Lie algebra grωC G is abelian. This simple observation will later on turn out to be very useful so that we record it as part of the subsequent lemma. Lemma 26.13. Suppose that (G, ω) is of finite rank ; for any 0 < C < 1 (ming=1 ω(g)) − p−1 we have: i. grωC G is an abelian Lie algebra; ii. rank(G, ωC ) = rank(G, ω); iii. suppose that there is an ordered basis (g1 , . . . , gr ) of (G, ω) such that max ω(gi ) − min ω(gi ) < 1; i

i

p then (G, ωC ) is saturated provided in addition C ≥ (maxi ω(gi )) − p−1 (and such C exists).

Proof. i. This was the observation (39). ii. It is clear that the Fp [P ]-modules grω G and grωC G coincide up to a shift of the degrees of their homogeneous components. iii. It also is clear that (g1 , . . . , gr ) is an ordered basis of (G, ωC ) as well. We have ωC (gj ) = ω(gj ) − C ≤ ω(gj ) − max ω(gi ) + i

p p ≤ . p−1 p−1

Hence Prop. 26.11 applies and shows that (G, ωC ) is saturated. Lemma 26.14. Suppose that (G, ω) is of finite rank and pick any ordered basis (g1 , . . . , gr ); for any ν ≥ maxi ω(gi ) and any sufficiently small > 0, ω  (g) := ω(g) − ν +

1 + p−1

is a p-valuation on Gν which defines the topology of Gν and such that (Gν , ω  ) is of finite rank (equal to rank(G, ω)) and is saturated.

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Proof. We have g1x1 · · · grxr ∈ Gν if and only if v(xi ) ≥ ν − ω(gi ) for any 1 ≤ i ≤ r. Itn follows that (h1 , . . . , hr ) is an ordered basis of (Gν , ω|Gν ) i where hi := gip and ni is the smallest integer greater or equal to ν − ω(gi ). Note that ni ≥ 0 since ν − ω(gi ) ≥ 0 by assumption. We have ν ≤ ω(gi ) + ni = ω(hi ) < ν + 1 so that ω|Gν satisfies the assumption in Lemma 26.13.iii. By this lemma (for 1 1 C := ν − − p−1 ) it therefore remains to observe that ν ≥ p−1 + and

 max ω(hi ) − i

p p <ν+1− − p−1 p−1 1 =ν− − p−1

 1 < min ω(hi ) − i p−1

whenever > 0 is sufficiently small. Formulated a little less technical the above lemma says that if (G, ω) is of finite rank then there is an open normal subgroup H ⊆ G and a p-valuation ω  on H which defines the topology of H and such that (H, ω  ) is of finite rank and saturated. The above two lemmas, which in themselves are rather simple, have important consequences. n

Proposition 26.15. If (G, ω) is of finite rank then the subgroups Gp , for n ≥ 1, form a fundamental system of open neighbourhoods of the identity element in G, and n

v([G : Gp ]) . n→∞ n

rank(G, ω) = lim

Proof. Step 1: We assume in addition that (G, ω) is saturated. Then the assertion is an immediate consequence of Remark 26.9 and Cor. 26.12.iii. In fact the second part of the assertion holds in the stronger form that n rank(G, ω) = v([G : Gp ])/n for any n ≥ 1. Step 2: According to Lemma 26.14 we find an open subgroup H ⊆ G together with a p-valuation ω  on H defining the topology and such that n+e ⊆ (H, ω  ) is saturated of rank equal to rank(G, ω). If [G : H] = pe then Gp n p  H for any n ≥ 1. This together with the first step for (H, ω ) implies the

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191

first part of the assertion for G as well as, for any n ≥ e, n

n

n

n

v([G : Gp ]) = v([H : H p ]) + v([G : H]) − v([Gp : H p ]) = n rank(H, ω  ) + e − v([Gp : H p ]) n

n

n

n

= n rank(G, ω) + e − v([Gp : H p ]) ≥ n rank(G, ω) + e − v([H p

n−e

n

: H p ])

= n rank(G, ω) + e − e rank(H, ω  ) and hence n

e v([G : Gp ]) e e − rank(H, ω  ) ≤ ≤ rank(G, ω) + . n n n n Passing to the limit with respect to n therefore results in the asserted equation. rank(G, ω) +

Proposition 26.16. Suppose that (G, ω) is of finite rank ; then any other p-valuation ω  on G defines the topology of G and satisfies rank(G, ω  ) = rank(G, ω). n

Proof. By Prop. 26.15 the Gp form a fundamental system of open neighbourhoods of the identity element. The axiom (d) therefore implies that the topology defined by ω  necessarily is coarser than the topology of G. But G is compact and the topology defined by ω  is Hausdorff. Hence the latter 1 must be equal to the topology of G. In particular, there is a ν0 > p−1 such that 

Gων0 ⊆ Gp

(40) 

where Gων0 denotes the filtration of G with respect to the p-filtration ω  .  If we establish that grω G is finitely generated over Fp [P ] then the limit formula in Prop. 26.15 shows that rank(G, ω  ) = rank(G, ω). By the remark  after Prop. 26.6, whether grω G is finitely generated or not, there is a family  {hi : i ∈ I} ⊆ G \ {1} such that the σ(hi ) form a basis of grω G over Fp [P ]. The argument in the proof of Cor. 26.8 shows that none of the hi is a p-th  power in G. Hence it follows from (40) that ωω (hi ) < ν0 for any i ∈ I. We see  that all σ(hi ) are contained in 0<ν<ν0 grν G, which is finite since G/Gων0 is finite. It follows that I has to be finite. As promised earlier we now have seen that the rank of (G, ω) actually is independent of the choice of ω. Hence, from now on we simply call G of finite rank and write rank G. The meaning of this invariant of G will become clearer in the next section. We also introduce the following terminology.

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Definition. A pro-p-group G is called p-valuable if there exists a p-valuation ω on G which defines the topology on G and such that the rank of G is finite.

27

Compact p-Adic Lie Groups

The question of the existence of p-valuable groups is overdue. It will be answered in this section with the help of the theory of p-adic Lie groups. By a p-adic Lie group G we always will mean a Lie group G over the field Qp of p-adic numbers (cf. Sect. 13). We recall that this means that G is a (locally analytic) manifold over Qp as well as a group such that the group multiplication mG : G × G −→ G (g, h) −→ gh is a map of manifolds. As a manifold G has a certain finite dimension dim G which also can be characterized as the dimension as a Qp -vector space of the tangent space Th (G) of G at any point h ∈ G. In the following it will be convenient to some times denote the unit element in G by e instead of 1. Theorem 27.1. Let G be any p-adic Lie group; then there exist a compact open subgroup G ⊆ G and an integral valued p-valuation ω on G defining the topology of G such that we have: i. (G , ω) is saturated ; ii. rank G = dim G. Proof. Let d := dim G. We pick a chart c = (U, ϕ, Qdp ) for G around the unit element e ∈ G such that ϕ(e) = 0. By the continuity of the multiplication map mG there is an open neighbourhood V ⊆ U of e such that mG (V ×V ) ⊆ U . Then (V, ϕ|V, Qdp ) is a chart around e as well, and the map ϕ ◦ mG ◦ (ϕ−1 × ϕ−1 ) : ϕ(V ) × ϕ(V ) −→ ϕ(U ) is locally analytic. By expanding this map around 0 we find a d-tuple (F1 , . . . , Fd ) of power series  Fi (X, Y ) = ci,α,β X α Y β α,β

in variables X = (X1 , . . . , Xd ) and Y = (Y1 , . . . , Yd ) with coefficients ci,α,β ∈ Qp (here, as usual, α, β run over all multi-indices and X α = X1α1 · · · Xdαd if α = (α1 , . . . , αd )) such that, for any sufficiently big n > 0, we have

27

Compact p-Adic Lie Groups

193

– pn Zdp ⊆ ϕ(V ),

 – lim|α|+|β|→∞ v(ci,α,β ) + n(|α| + |β|) = ∞ for any 1 ≤ i ≤ d, and – (F1 (x, y), . . . , Fd (x, y)) = ϕ(ϕ−1 (x)ϕ−1 (y)) for any x = (x1 , . . . , xd ) and y = (y1 , . . . , yd ) in pn Zdp (here |α| = α1 + · · · + αd ). By possibly increasing n further one has (Lemma 18.7) v(ci,α,β ) + n(|α| + |β|) ≥ n for any α, β, and i. This implies that the ϕ−1 (pn Zdp ), for sufficiently big n, are compact open subsets in G which are multiplicatively closed. By repeating this reasoning for the power series expansion of the inverse map g −→ g −1 around e one sees that these ϕ−1 (pn Zdp ) for large n actually are subgroups of G. We now pick any of these large n and replace the chart ϕ by the chart ψ := p−n ϕ. This has the effect that the new coefficients ci,α,β as well as the new coefficients of the power series for g −→ g −1 lie in Zp . By composition of power series then also the maps g −→ g p and (g, h) −→ [g, h] have power series expansions (with respect to the chart ψ) converging on all of Zdp with coefficients in Zp (Prop. 5.4). We put G := ψ −1 (p2 Zdp ) and ω(g) := + δ

if g ∈ ψ −1 (p+1 Zdp ) \ ψ −1 (p+2 Zdp )

with δ := 1 for p = 2 and δ := 0 for p = 2. Obviously ω satisfies the axioms (a) and (b) of a p-valuation, the latter since we already know that G+δ = ψ −1 (p+1 Zdp ) is a subgroup of G for any ∈ N. We note that Fi (X, Y ) = Xi + Yi + terms of degree ≥ 1 both in X and Y for any 1 ≤ i ≤ d (Prop. 18.6.i.). This implies that ψ actually induces isomorphisms of groups gr+δ G ∼ = p+1 Zdp /p+2 Zdp and, in particular, that [G : G+δ ] = pd(−1) for any ∈ N. It also shows that power series expansion of g −→ g p = pX + terms of degree ≥ 2

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from which one deduces immediately the axiom (d) for ω. Finally, the fact that the commutator map (g, h) −→ [g, h] has value e if one of the components is equal to e translates into its power series expansion consisting entirely of terms which are of degree ≥ 1 in both X and Y . This implies the axiom (c). (We point out that it was here in case p = 2 that the p2 in the definition of G is needed.) Hence we have established that ω is an integral valued p-valuation on G . It obviously defines the topology of G . For any ∈ N the operator P : gr+δ G −→ gr+δ+1 G is an injective map between finite abelian groups of the same cardinality. Hence it is bijective. This shows that gr G is finitely generated by gr1+δ G as an Fp [P ]module. It also allows for applying Lemma 26.10 to conclude that (G , ω) is saturated. At last, we use Prop. 26.15 in order to see that v([G : Gp ]) n→∞ n v([G : Gn+1+δ ]) = lim n→∞ n dn = lim n→∞ n = d = dim G. n

rank G = lim

This result gives us, of course, an abundance of p-valuable groups. In fact, it gives all of them. We will show in Sect. 29 that any p-valuable group has a natural structure of a p-adic Lie group.

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Chapter VI

Completed Group Rings of p-Valued Groups Throughout this part we fix a p-valuable group G together with a p-valuation ω on it. We recall that O is our complete discrete valuation ring. We always will assume that p is a prime element of O. In particular, Zp ⊆ O and the discrete valuation of K extends v. It therefore will also be denoted by v.

28

The Ring Filtration

Let us pick an ordered basis (g1 , . . . , gr ) of (G, ω). We then have the homeomorphism ∼

Zrp −−→ G

c:

(x1 , . . . , xr ) −→ g1x1 · . . . · grxr . It induces, by pulling back functions, an isomorphism of O-modules 

c∗ : C(G) −−→ C(Zrp ) and hence dually, by Lemma 21.1, an isomorphism of O-modules 

c∗ : Λ(Zrp ) −−→ Λ(G). Of course, this is not a ring isomorphism in general. Alternatively it can be obtained by applying the universal property Prop. 19.2 to the maps c and c−1 . This shows that c∗ is a topological isomorphism of pseudocompact O-modules. We have computed Λ(Zrp ) explicitly in Prop. 20.1 as the formal power series ring in variables X1 , . . . , Xr over O. This leads to the following description of Λ(G). We introduce the elements bi := gi −1 and bα := bα1 1 ·. . .·bαr r , for any multi-index α = (α1 , . . . , αr ), in Λ(G). Then 

 c : O[[X1 , . . . , Xr ]] −−→ Λ(G)   cα X α −→ cα bα α

α

P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8 6, © Springer-Verlag Berlin Heidelberg 2011

195

196

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Completed Group Rings of p-Valued Groups

is a topological isomorphism of pseudocompact O-modules. We emphasize that both series constitute convergent expansions in the respective pseudocompact ring. This allows us to define a function ω  : Λ(G) \ {0} −→ [0, ∞) by

 ω 

 α

 cα bα

 := inf α

v(cα ) +

r 

 αi ω(gi ) .

i=1

With the convention that ω  (0) := ∞ it obviously satisfies ω (λ1 ), ω  (λ2 )) ω  (λ1 + λ2 ) ≥ min(  lie in the additive submonoid of for any λ1 , λ2 ∈ Λ(G). The values of ω [0, ∞) generated by 1 and the values of ω. Because of Lemma 26.1 this set of values is a discrete subset of [0, ∞). In particular, we have     r   α ω  = min v(cα ) + cα b αi ω(gi ) . α

α

i=1

We introduce, for any real number ν ≥ 0, the O-submodules  (λ) ≥ ν} Λ(G)ν := {λ ∈ Λ(G) : ω and  (λ) > ν} Λ(G)ν+ := {λ ∈ Λ(G) : ω in Λ(G). Remark 28.1. The Λ(G)ν form a fundamental system of open neighbourhoods of zero in the pseudocompact topology of Λ(G). Proof. Let m denote the maximal ideal in the ring O[[X1 , . . . , Xr ]], and recall that its pseudocompact topology is the m-adic one. We pick any integer M ≥ maxi ω(gi ). Then        1 v(cα ) + αi ≥ v(cα ) + αi ω(gi ) ≥ αi . M v(cα ) + p−1 i

i

i

It follows that c(mn )  c(mM (p−1)n ) ⊆ Λ(G)M n ⊆ 

for any n ∈ N.

28

The Ring Filtration

197

In order to understand this filtration we compare it with a less explicit but seemingly more conceptual filtration (Jν )ν≥0 where Jν is defined to be the smallest closed O-submodule of Λ(G) which contains all elements p (h1 − 1) · · · (hs − 1) for any , s ≥ 0 and h1 , . . . , hs ∈ G with  + ω(h1 ) + · · · + ω(hs ) ≥ ν. The density of O[G] in Λ(G) implies J0 = Λ(G). We obviously have Jν ⊇ Jν  , if ν ≤ ν  ,

and Jν · Jν  ⊆ Jν+ν  .

In particular, each Jν is a two-sided ideal in Λ(G) which depends on ω but not on the choice of the ordered basis (g1 , . . . , gr ). It also is clear that Λ(G)ν ⊆ Jν . In particular, by Remark 28.1, the Jν are open in Λ(G). By defining  Jν  and grν Λ(G) := Jν /Jν+ Jν+ := ν  >ν

we obtain the graded algebra gr Λ(G) :=



grν Λ(G)

ν≥0

over the graded ring gr O :=



pn O/pn+1 O.

n≥0

We point out that the discreteness of the set of values of ω implies that the set {ν : grν Λ(G) = 0} is discrete as well. Next we observe that for g, h ∈ Gν we have g − 1, h − 1 ∈ Jν as well as (gh − 1) + Jν+ = (g − 1) + (h − 1) + (g − 1)(h − 1) + Jν+ = (g − 1) + (h − 1) + Jν+ .

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Hence Lν : grν G −→ grν Λ(G) gGν+ −→ (g − 1) + Jν+ is a homomorphism of abelian groups. We put  L := Lν : gr G −→ gr Λ(G). ν

Secondly, for g = 1, we observe that g p − 1 + J(ω(g)+1)+ = ((g − 1) + 1)p − 1 + J(ω(g)+1)+ p   p (g − 1)j + J(ω(g)+1)+ = j j=1

= p(g − 1) + p(g − 1)

p−2 

Z(g − 1)j + (g − 1)p + J(ω(g)+1)+

j=1

= p(g − 1) + J(ω(g)+1)+ , where, for the last identity, we have used that pω(g) > ω(g) + 1 by axiom (a). This shows that L is a homomorphism of graded Fp [P ]-modules with respect to the Fp -algebra map Fp [P ] → gr O which sends P to p + p2 O ∈ gr1 O. Thirdly we consider, for any g, h = 1, the identity (g − 1)(h − 1) − (h − 1)(g − 1) = gh − hg = ([g, h] − 1)hg. Since ([g, h] − 1)(hg − 1) ∈ Jω([g,h]) · Jω(hg) ⊆ Jω(g)+ω(h) · Jmin(ω(g),ω(h)) ⊆ J(ω(g)+ω(h))+ we deduce that (g − 1)(h − 1) − (h − 1)(g − 1) + J(ω(g)+ω(h))+ = ([g, h] − 1)hg + J(ω(g)+ω(h))+ = [g, h] − 1 + J(ω(g)+ω(h))+ . This finally implies that L is a homomorphism of graded Lie algebras over Fp [P ]. By the universal property of the universal enveloping algebra U (gr G) ([B-LL] Chap. I §2.1 Prop. 1 or §14) the map L therefore extends uniquely to a homomorphism of graded associative gr O-algebras L : gr O ⊗Fp [P ] U (gr G) −→ gr Λ(G).

28

The Ring Filtration

Lemma 28.2.



199

ν≥0 Jν

= 0.

Proof. By Prop. 19.7 the pseudocompact topology of Λ(G) coincides with the m(G)-adic topology. Let n ∈ N be arbitrary but fixed. We find an N ∈ N (G) such that Jn,N (G) ⊆ m(G)n . We note that h − 1 ∈ Jn,N (G) for any h ∈ N . Next we choose a ν0 > 0 such that Gν0 ⊆ N . Then h − 1 ∈ m(G)n whenever ω(h) ≥ ν0 . We put ν := n(ν0 + 1) and claim that Jν ⊆ m(G)n . Let p (h1 −1) · · · (hs −1) be any element with +ω(h1 )+· · ·+ω(hs ) ≥ ν. We may assume that  < n and ω(h1 ), . . . , ω(hs ) < ν0 . But then n + sν0 > ν = n + nν0 . Hence we have s > n and therefore p (h1 − 1) · · · (hs − 1) ∈ m(G)n . Theorem 28.3.

i. The map ∼

= L : gr O ⊗Fp [P ] U (gr G) −−→ gr Λ(G)

is an isomorphism. ii. Jν = Λ(G)ν for any ν ≥ 0.  By Thm. 22.3.ii. the sum of Proof. We first establish the surjectivity of L. two closed submodules in a pseudocompact module is closed. This implies that we have  Op (h1 − 1) · · · (hs − 1) Jν = Jν+ + +ω(h1 )+···+ω(hs )=ν

for any ν ≥ 0. It follows that gr Λ(G) as a gr O-algebra is generated by the  elements (g − 1) + Jω(g)+ , for g ∈ G \ {1}, which all lie in the image of L. The injectivity will be shown alongside with the assertion ii. By Prop. 26.5 the elements σ(g1 ), . . . , σ(gr ) form a basis of the Fp [P ]-module gr G. The Poincar´e-Birkhoff-Witt theorem (cf. [B-LL] Chap. I §2.7 Cor. 3) then implies that the (ordered) monomials σ(g1 )α1 · . . . · σ(gr )αr with α = (α1 , . . . , αr ) running over all multi-indices form an Fp [P ]-basis of U (gr G). This implies, by the surjectivity which we know already, that grν Λ(G) is generated as a O/pO-vector space by the finitely many elements p (g1 − 1)α1 · · · (gr − 1)αr + Jν+

for  + α1 ω(g1 ) + · · · + αr ω(gr ) = ν.

For the injectivity we therefore must see that these elements are O/pOlinearly independent. But first we draw, from this generation property, the conclusion that  Jν = Jν+ + Op (g1 − 1)α1 · · · (gr − 1)αr +α1 ω(g1 )+···+αr ω(gr )=ν

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and hence Jν = Jν+ + Λ(G)ν holds true for any ν ≥ 0. But the set of all ν such that grν Λ(G) = 0 is discrete in [0, ∞). We therefore get by an inductive argument that Jν = Jν  + Λ(G)ν

for any ν  > ν.

Since Λ(G)ν is closed in Λ(G) by Remark 28.1 we may apply Lemma 22.2 and, moreover using Lemma 28.2, obtain    Jν = Jν  + Λ(G)ν = Λ(G)ν . ν  >ν

Now suppose that we have a relation of the form  cα p (g1 − 1)α1 · · · (gr − 1)αr ∈ Jν+ = Λ(G)ν+ +α1 ω(g1 )+···+αr ω(gr )=ν

for some cα ∈ O. Then  ν<ω 





cα p (g1 − 1) 

α1

· · · (gr − 1)

α1 ω(g1 )+···+αr ω(gr )=ν−



=

min α1 ω(g1 )+···+αr ω(gr )=ν−

v(cα ) +  +

r 

αr

 αi ω(gi )

i=1

= ν + min v(cα ). α

It follows that all cα lie in pO which is the linear independence over O/pO which we wanted. Corollary 28.4. i. The function ω  only depends on ω and not on the choice of the ordered basis (g1 , . . . , gr ). ii. ω  (g − 1) = ω(g) for any g ∈ G. Proof. i. This follows from Thm. 28.3.ii. ii. The map L : gr G −→ ⊕ν Λ(G)ν /Λ(G)ν+ is injective as a consequence of Thm. 28.3. Corollary 28.5.

i. gr Λ(G) is an integral domain.

 (λ1 ) + ω  (λ2 ) for any λ1 , λ2 ∈ Λ(G). ii. ω  (λ1 λ2 ) = ω

29

Analyticity

201

iii. Λ(G) is an integral domain. Proof. i. Using Thm. 28.3.i. this follows from the fact that the universal enveloping algebra of a Lie algebra which is free over the base ring is an integral domain (cf. [B-LL] Chap. I §2.7 Cor. 7) provided the base ring is an integral domain. ii. Because, by Thm. 28.3.ii., the Λ(G)ν satisfy Λ(G)ν ·Λ(G)ν  ⊆ Λ(G)ν+ν  we have ω  (λ1 λ2 ) ≥ ω  (λ1 ) + ω  (λ2 ). The equality then is an immediate consequence of the first assertion. iii. This is a direct consequence of the second assertion. Corollary 28.6. If the p-valuation ω is such that the Lie algebra gr G is abelian then gr Λ(G) is a polynomial ring in 1+rank G variables over O/pO. Proof. By construction, the universal enveloping algebra of an abelian Lie algebra g is the symmetric algebra over g. In our case gr G is free over Fp [P ] of rank equal to rank G. Hence U (gr G) is a polynomial ring in rank G variables over Fp [P ]. We now apply Thm. 28.3.i.

29

Analyticity

As an application of the methods of computation in the completed group ring Zp [[G]], which we have established in the last section, we now will be able to see that our p-valuable group G naturally is a p-adic Lie group. As before we pick an ordered basis (g1 , . . . , gr ) of (G, ω) so that we have the homeomorphism c:

∼

Zrp −−→ G (x1 , . . . , xr ) −→ g1x1 · . . . · grxr .

If ϕ denotes the inverse of c then we may view the triple (G, ϕ, Qrp ), which by abuse of notation we also denote by c, as a “global” chart for G. By equipping G with the maximal atlas equivalent to the atlas consisting of the single chart c we obtain the structure of an r-dimensional manifold over Qp on G (cf. Sects. 7 and 8). Equivalently this structure is characterized by the fact that it makes the original map c into an isomorphism of manifolds. We also will need the individual “coordinate functions” ϕi : G −→ Qp g −→ xi

if g = g1x1 · . . . · grxr

for 1 ≤ i ≤ r. They, by construction, are locally analytic.

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Remark 29.1. For any h ∈ G and any y ∈ Zp we have the convergent expansion  y y (h − 1)n h = n n≥0

in Zp [[G]]. Proof. We recall the definition  y y(y − 1) · · · (y − n + 1) := for n ≥ 1 n! n and y0 := 1. Because of the integrality of the usual binomial coefficients and the density of Z in Zp we have ny ∈ Zp . We write y = limj→∞ mj as the limit of a sequence of integers mj ∈ Z. Then hy = hlim mj = lim hmj (in G) = lim hmj (in Zp [[G]]) j→∞

j→∞

= lim ((h − 1) + 1) j→∞  mj = lim (h − 1)n j→∞ n n≥0   mj (h − 1)n = lim j→∞ n n≥0  y (h − 1)n = n mj

n≥0

where, in order to exchange the limit and the sum, we have used that limn→∞ ω  ((h − 1)n ) = ω(h) limn→∞ n = ∞. Proposition 29.2. Let h1 , . . . , hs be any elements then, for any  in G; α in s variables 1 ≤ i ≤ r, there is a power series Fi (Y ) = α ci,α Y Y = (Y1 , . . . , Ys ) with coefficients in Qp such that we have: i. lim|α|→∞ v(ci,α ) = ∞; ii. Fi (y1 , . . . , ys ) ∈ Zp for any (y1 , . . . , ys ) ∈ Zsp ; F (y1 ,...,ys )

iii. hy11 · · · hyss = g1 1

In particular, the functions

F (y1 ,...,ys )

· · · gr r

for any (y1 , . . . , ys ) ∈ Zsp .

29

Analyticity

203

Zsp −→ Qp (y1 , . . . , ys ) −→ ϕi (hy11 · · · hyss ), for 1 ≤ i ≤ s, are locally analytic. Proof. We note that the condition i. assures that the power series Fi converges on Zsp so that the assertion in ii. makes sense. By multiplying the expansions in Remark 29.1 we obtain    x1 xr α x1 xr (41) g 1 · · · gr = ··· b α1 αr α and in the same way (42)

hy11

· · · hyss

=

  y1 β1

β

 ys ··· (h1 − 1)β1 · · · (hs − 1)βs . βs

We now insert into the right hand side of (42) the “standard” expansions  cα,β bα (43) (h1 − 1)β1 · · · (hs − 1)βs = α

in Zp [[G]] and obtain (44)

hy11

· · · hyss

=

    y1 α

β1

β

 ys ··· cα,β bα . βs

By comparing (41) and (44) for xi = ϕi (hy11 · · · hyss ) we deduce     ϕ1 (hy11 · · · hyss ) ϕr (hy11 · · · hyss ) y1 ys ··· = ··· cα,β . β1 βs α1 αr β

Moreover, applying Cors. 28.4.ii. and 28.5.ii. to (43) gives   r s   β1 βs αi ω(gi ) = ω  ((h1 − 1) · · · (hs − 1) ) = βj ω(hj ). min v(cα,β ) + α

i=1

j=1

In particular, we see that, for any fixed α, we indeed have lim|β|→∞ v(cα,β ) = ∞. We now specialize to the multi-index α = i := (. . . , 0, 1, 0, . . .) which has a one in the i-th place, and we get    y1 ys y1 ys ϕi (h1 · · · hs ) = ··· ci,β β1 βs β

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and v(ci,β ) ≥ −ω(gi )+

s  j=1

Using that v(n!) ≤ 

 s s   βj 1 . βj ω(hj ) = −ω(gi )+ βj ω(hj )− + p−1 p−1

n p−1

j=1

j=1

(cf. Lemma 2.2) we obtain the estimate

ci,β v β1 ! · · · βs !

≥ −ω(gi ) +

s  j=1

 βj

1 . ω(hj ) − p−1

It therefore follows from axiom (a) that  ci,β lim v = ∞. β1 ! · · · βs ! |β|→∞ On the other hand Fβ (Y1 , . . . , Ys ) :=

s 

Yj (Yj − 1) · · · (Yj − βj + 1)

j=1

is a polynomial of degree |β| with coefficients in Z satisfying   y1 ys Fβ (y1 , . . . , ys ) = ··· β1 ! · · · βs !. β1 βs We conclude that the power series  Fi := β

ci,β Fβ β1 ! · · · βs !

has all the asserted properties. Remark 29.3. If (G, ω) is saturated then the power series F1 , . . . , Fr in Prop. 29.2 have coefficients in Zp . Proof. From the proof of Prop. 29.2 (and with its notations) we know that  s s   ci,β ≥ −ω(gi ) − v(βj !) + βj ω(hj ) v β1 ! · · · βs ! j=1

and that it suffices to show that  ci,β ≥ 0. v β1 ! · · · βs !

j=1

29

Analyticity

205

But the left hand side is an integer so that we, in fact, need only to establish that  ci,β > −1. v β1 ! · · · βs ! Moreover, by looking at (43) we see that ci,0 = 0. Hence we may assume that 1 . By βj0 = 0 for some 1 ≤ j0 ≤ s. Axiom (a) implies M := minj ω(hj ) > p−1 p Prop. 26.11 we have ω(gi ) ≤ p−1 . Therefore showing that −

  p v(βj !) + M βj > −1 − p−1 j

j

or, equivalently, that −

  1 v(βj !) + M βj > 0 − p−1 j

j

is sufficient. The left hand side is equal to      βj βj0 − 1 1 βj + − v(βj !) + − v(βj0 !) M− p−1 p−1 p−1 j

j =j0

1 )βj0 > 0 since which is greater or equal to (M − p−1

and v(βj0 !) ≤

βj0 −1 p−1

1 p−1

< M , v(βj !) ≤

βj p−1 ,

(cf. Lemma 2.2).

Corollary 29.4. The structure of G as a manifold over Qp is independent of the choice of ω and the choice of an ordered basis (g1 , . . . , gr ). Proof. Apply Prop. 29.2 to any ordered basis (h1 , . . . , hr ) with respect to a possibly different p-valuation. It follows that the coordinate changes between any two ordered bases are locally analytic maps. Hence the corresponding two global charts belong to the same maximal atlas. Corollary 29.5. The multiplication map mG is a morphism of manifolds. Proof. This follows by applying Prop. 29.2 to the sequence of elements (h1 , . . . , h2r ) := (g1 , . . . , gr , g1 , . . . , gr ). Corollary 29.6. Any p-valuable pro-p-group G carries a natural structure of a compact p-adic Lie group, and dim G = rank G. In fact, as we will see presently, a much stronger statement holds true.

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Lemma 29.7. Let G be a p-adic Lie group which is pro-p; for any g ∈ G we have: i. The map cg : Zp −→ G x −→ g x is a homomorphism of Lie groups; ii. if g is the image of y ∈ Lie(G) under some exponential map for G (cf. Cor. 18.19) then T0 (cg ) : T0 (Zp ) = Qp −→ Lie(G) x −→ xy. Proof. In Sect. 26 we have seen that cg is a well defined continuous homomorphism of groups. It therefore suffices to show that its restriction cg |pn Zp , for some n ≥ 0, is a locally analytic map. Equivalently it suffices to show n that cg is a locally analytic map after we have replaced g by g p for some n ≥ 0. The continuity of cg implies n

lim g p = cg ( lim pn ) = cg (0) = 1.

n→∞

n→∞

Using Cor. 18.19 we therefore may replace G by Gε , for some sufficiently small ε > 0, where {Gε }ε denotes the Campbell-Hausdorff Lie group germ of Lie(G). But in this case it follows from Prop. 17.2 by induction that g m = mg for any integer m ≥ 0. Since N0 is dense in Zp we obtain by continuity that, for Gε , the map cg coincides with the obviously locally analytic map Zp −→ Gε ⊆ Lie(G) x −→ xg. It remains to note that the associated map between tangent spaces at 0 simply is Qp −→ Lie(G) x −→ xg.

Theorem 29.8. Any p-valuable pro-p-group G carries a unique structure of manifold over Qp which makes it into a p-adic Lie group. Proof. Let G1 denote G viewed as a p-adic Lie group according to Cor. 29.6. We write G2 for G equipped with any other (fixed) structure of a p-adic Lie

29

Analyticity

207

group. We emphasize that our only assumption is that the identity map idG → G2 is an isomorphism of abstract groups. The assertion claims G1 −−− that idG then necessarily is an isomorphism of Lie groups. We begin by picking an ordered basis (g1 , . . . , gr ) of G1 . Then, by construction, ∼

Zrp −−→ G1

c1 :

(x1 , . . . , xr ) −→ g1x1 · . . . · grxr is an isomorphism of manifolds. Viewed as a map c1 : Zrp −→ G2 it at least is locally analytic as a consequence of Lemma 29.7.i. This implies that idG → G2 is a homomorphism of p-adic Lie groups. Next we will show G1 −−− that the tangent map Lie(idG ) is bijective. We also pick a Qp -basis y1 , . . . , ys of Lie(G2 ). By replacing the yj by appropriate nonzero scalar multiples we may assume that this basis lies in the domain of definition of some exponential map expG2 ,ε for G2 . We put hj := expG2 ,ε (yj ). According to Lemma 29.7 the map Zsp −→ G2

c2 :

(y1 , . . . , ys ) −→ hy11 · . . . · hyss is locally analytic with associated tangent map (cf. Cor. 13.5) T0 (c2 ) : T0 (Zsp ) = Qsp −→ Lie(G2 ) (y1 , . . . , ys ) −→ y1 y1 + · · · + ys ys . In particular, T0 (c2 ) is bijective. Since, by the same lemma, c2 also can be viewed as a locally analytic map Zsp −→ G1 there exists, by Prop. 29.2, a locally analytic map ψ : Zsp −→ Zrp such that the diagram Zrp

c1

G1

ψ

Zsp

idG c2

G2

is commutative. On tangent spaces we obtain Qrp

∼ =

Lie(G1 )

T0 (ψ)

Qsp

Lie(idG ) ∼ =

Lie(G2 )

which shows that Lie(idG ) is surjective.

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Completed Group Rings of p-Valued Groups

On the other hand, by Cor. 29.6 we have r = dim G1 = rank G1 . Moreover, by Thm. 27.1, there exists an open subgroup G ⊆ G which is p-valuable such that s = dim G2 = rank G . Using Prop. 26.16 and Ex. 26.2 we conclude that s = rank G = rank G1 = r. This establishes that Lie(idG ) is an isomorphism of Lie algebras. We finally use Prop. 18.17 (consult the proof of Cor. 18.18) in order to see that there is an open subgroup U ⊆ G such that idG |U is an isomorphism of Lie groups. It follows immediately that idG has to be an isomorphism of Lie groups as well.

30

Saturation

Our goal in this section is to show that the pair (G, ω) always can be embedded as an open subgroup into another pair (G , ω  ) which is saturated. For this purpose it suffices to consider Zp as the basic discrete valuation ring. Therefore, throughout this section we let O = Zp . To motivate the later strategy we begin with the following observation. We have three continuous group homomorphisms ι1 : G −→ G × G,

Δ : G −→ G × G, g −→ (g, g)

ι2 : G −→ G × G.

g −→ (g, 1)

g −→ (1, g)

They induce corresponding continuous monomorphisms between completed group algebras Δ∗ , ι1∗ , and ι2∗ : Λ(G) −→ Λ(G × G). Remark 30.1. G = {λ ∈ Λ(G) \ {0} : Δ∗ (λ) = ι1∗ (λ) · ι2∗ (λ)}. Proof. Obviously G is contained in the right hand side. In order to show equality we may  assume, by a projective limit argument, that G is finite. Let 0 = λ = g∈G cg g be any element in the right hand side. We then have  g

 cg (g, g) =

 g

 cg (g, 1)

 g

 cg (1, g)

=



cg ch (g, h).

g,h

Comparing the coefficients gives cg = c2g and cg ch = 0 if g = h. Since λ = 0 there must be at least one nonzero coefficient cg0 . The first identity then implies that cg0 = 1 and the second that ch = 0 for any h = g0 . Hence λ = g0 .

30

Saturation

209

The strategy in the following will be to embed Λ(G) into a bigger algebra and, by turning, for this bigger algebra, the Remark 30.1 into a definition, to obtain a bigger group. We start with the Qp -algebra Qp ⊗Zp Λ(G) any element of which can be written as a ⊗ λ with a ∈ Q× p and λ ∈ Λ(G). This allows us to extend our function ω  to Qp ⊗Zp Λ(G) by the rule ω  (a ⊗ λ) := v(a) + ω  (λ). Using for a moment multiplicative notation we set a ⊗ λ ω := p−eω(a⊗λ) . Then the pair (Qp ⊗Zp Λ(G), ω ) is a normed Qp -algebra whose norm ω , by Cor. 28.5.ii., in fact is multiplicative. We let (ΛQp (G, ω), ω ) denote the corresponding completion which then is a Qp -Banach algebra with multiplicative norm. We emphasize that, by Remark 28.1, the norm topology on ΛQp (G, ω) induces on Λ(G) the compact topology. Having said this we revert to additive notation and we keep writing ω  also for its continuous extension to ΛQp (G, ω). We may visualize ΛQp (G, ω) as a normed vector space more concretely by picking an ordered basis (g1 , . . . , gr ) of (G, ω). Then ΛQp (G, ω) is the Qp -vector space of all expansions   r   α v(cα ) + cα b with cα ∈ Qp and lim αi ω(gi ) = ∞, |α|→∞

α



and ω 



 cα bα

i=1

 = min v(cα ) + α

α

r 

 αi ω(gi ) .

i=1

We point out that, in general, the set of values of ω  on ΛQp (G, ω) is no longer discrete. Example. Let G = Zp , for some p = 2, be the additive group of p-adic integers with the p-valuation ω(pn u) := n + e for n ≥ 0 and u ∈ Z× p , where e ≥ 1 is a fixed integer. According to Prop. 20.1 we may view Λ(Zp ) as the formal power series ring Zp [[X]] in one variable X over Zp . In this picture we  have ω  ( n≥0 cn X n ) = minn (v(cn ) + ne). Phrased more analytically Zp [[X]] is the ring of rigid Qp -analytic functions on the open unit disk which are bounded by one. On the other hand    ΛQp (Zp , ω) = cn X n ∈ Qp [[X]] : lim (v(cn ) + ne) = ∞  =

n→∞

n

 n

cn



1 X pe



n ∈ Qp

1 X p



 : lim v(cn ) = ∞ n→∞

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Completed Group Rings of p-Valued Groups

can be seen as the ring of all rigid Qp -analytic functions on the smaller closed disk of radius p1e . We equip the group G × G with the p-valuation ω2 (g, h) := min(ω(g), ω(h)). Exercise. G × G is of finite rank equal to twice the rank of G. The group homomorphisms Δ, ι1 , and ι2 are isometric with respect to ω and ω2 . We claim that the ring homomorphisms Δ∗ , ι1∗ , and ι2∗ are isometric for ω  and ω 2 . To see this let (g1 , . . . , gr ) be an ordered basis of (G, ω). We will take ((g1 , 1), . . . , (gr , 1)(1, g1 ), . . . , (1, gr )) as an ordered basis for G × G. It then is immediately visible that ι1∗ and ι2∗ are isometries. To treat Δ∗ we first have to compute it explicitly. For this purpose we recall the following notational conventions about multi-indices: β + α := (β1 + α1 , . . . , βr + αr ), α≤β

(45)

if αi ≤ βi for any 1 ≤ i ≤ r,

β − α := (β1 − α1 , . . . , βr − αr ) if α ≤ β, and α! := α1 ! . . . αr !.

Lemma 30.2. We have Δ∗ (bα ) =

 β,γ≤α

α! ι1∗ (bβ )ι2∗ (bγ ) (α − β)!(α − γ)!(β + γ − α)!

β+γ≥α

for any α. Proof. First of all, for any g ∈ G, we compute Δ∗ (g − 1) = (g, g) − 1 = (g, 1)(1, g) − 1 = ((g, 1) − 1)((1, g) − 1) + ((g, 1) − 1) + ((1, g) − 1) = ι1∗ (g − 1)ι2∗ (g − 1) + ι1∗ (g − 1) + ι2∗ (g − 1). Next, quite generally, one checks, either by applying the binomial formula twice or by induction with respect to n, that (46)

(xy + x + y)n =

 0≤j,≤n

j+≥n

n! xj y  (n − j)!(n − )!(j +  − n)!

30

Saturation

211

holds true for any n ≥ 0 (the coefficients being integers of course). We deduce the formula Δ∗ (bαi ) =

 0≤βi ,γi ≤n

αi ! ι1∗ (bβi )ι2∗ (bγi ) (αi − βi )!(αi − γi )!(βi + γi − αi )!

βi +γi ≥n

for any 1 ≤ i ≤ r. Multiplying these latter formulae together gives the assertion. Observe that we constantly are using that any element in the image of ι1∗ commutes with any element in the image of ι2∗ . By this lemma we obtain the explicit formula        (47) Δ∗ cα bα = mβ,γ,α cα ι1∗ (bβ )ι2∗ (bγ ) α

β,γ

β,γ≤α≤β+γ

 for any λ = α cα bα ∈ Λ(G) where the mα,β,γ are certain integers (not depending on λ). We see that   ω 2 (Δ∗ (λ)) = min v β,γ

 

mβ,γ,α cα

β,γ≤α≤β+γ

≥ min v cα + α



 r 





r  + (βi + γi )ω(gi ) i=1

αi ω(gi )

i=1

=ω  (λ). It remains to observe that for β = 0 the only possibility for α is α = γ in which case the coefficient is m0,γ,γ = 1. Hence we actually have ω (λ), min (· · · )) ≤ ω  (λ) ω 2 (Δ∗ (λ)) = min( β =0,γ

which shows that ω 2 (Δ∗ (λ)) = ω  (λ). We now have established that the ring homomorphism Δ∗ is isometric as well. This enables us to extend Δ∗ , ι1∗ , and ι2∗ first by linearity to Qp ⊗Zp Λ(G) and then by density to isometric homomorphisms of Qp -Banach algebras from ΛQp (G, ω) into ΛQp (G × G, ω2 ) which we will denote by the same symbols. The identity (47) remains valid for λ ∈ ΛQp (G).

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Completed Group Rings of p-Valued Groups

Definition. We put σω (G) := {λ ∈ ΛQp (G, ω) \ {0} : Δ∗ (λ) = ι1∗ (λ)ι2∗ (λ)}. Let us examine the expansions of elements in σω (G) more closely. For this we will need the following elementary identity. Remark 30.3. For any integers j,  ≥ 0 and any x ∈ Qp we have     x x x n! . =  j (n − j)!(n − )!(j +  − n)! n j,≤n≤j+

Proof. First we assume that x ≥ 0 is an integer. Then, using the binomial formula together with (46), we compute the identity  x x Y j Z  = (Y + 1)x (Z + 1)x = (Y Z + Y + Z + 1)x  j 0≤j,≤x  x (Y Z + Y + Z)n = n n≥0  x  n! = Y jZ (n − j)!(n − )!(j +  − n)! n n≥0

j,≤n≤j+

between polynomials in two variables Y and Z. Comparing coefficients we obtain     x x x n! = (n − j)!(n − )!(j +  − n)! n j  j,≤n≤j+

for any integer x ≥ 0. But this means we actually have the identity     X X X n! = (n − j)!(n − )!(j +  − n)! n j  j,≤n≤j+

between polynomials in one variable X with rational coefficients where  X X(X − 1) · · · (X − n + 1) := . n! n Of course, we may insert into this latter identity now any x ∈ Qp . We recall that i = (. . . , 0, 1, 0, . . .) denotes the multi-index with the entry 1 in the i-th place and zero elsewhere. Let e1 , . . . , er ≥ 0 be the unique integers such that 1 1 < ω(gi ) − ei ≤ 1 + . p−1 p−1

30

Saturation

213

Proposition 30.4. The map {(y1 , . . . , yr ) ∈ Qrp : v(yi ) ≥ −ei for any i} −→ σω (G)    y1 yr y = (y1 , . . . , yr ) −→ λy := ··· bα α α 1 r α is a bijection which satisfies ω  (λy − 1) = min (v(yi ) + ω(gi )) > 1≤i≤r

1 . p−1

Proof. We recall once more (cf. Lemma 2.2) that v(n!) ≤ n−1 p−1 for any n ∈ N.y1 Let yyr=

(y1 , . . . , yr ) be a point in the left hand side, and put cα := · · · α1 αr . If v(yi ) < 0 then  yi v + αi ω(gi ) = αi (v(yi ) + ω(gi )) − v(αi !) αi  αi − 1 1 + v(yi ) + ω(gi ) = − v(αi !) + (αi − 1) v(yi ) + ω(gi ) − p−1 p−1  1 + v(yi ) + ω(gi ) ≥ (αi − 1) ω(gi ) − ei − p−1 for any αi ≥ 1. If v(yi ) ≥ 0 then  yi v + αi ω(gi ) ≥ v(yi ) + αi ω(gi ) − v(αi !) αi  αi − 1 1 = v(yi ) + ω(gi ) + − v(αi !) + (αi − 1) ω(gi ) − p−1 p−1  1 + v(yi ) + ω(gi ) ≥ (αi − 1) ω(gi ) − ei − p−1 for any αi ≥ 1. Since ω(gi ) − ei −

1 p−1

 lim

|α|→∞

v(cα ) +

> 0 this implies that

r 

 αi ω(gi )

=∞

i=1

and that v(cα ) +

r  i=1

αi ω(gi ) ≥ min (v(yi ) + ω(gi )) > 1≤i≤r

1 p−1

for any α = 0.

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Completed Group Rings of p-Valued Groups

Hence λy is a well defined nonzero (as c0 = 1) element in ΛQp (G, ω) with ω  (λy − 1) = min (v(yi ) + ω(gi )) > 1≤i≤r

1 . p−1

The Remark 30.3 together with (47) implies Δ∗ (λy ) = ι1∗ (λy )ι2∗ (λy ). Hence clear since ci = yi . λy lies in σω (G). The injectivity of the map is  α For the surjectivity of the map let λ = α cα b be any element in σω (G). Because of (47) the coefficients cα satisfy the relation  α! cα = cβ cγ . (α − β)!(α − γ)!(β + γ − α)! β,γ≤α≤β+γ

For β = γ = 0 we in particular obtain c20 = c0 . Moreover, for β = 0 the only possibility for α in the left hand sum is α = γ. It follows that cγ = c0 cγ for any γ. Since λ = 0 by assumption we cannot have c0 = 0. Therefore c0 = 1. For β = i the above identity reads γi cγ + (γi + 1)cγ+i = ci cγ or, equivalently,

ci − γi cγ . γi + 1 From this it follows inductively that we have   c1 cr cα = ··· α1 αr cγ+i =

for any α. In order to verify that v(ci ) ≥ −ei we observe that cni is the coefficient for the multi-index (. . . , 0, n, 0, . . .) with n in the i-th place. We therefore have  ci lim v + nω(gi ) = ∞. n→∞ n The asserted ci inequality being trivial otherwise we may assume that v(ci ) < 0. Then v( n ) = nv(ci ) − v(n!) and hence lim n(v(ci ) + ω(gi )) − v(n!) = ∞.

n→∞

For n = pm we have v(pm !) =

pm −1 p−1

and therefore obtain

pm − 1 ∞ = lim pm (v(ci ) + ω(gi )) − m→∞ p−1  1 m . = lim p v(ci ) + ω(gi ) − m→∞ p−1

30

Saturation

215

1 It follows that v(ci ) > p−1 − ω(gi ). Since v(ci ) is an integer we must have v(ci ) ≥ −ei . This shows that λ = λy for y := (c1 , . . . , cr ).

Since Δ∗ , ι1∗ , and ι2∗ are ring homomorphisms the set σω (G) is invariant under multiplication and under passing to the multiplicative inverse (if  1 it exists). If ω  (λ − 1) > p−1 then the geometric series n≥0 (1 − λ) converges in ΛQp (G, ω) and provides a multiplicative inverse λ−1 for λ. Since the λ ∈ σω (G) satisfy this assumption by Prop. 30.4 we see that σω (G) is a subgroup of the group of units in ΛQp (G, ω). We equip σω (G) with the topology induced by the norm topology of ΛQp (G, ω). The multiplication in σω (G) is continuous since it is the multiplication in the Banach algebra ΛQp (G). As a consequence of Cor. 28.5.ii. any unit λ ∈ ΛQp (G, ω)× satisfies ω  (λ−1 ) = − ω (λ) = 0. Hence ω  (λ−1 − λ−1 ) = ω  (λ−1 ) + ω  (λ−1 ) + ω  (λ − λ ) = ω  (λ−λ ) for these units. We see that σω (G) is a Hausdorff topological group. Of course it contains G as a compact subgroup. We define the function ω on σω (G) by ω(λ) := ω  (λ − 1). By Prop. 30.4 it satisfies axiom (a) for a p-valuation. The other axioms (b) to (d) follow by a computation of exactly the same type as we have done already twice, in Example 23.2 and before Lemma 28.2, and which we will not repeat here. That ω defines the topology of σω (G) is clear. Finally, by Cor. 28.4.ii., this function ω restricts to the original p-valuation ω on G. We collect the results of this discussion so far. Lemma 30.5. σω (G) is a Hausdorff topological group carrying the p-valuation ω which defines its topology and which restricts to the p-valuation ω on the compact subgroup G. By Prop. 30.4 we have in σω (G) the elements hi :=

 p−ei n≥0

n

bni

Lemma 30.6. For any 1 ≤ i ≤ r we have: i.

1 p−1 ei

< ω(hi ) ≤

ii. hpi = gi .

p p−1 ;

for 1 ≤ i ≤ r.

216

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Completed Group Rings of p-Valued Groups

Proof. i. This is immediate form the definition of the ei and either the formula for ω  in Prop. 30.4 or the second assertion by axiom (d). ii. By ei Prop. 30.4 the element hpi is completely determined by the coefficient of bi in its expansion. This coefficient is easily computed to be one which also is the coefficient belonging to gi . Hence the asserted equality. Theorem 30.7. of G.

i. σω (G) is p-valuable of finite rank equal to the rank

ii. h1 , . . . , hr is an ordered basis of (σω (G), ω). iii. G is an open subgroup of σω (G). iv. (σω (G), ω) is saturated. v. If (G, ω) is saturated then σω (G) = G. Proof. If y ∈ Zrp then, by definition, λy lies in Λ(G) and hence, by Remark 30.1, in G. The formula for ω  in Prop. 30.4 then implies that σω (G)ν ⊆ G for any sufficiently big ν. In particular, G is open in σω (G). Let now Hi ⊆ σω (G), for any 1 ≤ i ≤ r, denote the closed subgroup topologically generated by hi . On the one hand Hi ∩ G is open in the commutative group Hi . On the other hand Hi /Hi ∩ G then is a finite cyclic group generated by the coset of hi (cf. Lemma 30.6). Therefore Hi is profinite and the map ⊆

Zp −→ Hi − → σω (G) x −→ hxi is well defined and continuous. In combination this gives the continuous map c:

Zrp −→ σω (G) (x1 , . . . , xr ) −→ hx1 1 · . . . · hxr r .

We claim that c((x1 , . . . , xr )) = λ(p−e1 x1 ,...,p−er xr ) holds true for any (x1 , . . . , xr ) ∈ Zrp . Since, by definition, we have λ(y1 ,...,yr ) = λ(y1 ,0,...,0) · . . . · λ(0,...,0,yr )

30

Saturation

217

it suffices to show that hxi = λ(...,0,p−ei x,0,...)

for any x ∈ Zp .

By Prop. 30.4 this amounts to showing that 0, . . . , 0, p−ei x, 0, . . . , 0 are the coefficients of  b1 , . . . , br , respectively, in the expansion of hxi . We observe that the map α cα bα −→ cβ , for any fixed β, is a continuous linear form on the Banach space ΛQp (G, ω). Hence we are reduced to computing these coefficients in the case when x ∈ Z is an integer. But then the asserted answer is clear from the defining expansion of hi . This establishes our claim. It now follows from Prop. 30.4 that c is a continuous bijection. In particular, σω (G) is compact with the open profinite subgroup G and therefore itself is profinite. Moreover, it follows that ω(hx1 1 · · · hxr r ) = ω  (λ(p−e1 x1 ,...,p−er xr ) − 1) = min (v(xi ) − ei + ω(gi )) 1≤i≤r

= min (v(xi ) + ω(hi )). 1≤i≤r

1 < Hence (h1 , . . . , hr ) is an ordered basis of (σω (G), ω). Because of p−1 p ω(hi ) ≤ p−1 Prop. 26.11 says that (σω (G), ω) is saturated. Finally, if (G, ω) already is saturated then ei = 0 and hence hi = gi which implies σω (G) = G.

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Chapter VII

The Lie Algebra We keep fixing a p-valuable group G with a p-valuation ω on it. We also continue to assume that O = Zp .

31

A Normed Lie Algebra

In Sect. 30 we have introduced the Qp -Banach algebra ΛQp (G, ω) with the (additively written) norm ω  on it. Inside the group of units of ΛQp (G, ω) we found the subgroup of group-like elements σω (G) = {λ ∈ ΛQp (G, ω) \ {0} : Δ∗ (λ) = ι1∗ (λ)ι2∗ (λ)}. In this section we will study the vector subspace Lω (G) := {λ ∈ ΛQp (G, ω) : Δ∗ (λ) = ι1∗ (λ) + ι2∗ (λ)} of primitive elements. The associative algebra ΛQp (G, ω) is a Lie algebra over Qp in the usual way by the Lie bracket being the additive commutator [λ, μ] := λμ − μλ. We have ω  ([λ, μ]) ≥ ω  (λ) + ω  (μ)

for any λ, μ ∈ ΛQp (G, ω).

Since elements in the image of ι1∗ and ι2∗ , respectively, commute with each other a one line computation shows that Lω (G) is a Lie subalgebra of ΛQp (G, ω). Of course, Lω (G) comes equipped with the norm ω . In order to investigate this normed Lie algebra we will use the fact that the exponential and logarithm power series converge on large parts of the Banach algebra ΛQp (G, ω). Let λ ∈ ΛQp (G, ω). 1 First we suppose that ω  (λ) > p−1 . Then     1 n 1 ω  = n ω (λ) − v(n!) ≥ n ω  (λ) − λ for any n ≥ 0 n! p−1 and, if λ = 0,   n−1 1 n = n ω (λ) − v(n!) ≥ n ω (λ) − ω  λ n! p−1   1 1 + =n ω  (λ) − >ω  (λ) p−1 p−1

for any n ≥ 2.

P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8 7, © Springer-Verlag Berlin Heidelberg 2011

219

220

VII

Hence exp(λ) :=

The Lie Algebra

 1 λn n!

n≥0

converges in ΛQp (G, ω) and satisfies ω  (exp(λ) − (1 + λ)) > ω  (λ) (if λ = 0). In particular, ω  (exp(λ) − 1) = ω  (λ). On the other hand suppose that ω  (λ − 1) > 0. Then (exercise!)   1 n→∞ (λ − 1)n = n ω (λ − 1) − v(n) −−−−→ ∞. ω  n Hence log(λ) :=

 (−1)n+1 n≥1

n

(λ − 1)n

1 converges in ΛQp (G, ω). If λ = 1 and ω  (λ − 1) > p−1 then   1 n−1 n (λ − 1) ω  = n ω (λ − 1) − v(n!) + v((n − 1)!) ≥ n ω (λ − 1) − n p−1   1 1 + >ω  (λ − 1) =n ω  (λ − 1) − p−1 p−1

for any n ≥ 2 and hence ω  (log(λ) − (λ − 1)) > ω  (λ − 1).

(48) In particular,

ω  (log(λ)) = ω  (λ − 1)

whenever ω  (λ − 1) >

1 . p−1

Exercise. Where they are defined the maps exp and log are continuous and, 1 if ω  (λ), ω  (λ ), ω  (μ − 1), ω  (μ − 1) > p−1 , they satisfy: – log(exp(λ)) = λ, and exp(log(μ)) = μ; – exp(λ + λ ) = exp(λ) exp(λ ) and log(μμ ) = log(μ) + log(μ ) whenever λλ = λ λ and μμ = μ μ, respectively; – exp(mλ) = exp(λ)m and log(μm ) = m log(μ) for any m ∈ Z.

31

A Normed Lie Algebra

221

1 Altogether, we see that exp and log define, for any ν > p−1 , homeomorphisms which are inverse to each other between the subsets

{λ : ω  (λ) ≥ ν}

exp

{λ : ω  (λ − 1) ≥ ν}

log

in ΛQp (G, ω). Lemma 31.1. The maps exp and log restrict to homeomorphisms Lω (G)

1 + p−1

 := λ ∈ Lω (G) : ω  (λ) >

1 p−1



exp

σω (G) log

which are inverse to each other ; we have for any h ∈ σω (G) and x ∈ Zp .

log(hx ) = x log(h)

Proof. By the above discussion we have 

1 λ:ω  (λ) > p−1



exp

 λ:ω  (λ − 1) >

log

 1 . p−1

The maps exp and log, of course, exist in the same way for ΛQp (G × G, ω2 ). By their very construction they commute with any isometric algebra homomorphism from ΛQp (G, ω) into ΛQp (G × G, ω2 ). Consider first an element λ ∈ Lω (G) 1 + . We compute p−1

Δ∗ (exp(λ)) = exp(Δ∗ (λ)) = exp(ι1∗ (λ) + ι2∗ (λ)) = exp(ι1∗ (λ)) · exp(ι2∗ (λ)) = ι1∗ (exp(λ)) · ι2∗ (exp(λ)) where the third identity uses the fact that ι1∗ (λ) and ι2∗ (λ) commute with each other. Hence λ ∈ σω (G). If, on the other hand, we start with an element 1  (λ − 1) > p−1 . We therefore λ ∈ σω (G) then we know from Prop. 30.4 that ω may compute, similarly as above, Δ∗ (log(λ)) = log(Δ∗ (λ)) = log(ι1∗ (λ) · ι2∗ (λ)) = log(ι1∗ (λ)) + log(ι2∗ (λ)) = ι1∗ (log(λ)) + ι2∗ (log(λ)), and we see that log(λ) ∈ Lω (G) 1 + . p−1 The second part of the assertion follows, by continuity, from the case x ∈ Z.

222

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The Lie Algebra

We note that Lω (G) 1 + is closed under the Lie bracket and therefore p−1 can be viewed as a Lie algebra over Zp . Proposition 31.2. For any ordered basis (g1 , . . . , gr ) of (G, ω) the elements log(g1 ), . . . , log(gr ) form a basis of the Qp -vector space Lω (G) satisfying  r  yi log(gi ) = min (v(yi ) + ω(gi )) ω  1≤i≤r

i=1

for any y1 , . . . , yr ∈ Qp . In particular, we have dimQp Lω (G) = rank G. Proof. By Lemma 31.1 the log(gi ) and therefore any linear combination

r y log(gi ) lie in Lω (G). Using (48) we have i i=1  r  yi log(gi ) = min min(v(yi ) − v(n) + nω(gi )) ω  1≤i≤r n∈N

i=1

= min (v(yi ) + min(nω(gi ) − v(n))) 1≤i≤r

n∈N

= min (v(yi ) + ω(gi )). 1≤i≤r

This in particular shows that the log(gi ) are Qp -linearly independent. Let

now λ = α cα bα be any element in Lω (G). From our explicit formula (47) we know that Δ∗ (λ) is equal to    α! cα ι1∗ (bβ )ι2∗ (bγ ) (α − β)!(α − γ)!(β + γ − α)! β,γ

β,γ≤α≤β+γ

= ι1∗ (bβ ) + ι2∗ (bγ ) − c0    α! + cα ι1∗ (bβ )ι2∗ (bγ ). (α − β)!(α − γ)!(β + γ − α)! β,γ =0

β,γ≤α≤β+γ

It follows that c0 = 0 and that  α! cα = 0 (α − β)!(α − γ)!(β + γ − α)! β,γ≤α≤β+γ

For β = i and γ = 0 this identity becomes γi cγ + (γi + 1)cγ+i = 0 or, equivalently, cγ+i = − We inductively conclude that

γi cγ . γi + 1

for any β, γ = 0.

31

A Normed Lie Algebra

223

cα = 0 whenever at least two different entries of the multi-index α are nonzero, and c(...,0,αi ,0,...) =

(−1)αi −1 αi

This amounts to λ =



ci for any 1 ≤ i ≤ r and αi ≥ 1.

1≤i≤r ci log(gi ).

Corollary 31.3. There is an ordered basis (h1 , . . . , hr ) of (σω (G), ω) such that the elements log(h1 ), . . . , log(hr ) form a Zp -basis of Lω (G) 1 + satisp−1 fying  r  xi log(hi ) = min (v(xi ) + ω(hi )) = ω(hx1 1 . . . hxr r ) ω  1≤i≤r

i=1

for any x1 , . . . , xr ∈ Zp . Proof. We pick an ordered basis (g1 , . . . , gr ) of (G, ω). In Thm. 30.7.ii. we have seen the existence of an ordered basis (h1 , . . . , hr ) of (σω (G), ω) such ei p 1 that hpi = gi for the unique integer ei

≥ 0 such that p−1 < ω(gi )−ei ≤ p−1 .

e By using Prop. 31.2 we obtain that i yi p i log(hi ) = i yi log(gi ) lies in 1 1 Lω (G) 1 + if and only if v(yi ) > p−1 −ω(gi ), resp. v(yi pei ) > p−1 −ω(gi )+ei , p−1

for any 1 ≤ i ≤ r. Since v(.) always is an integer and −1 ≤ 0 the latter is equivalent to v(yi pei ) ≥ 0.

1 p−1 −ω(gi )+ei

<

Corollary 31.4. The map Lω (G)

1 + p−1

⊕ Lω (G)

∼ =

1 + p−1

−−−−→ Lω2 (G × G) ι1∗ +ι2∗

1 + p−1

is an isomorphism of Lie algebras over Zp . Proof. We pick an ordered basis (g1 , . . . , gr ) of (G, ω), and we define the intep 1 gers ei ≥ 0 by p−1 < ω(gi ) − ei ≤ p−1 . In the proof of Cor. 31.3 we have seen 1 1 that pe1 log(g1 ), . . . , per log(gr ) is a Zp -basis of Lω (G) 1 + . The exactly same p−1

reasoning applies to the ordered basis ((g1 , 1), . . . , (gr , 1), (1, g1 ), . . . , (1, gr )) of (G × G, ω2 ) and the corresponding integers (e1 , . . . , er , e1 , . . . , er ). Hence we see that     1 1 1 1 log(ι (g )) = ι log(g ) , . . . , log(ι (g )) = ι log(g ) , 1 1 1∗ 1 1 r 1∗ r pe1 pe1 per per     1 1 1 1 log(ι2 (g1 )) = ι2∗ e1 log(g1 ) , . . . , er log(ι2 (gr )) = ι2∗ er log(gr ) pe1 p p p is a Zp -basis of Lω2 (G × G)

1 + p−1

.

224

VII

The Lie Algebra

By the universal property of the universal enveloping algebra (cf. Sect. 14) the inclusion Lω (G) ⊆ ΛQp (G, ω) extends uniquely to a homomorphism of associative Qp -algebras I : U (Lω (G)) −→ ΛQp (G, ω). Theorem 31.5.

i. The map I is injective with dense image.

ii. Let (g1 , . . . , gr ) be an ordered basis of (G, ω); any λ ∈ ΛQp (G, ω) has a unique convergent expansion of the form  λ= dα log(g1 )α1 · · · log(gr )αr with dα ∈ Qp α



such that lim

|α|→∞

v(dα ) +

r 

αi ω(gi )

= ∞,

i=1



and

ω  (λ) = min v(dα ) + α

r 

αi ω(gi ) ;

i=1

conversely, any series as above converges in ΛQp (G, ω). Proof. According to Prop. 31.2 the elements log(g1 ), . . . , log(gr ) form a basis of Lω (G). Hence, by the Poincar´e-Birkhoff-Witt theorem (cf. [B-LL] Chap. I §2.7 Cor. 3), the (ordered) monomials log(g• )α := log(g1 )α1 · · · log(gr )αr , with α = (α1 , . . . , αr ) running over all multi-indices, form a basis of the universal enveloping algebra U (Lω (G)). For the injectivity of I we therefore have to show that these monomials, when viewed in ΛQp (G, ω), are Qp linearly independent. We have the expansions log(gi ) =

 (−1)n+1 n≥1

n

bni

and hence log(g• )α = bα + b-terms of degree > |α|.

Now suppose that there is a finite relation of the form α cα log(g• )α = 0 with m := min{|α| : cα = 0} < ∞. We then have   cα log(g• )α = cα bα + b-terms of degree > m, 0= α

|α|=m

31

A Normed Lie Algebra

225

and we deduce the contradiction that cα = 0 whenever |α| = m. This establishes the claimed linear independence and hence the injectivity of I. Let B ⊆ ΛQp (G, ω) denote the closure of the image of I. With log(g) ∈ L any g ∈ G, by Lemma 31.1 we have g = exp(log(g)) =

ω (G) 1⊆ B, for n ∈ B. Hence Z [G] ⊆ B. Since Z [G] is dense in Λ(G) we log(g) p p n≥0 n! obtain Qp ⊗Zp Λ(G) ⊆ B and therefore B = ΛQp (G, ω). Knowing now that ΛQp (G, ω) is the completion of U (Lω (G)) with respect to ω  it suffices, for the proof of ii., to check that   r   ω  dα log(g• )α = min v(dα ) + αi ω(gi ) α

α

holds true for any finite sum



i=1

α dα log(g•

)α .

We have

 (bi ) = ω(gi ) ω  (log(gi ) − bi ) > ω for any 1 ≤ i ≤ r by (48). It follows inductively that  (b ) = ω  (log(g• ) − b ) > ω α

α

α

r 

αi ω(gi )

i=1

for any α. We now compute       dα log(g• )α − dα bα = ω dα (log(g• )α − bα ) ω  α

α

α





 (log(g• )α − bα ) ≥ min v(dα ) + ω α   (bα ) > min v(dα ) + ω α   α dα b . =ω  α

This implies    r    α α ω  =ω  = min v(dα ) + dα log(g• ) dα b αi ω(gi ) . α

α

α

i=1

We now can prove a converse to Cor. 31.3. Proposition 31.6. Let λ1 , . . . , λr be a Zp -basis of Lω (G) 1 + satisfying p−1  r  xi λi = min (v(xi ) + ω  (λi )) ω  i=1

1≤i≤r

226

VII

The Lie Algebra

for any x1 , . . . , xr ∈ Zp ; we then have: i. Any λ ∈ ΛQp (G, ω) has a unique convergent expansion of the form λ=



dα λα1 1 · · · λαr r

with dα ∈ Qp

α

and

 ω  (λ) = min v(dα ) + α

r 

αi ω  (λi ) ;

i=1

ii. (exp(λ1 ), . . . , exp(λr )) is an ordered basis of (σω (G), ω). Proof. i. By Cor. 31.3 and Thm. 31.5.ii. there exists at least one Zp -basis μ1 , . . . , μr of Lω (G) 1 + which satisfies the assertion. According to the p−1

Poincar´e-Birkhoff-Witt theorem both sets of (ordered) monomials μα• := μα1 1 · · · μαr r

λα• := λα1 1 · · · λαr r

and

form a Qp -basis of the universal enveloping algebra U (Lω (G)). Since U (Lω (G)), by Thm. 31.5.i.,

is dense in the Banach space ΛQp (G, ω) it suffices to show that any λ = α dα λα• ∈ U (Lω (G)) satisfies  (49)

ω  (λ) = min v(dα ) + α

r 

αi ω  (λi )

i=1



= min(v(dα ) + ω  (λα• )) . α

For trivial reasons the left hand side is ≥ the right hand side. We therefore have to establish the reverse inequality ≤. By assumption it holds for all elements λ ∈ Lω (G). We also make the easy observation that if λ = c1 ν1 + · · · + cm νm is a linear combination of elements νj ∈ U (Lω (G)) such that ω  (λ) ≤ min1≤j≤m (v(cj ) + ω  (νj )) then λ satisfies (49) if ν1 , · · · , νm satisfy (49). This applies, for example, to any λ written as a linear combination of the μα• . We consequently are reduced to showing that (49) holds true for all monomials λ := μα1 1 · · · μαr r . If μi = ai1 λ1 + · · · + air λr

with aij ∈ Qp

we have ω  (μi ) = min(v(aij + ω  (λj )) j

31

A Normed Lie Algebra

227

by assumption. It follows that α1 αr  r  r   a1j λj ··· arj λj λ = μα1 1 · · · μαr r = j=1 r 

=

j=1

a1j11 · · · a1j1α1 a2j21 · · · a2j2α2 · · · arjrαr λj11 · · · λjrαr

j11 ,··· ,jrαr =1

with ω  (λ) =

r 

αi ω  (μi ) =

i=1



r  i=1

αi min(v(aij ) + ω  (λj )) j



v(a1j11 ) + · · · + v(arjrαr ) + ω  (λj11 ) + · · · + ω  (λjrαr ) j11 ,...,jrαr  v(a1j11 · · · arjrαr ) + ω  (λj11 · · · λjrαr ) = min =

min

j11 ,...,jrαr

(recall that ω  is “multiplicative”). Hence our easy observation applies and further reduces us to showing that arbitrary products of the form λ := λi1 · · · λin satisfy (49). We will argue by induction with respect to n ≥ 2 that the following two assertions hold true: a) For any 1 ≤ i1 , . . . , in ≤ r and any permutation σ of {1, . . . , n}, if  λiσ(1) · · · λiσ(n) − λi1 · · · λin = dα λα• , α

then min(v(dα ) + ω  (λα• )) ≥ ω  (λi1 ) + · · · + ω  (λin ). α

b) For any 1 ≤ i1 , . . . , in ≤ r the product λ := λi1 · · · λin satisfies (49). First of all we note that a) implies b). If we reorder the product λi1 · · · λin in such a way that the indices are increasing we obtain an element λβ• for some appropriate multi-index β. Let  λi1 · · · λin − λβ• = dα λα• . α

By a) we have  (λα• )). ω  (λi1 · · · λin ) ≤ min(v(dα ) + ω α

228

VII

The Lie Algebra

Since ω  (λi1 · · · λin ) = ω  (λi1 ) + · · · + ω  (λin ) = ω  (λβ• ) it follows that   (λβ• ), min(v(dα ) + ω  (λα• )) . ω  (λi1 · · · λin ) ≤ min v(dβ + 1) + ω α =β

For n = 2 we have λi2 λi1 − λi1 λi2 = [λi2 , λi1 ] ∈ Lω (G) and ω  ([λi2 , λi1 ]) ≥ ω  (λi1 ) + ω  (λi2 ) so that a) holds by assumption. For general n we may assume by another induction with respect to the length of the permutation σ that σ is an elementary transposition. In this case we are looking at a difference of the form λ := λi1 · · · λim−1 (λim+1 λim )λim+2 · · · λin − λi1 · · · λin = λi1 · · · λim−1 [λim+1 , λim ]λim+2 · · · λin . Let [λim+1 , λim ] = and we obtain

r

j=1 dj λj .

λ=

r 

Then ω  ([λim+1 , λim ]) = minj (v(dj ) + ω  (λj )),

dj λi1 · · · λim−1 λj λim+2 · · · λin

j=1

with  ([λim+1 , λim ]) + ω  (λim+2 · · · λin ) ω  (λ) = ω  (λi1 · · · λim−1 ) + ω   (λi1 · · · λim−1 ) + v(dj ) + ω  (λj ) + ω  (λim+2 · · · λin ) = min ω j

 (λi1 · · · λim−1 λj λim+2 · · · λin )). = min(v(dj ) + ω j

By the induction hypothesis all products λi1 · · · λim−1 λj λim+2 · · · λin satisfy (49). Hence our easy observation at the beginning shows that λ satisfies (49) and a fortiori a). ii. Let g ∈ σω (G) be an arbitrary element. According to i. we have an expansion  g= dα λα• , α

of which we will compute the image Δ∗ (g) in two different ways. First of all the definition of σω (G) says that     dβ ι1∗ (λ• )β dγ ι2∗ (λ• )γ Δ∗ (g) = ι1∗ (g)ι2∗ (g) = =

 β,γ

γ

β β

dβ dγ ι1∗ (λ• ) ι2∗ (λ• )

γ

31

A Normed Lie Algebra

229

with the notational convention that ιj∗ (λ• )β := ιj∗ (λ1 )β1 · · · ιj∗ (λr )βr . Secondly, since Δ∗ is a continuous algebra homomorphism we compute (recall the conventions (45)) Δ∗ (g) =



dα Δ∗ (λ1 )α1 · · · Δ∗ (λr )αr

α

=



dα (ι1∗ (λ1 ) + ι2∗ (λ1 ))α1 · · · (ι1∗ (λr ) + ι2∗ (λr ))αr

α

=



dα

α

=

 β+γ=α

 (β + γ)! β,γ

β!γ!

α! ι1∗ (λ• )β ι2∗ (λ• )γ β!γ!

dβ+γ ι1∗ (λ• )β ι2∗ (λ• )γ .

As a consequence of Cor. 31.4 and the unicity part in i. (applied to the Banach algebra ΛQp (G × G, ω2 )) the above two expansions of Δ∗ (g) must coincide termwise. Hence dβ+γ =

β!γ! dβ dγ (β + γ)!

for any β, γ.

In particular, dβ = dβ d0

and dβ+i =

1 dβ di βi + 1

for any β.

Since g = 0 the first identity implies d0 = 1. The second identity then can be used to obtain inductively the formula 1 αi di α! r

dα =

for any α.

i=1

Since ω  (g − 1) = minα =0 (v(dα ) + ω  (λα• )) by i. and ω  (g − 1) > Lemma 31.1 we have v(di ) + ω  (λi ) >

1 p−1

satisfy

<ω  (λi ) ≤ 1 +

1 p−1 .

1 + p−1

necessarily

It follows that v(di ) > −1 and hence that

yi := di ∈ Zp

by

for any 1 ≤ i ≤ r.

On the other hand the members of a Zp -basis of Lω (G) 1 p−1

1 p−1

for any 1 ≤ i ≤ r.

230

VII

The Lie Algebra

Inserting this information into the initial expansion of g gives  1 1 (y1 λ1 )α1 · · · (yr λr )αr α ! α ! 1 r α    1  1 n n = (y1 λ1 ) · · · (yr λr ) n! n!

g=

n≥1

n≥1

= exp(y1 λ1 ) · · · exp(yr λr ) = exp(λ1 )y1 · · · exp(λr )yr . The last identity follows by continuity from the case where the yi are integers. Moreover, we have ω(g) = ω  (g − 1) = min(v(dα ) + ω  (λα• )) α =0      (y1 λ1 )α1 (yr λr )αr = min ω  + ··· + ω  α =0 α1 ! αr ! = min ω  (yi λi ) = min(v(yi ) + ω  (λi )) i

i

 (exp(λi ) − 1)) = min(v(yi ) + ω i

= min(v(yi ) + ω(exp(λi ))). i

Here the fourth identity is a consequence of the fact, which we have discussed n 1  (λ) whenever n ≥ 2 and ω  (λ) > p−1 . earlier, that ω  ( λn! ) > ω Corollary 31.7. If (G, ω) is saturated then the map log induces a bijection between the set of all ordered bases of (G, ω) and the set of all ordered Zp bases (λ1 , . . . , λr ) of Lω (G) 1 + satisfying p−1

ω 

 r  i=1

xi λi

= min (v(xi ) + ω  (λi )) 1≤i≤r

for any x1 , . . . , xr ∈ Zp .

Proof. This is Cor. 31.3 (which, by its proof, applies to any ordered basis of the saturated (G, ω)) and Prop. 31.6.ii. If we define a multiplication • on Lω (G)

1 + p−1

by

λ • μ := log(exp(λ) exp(μ))

31

A Normed Lie Algebra

231

then, for the restriction to Lω (G)

1 + p−1

of the natural topology of the finite

dimensional Qp -vector space Lω (G), the pair Lω (G) 1 + is a topological p−1 group such that exp  σω (G) Lω (G) 1 + , • p−1

log

are isomorphisms of topological groups (inverse to each other). Letting λ•x , for λ ∈ Lω (G) 1 + and x ∈ Zp , denote the p-adic power formed with respect p−1 to the multiplication • then we have λ•x = log(exp(λ)x ) = log(exp(xλ)) = xλ. In particular, λ•(−1) = −λ. Moreover, since ω(g) = ω  (g − 1) = ω  (log(g))

for any g ∈ σω (G)

the maps 

Lω (G)



1 + p−1

,•

exp

σω (G) log

in fact are isomorphisms of topological groups with a p-valuation defining the topology and hence (cf. Thm. 29.8) are isomorphisms of p-adic Lie groups. As such they induce an isomorphism between Lie algebras  Lie Lω (G) 1 + , • ∼ = Lie(σω (G)) = Lie(G) p−1

(the latter identification holds since G is open in σω (G)). On the other hand it follows from Prop. 17.6 that (Lω (G) 1 + , •) is nothing else but the p−1

Campbell-Hausdorff Lie group germ {Gε }ε of the Lie algebra Lω (G). More precisely, let us choose a Zp -basis of Lω (G) 1 + as a basis for Lω (G). The p−1 1

constant ε0 in Prop. 17.6 then is equal to |p| p−1 and   Gε . Lω (G) 1 + , • ⊇ p−1

ε<ε0

We therefore may apply Prop. 17.3 to get  Lie Lω (G) 1 + , • ∼ = Lω (G). p−1

Together we obtain an isomorphism of Lie algebras Lω (G) ∼ = Lie(G).

232

VII

The Lie Algebra

Viewing the latter as an identification the map exp : Lω (G)

1 + p−1

−→ σω (G)

becomes an exponential map for σω (G) in the sense of Cor. 18.19.

32

The Hausdorff Series

Of course, the maps exp

Lω (G)

1 + p−1

σω (G) log

do not, in general, respect the algebraic structures on the two sides, which are of a different nature anyway: Lie algebra versus group. Nevertheless, in a rather subtle way they can be used to transport important information from one side to the other. This comes from a careful application of the p-adic properties of the Hausdorff series.  (d) Let MQ = d≥0 MQ denote the Magnus algebra of associative formal power series in the variables X and Y with coefficients in Q. We will use the same notations as in Sect. 24 except indicating by a subscript Q that now we work with rational coefficients. Inside MQ we have the Lie subalgebra  (d) LQ which is free on {X, Y } and which is graded by LQ = d≥1 LQ ∩ MQ .

1 We view the formal power series exp(X) = d≥0 d! X d and exp(Y ) =

1 d d≥0 d! Y as elements of MQ and form their product exp(X) exp(Y ) ∈ MQ . It is straightforward to see (cf. Prop. 16.2) that then H(X, Y ) =

 (−1)n+1 n≥1

n

(exp(X) exp(Y ) − 1)n

is a well defined element in MQ called the Hausdorff series. If H(X, Y ) = H1 (X, Y ) + · · · + Hd (X, Y ) + · · · (d)

the calculation of its degree d component Hd ∈ MQ only involves finitely many components of exp(X) and exp(Y ). It follows from the CampbellHausdorff formula (cf. Thm. 16.5) that (d)

Hd ∈ LQ ∩ MQ

for any d ≥ 1.

For example, H1 = X + Y

and

1 H2 = (XY − Y X). 2

32

The Hausdorff Series

233 (d)

Each noncommutative polynomial Q(X, Y ) ∈ MQ , for any d ≥ 0, induces, in an obvious way, a well defined map ΛQp (G, ω) × ΛQp (G, ω) −→ ΛQp (G, ω) (λ, λ ) −→ Q(λ, λ ).

 (λ), ω  (λ ) > Theorem 32.1. For any λ, λ ∈ ΛQp (G, ω) such that ω have the convergent expansion  Hd (λ, λ ) log(exp(λ) exp(λ )) =

1 p−1

we

d≥1

in ΛQp (G, ω). Proof. First of all we note that ω  (exp(λ) exp(λ ) − 1)  =ω  (exp(λ) − 1)(exp(λ ) − 1) + (exp(λ) − 1) + (exp(λ ) − 1)   (exp(λ) − 1), ω  (exp(λ ) − 1) ≥ min ω  (exp(λ) − 1) + ω  (exp(λ ) − 1), ω  (λ), ω  (λ )) = min( ω (λ) + ω  (λ ), ω 1 > . p−1 Hence the left hand side in the assertion is well defined. 1 > 0. We more precisely will establish Let C := min( ω (λ), ω  (λ )) − p−1 that  D    ω  log(exp(λ) exp(λ )) − Hd (λ, λ ) ≥ (D + 1)C d=1

holds true for any D ≥ 1. In a first step we use the convergent expansion log(exp(λ) exp(λ )) =

 (−1)n+1 n≥1

n

(exp(λ) exp(λ ) − 1)n .

By the first inequality above and Lemma 2.2 we have   n (−1)n+1  n (exp(λ) exp(λ ) − 1) = nC. ω  ≥ n min( ω (λ), ω  (λ )) − n p−1 This reduces us to showing that  D D  (−1)n+1   n  ω  Hd (λ, λ ) ≥ (D + 1)C. (exp(λ) exp(λ ) − 1) − n n=1

d=1

234

VII

By multiplying together the convergent expansions exp(λ) =

exp(λ ) = s≥0 s!1 λs we obtain the convergent expansions (−1)n+1 (exp(λ) exp(λ )−1)n = n



The Lie Algebra



1 r r≥0 r! λ

and

(−1)n+1 λr1 λs1 λrn λsn ··· . n r1 ! s1 ! rn ! sn !

r1 ,...,rn ≥0 s1 ,...,sn ≥0 r1 +s1 ≥1,...,rn +sn ≥1

On the other hand 

Hd (λ, λ ) =

1≤e≤d r1 ,...,re ,s1 ,...,se ≥0 r1 +s1 ≥1,...,re +se ≥1 r1 +s1 +···+re +se =d

λre λse (−1)e+1 λr1 λs1 ··· . e r1 ! s1 ! re ! se !

Hence D  (−1)n+1

n

n=1

=

(exp(λ) exp(λ ) − 1)n − 

1≤n≤D r1 ,...,rn ,s1 ,...,sn ≥0 r1 +s1 ≥1,...,rn +sn ≥1 r1 +s1 +···+rn +sn >D

D 

Hd (λ, λ )

d=1

λrn λsn (−1)n+1 λr1 λs1 ··· . n r1 ! s1 ! rn ! sn !

It remains to notice that  λrn λsn (−1)n+1 λr1 λs1 ω  ··· n r1 ! s1 ! rn ! sn !

n n n + i=1 (ri + si − 1)  (ri + si ) min( ω (λ), ω  (λ )) ≥− + p−1 i=1

≥

n 

(ri + si )C

i=1

(compare the computation in the proof of Prop. 17.6). For Q(X, Y ) ∈ LQ ∩ MQ the map (λ, λ ) −→ Q(λ, λ ) restricts to a map Lω (G) × Lω (G) −→ Lω (G) (cf. the discussion after Cor. 16.11). In particular, if λ, λ ∈ Lω (G) 1 + then the convergent expansion in Thm. 32.1 (d)

p−1

is an expansion in the finite dimensional Qp -vector space Lω (G).

32

The Hausdorff Series

235

To state finer p-adic properties of the Hd we need to recall certain technical facts about L. Let M(d), for any d ≥ 1 denote the set of nonassociative and noncommutative monomials of degree d in the variables X and Y . It is defined inductively by M(1) := {X, Y } and M(d) := disjoint union of all M(a) × M(b) for a + b = d. We inductively construct maps M(d) −→ L ∩ M(d) x −→ ex by eX := X, eY := Y , and ex := [ey , ez ] if x = (y, z) ∈ M(a) × M(b) ⊆ M(a + b). Since L is the free Lie algebra on eX and eY each L ∩ M(d) , as an abelian group, is generated by {ex }x∈M(d) . The important technical point is (cf. [B-LL] Chap. II §2.10-11 and §8.1 Prop. 1, resp. Prop. 17.4 and the proof of Prop. 17.6) that one can choose subsets B(d) ⊆ M(d) such that: (d)

(H1) {ex }x∈B(d) is a Z-basis of L ∩ M(d) and hence a Q-basis of LQ ∩ MQ ; (H2) if Hd =



x∈B(d) ad,x ex

then v(ad,x ) ≥ − d−1 p−1 for any x ∈ B(d).

We now consider any function  w : Lω (G)

1 + p−1

\ {0} −→



1 ,∞ p−1

which satisfies (with the usual convention that w(0) := ∞) (b+) w(λ − λ ) ≥ min(w(λ), w(λ )), (c+) w([λ, λ ]) ≥ w(λ) + w(λ ), and (d+) w(cλ) = v(c) + w(λ) for any λ, λ ∈ Lω (G) 1 + and c ∈ Zp . Of course, w = ω  is a possible p−1 choice. But for our later applications it is crucial to allow an arbitrary such w. Clearly, w extends uniquely to a function w : Lω (G) \ {0} −→ R which satisfies (b+) – (d+) for any λ, λ ∈ Lω (G) and any c ∈ Qp . Lemma 32.2. For any x ∈ M(d) and any λ, λ ∈ Lω (G)

1 + p−1

we have:

236

VII

i. w(ex (λ, λ )) > w(λ + λ ) + ii. w(ex (λ, λ )) > w([λ, λ ]) +

d−1 p−1 d−2 p−1

The Lie Algebra

if d ≥ 2; if d ≥ 3.

Proof. i. We proceed by induction with respect to d. If d = 2 then ex = 0 or ±ex = XY − Y X, and we have w(λλ − λ λ) = w(λ(λ + λ ) − (λ + λ )λ) = w([λ, λ + λ ]) ≥ w(λ) + w(λ + λ ) 1 > w(λ + λ ) + . p−1 Now suppose that a + b = d > 2 and x = (y, x) ∈ M(a) × M(b). Then ex = [ey , ez ]. Since a, b < d we may assume the assertion to hold, by induction, for ey and ez . In case a, b ≥ 2 we compute w(ex (λ, λ )) = w([ey (λ, λ ), ez (λ, λ )]) ≥ w(ey (λ, λ )) + w(ez (λ, λ )) a−1 b−1 + w(λ + λ ) + > w(λ + λ ) + p−1 p−1 1 b−1 a−1 + + > w(λ + λ ) + p−1 p−1 p−1 d−1 . = w(λ + λ ) + p−1 In case a = d − 1 ≥ 2 and b = 1 we have w(ex (λ, λ )) = w([ey (λ, λ ), ez (λ, λ )]) ≥ w(ey (λ, λ )) + w(ez (λ, λ )) 1 a−1 + > w(λ + λ ) + p−1 p−1 d −1 , = w(λ + λ ) + p−1 and for a = 1 the computation is entirely analogous. ii. Again proceeding by induction let x = (y, z) ∈ M(a) × M(b) with a + b = d ≥ 3. In case a ≥ 3 and b ≥ 2 we have w(ex (λ, λ )) = w([ey (λ, λ ), ez (λ, λ )])

32

The Hausdorff Series

237

≥ w(ey (λ, λ )) + w(ez (λ, λ )) a−2 b−1 > w([λ, λ ]) + + w(λ + λ ) + p−1 p−1 a − 2 1 b − 1 + + > w([λ, λ ]) + p−1 p−1 p−1 d−2 , = w([λ, λ ]) + p−1 where in the second inequality we have used i. for the second summand. In case a ≥ 3 and b = 1 we have w(ex (λ, λ )) = w([ey (λ, λ ), ez (λ, λ )]) ≥ w(ey (λ, λ )) + w(ez (λ, λ )) a−2 1 > w([λ, λ ]) + + p−1 p−1 d −2 . = w([λ, λ ]) + p−1 The case b ≥ 3 follows by symmetric computations. If a = b = 2 then necessarily ex = 0 and there is nothing to prove. By symmetry it remains to consider the case a = 2 and b = 1. If ex = 0 then ey (λ, λ ) = ±[λ, λ ] and hence w(ex (λ, λ )) = w([ey (λ, λ ), ez (λ, λ )]) ≥ w(ey (λ, λ )) + w(ez (λ, λ )) 1 > w([λ, λ ]) + p−1 d −2 . = w([λ, λ ]) + p−1 Proposition 32.3. For any λ, λ ∈ Lω (G)

1 + p−1

we have

 w log(exp(λ) exp(λ )) = w(λ + λ ). Proof. By Thm. 32.1 we have the convergent expansion  Hd (λ, λ ) log(exp(λ) exp(λ )) = d≥1

in Lω (G). Since H1 (λ, λ ) = λ + λ it suffices to show that

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VII

w

 

The Lie Algebra



Hd (λ, λ )

> w(λ + λ ).

d≥2

The function w necessarily is continuous on the finite dimensional Qp -vector space Lω (G) (cf. [NFA] Prop. 4.13). This further reduces us to the inequalities w(Hd (λ, λ )) > w(λ + λ ) for any d ≥ 2. They follow from (H2) and Lemma 32.2.i. Lemma 32.4. If Hd (X, Y ) − Hd (Y, X) = − d−2 p−1 for any x ∈ B(d).



x∈B(d) cd,x ex

then v(cd,x ) ≥

1 Proof. Let K := Qp (π) where π p−1 = −p. In particular, v(π) = p−1 . We consider the Hausdorff series H(X, Y ) in the Magnus algebra MK with coefficients

in K. As a consequence of Lemma 2.2 the formal power series n−1 e(X) := n≥1 π n! X n has coefficients in the ring of integers O of K. In MK we have

1  (−1)n+1 1 H(πX, πY ) = (exp(πX) exp(πY ) − 1)n π π n n≥1  (−π)n−1  exp(πX) exp(πY ) − 1 n = n π n≥1

=

 (−π)n−1  n≥1

Exercise. v(n) <

n−1 p−1

n

n πe(X)e(Y ) + e(X) + e(Y ) .

for any n ≥ 2 such that n = p.

It follows that the associative formal power series cients in O and satisfies

1 π H(πX, πY

1 H(πX, πY ) π ≡ πe(X)e(Y ) + e(X) + e(Y ) − (πe(X)e(Y ) + e(X) + e(Y ))p ≡ e(X) + e(Y ) + (e(X) + e(Y ))

p

mod πO.

The symmetry in X and Y of this latter expression shows that 1 1 H(πX, πY ) − H(πY, πX) ≡ 0 π π

mod πO.

) has coeffi-

mod πO

32

The Hausdorff Series

239

This means that the associative polynomials π d−2 (Hd (X, Y ) − Hd (Y, X)) have coefficients in O. Since (L ∩ M(d) ) ⊗Z K ∩ M(d) ⊗Z O = (L ∩ M(d) ) ⊗Z O by [B-LL] Chap. II §3.1 Remark 1) this implies π d−2 cd,x ∈ O. Proposition 32.5. For any λ, λ ∈ Lω (G)

1 + p−1

we have

  w log exp(λ) exp(λ ) exp(−λ) exp(−λ ) = w([λ, λ ]). Proof. Using Prop. 32.3 and Thm. 32.1 we obtain   w log exp(λ) exp(λ ) exp(−λ) exp(−λ )     = w log exp log(exp(λ) exp(λ )) exp log(exp(−λ) exp(−λ ))  = w log(exp(λ) exp(λ )) + log(exp(−λ) exp(−λ ))  = w log(exp(λ) exp(λ )) − log(exp(λ ) exp(λ))     =w Hd (λ, λ ) − Hd (λ , λ) 

d≥1

= w [λ, λ ] +







Hd (λ, λ ) − Hd (λ , λ)

d≥3

By the continuity of w it therefore suffices to show that w(Hd (λ, λ ) − Hd (λ , λ)) > w([λ, λ ])

for any d ≥ 3.

This follows from Lemma 32.4 and Lemma 32.2.ii. This analysis of the Hausdorff series now allows us to transport the functions w on Lω (G) 1 + to p-valuations on σω (G). p−1

Proposition 32.6. For any function w : Lω (G)

1 + p−1

1 \ {0} −→ ( p−1 , ∞)

which satisfies (b+), (c+), and (d+) the function ω  (g) := w(log(g)) is a pp valuation on σω (G). Moreover, if in addition w−1 (( p−1 , ∞)) ⊆ pLω (G) 1 + holds true then (σω (G), ω  ) is saturated.

p−1

240

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The Lie Algebra

Proof. The axiom (a) for ω  holds by construction. The axiom (d) translates (d+). Using Prop. 32.3 and (b+) we compute  ω  (g −1 h) = w(log(g −1 h)) = w log(exp(log(g))−1 exp(log(h)))  = w log(exp(− log(g)) exp(log(h))) = w(− log(g) + log(h)) ≥ min(w(log(g)), w(log(h))) = min(ω  (g), ω  (h)). The axiom (c) follows in an analogous way from (c+) and Prop. 32.5. For the second part of the assertion let g ∈ σω (G) be any element such that ω  (g) > p p−1 . Then, by assumption, log(g) = pλ for some λ ∈ Lω (G) 1 + . Setting p−1

h := exp(λ) ∈ σω (G) we obtain hp = exp(λ)p = exp(pλ) = exp(log(g)) = g. Hence (σω (G), ω  ) is saturated. For technical reasons we later will need the following generalization of Thm. 32.1. Let r ≥ 2 be an integer. We consider the iterated Hausdorff series H(X1 , . . . , Xr ) := log(exp(X1 ) · · · exp(Xr )) = H(· · · (H(H(X1 , X2 ), X3 ), . . .), Xr ) as an associative formal power series in the variables X1 , . . . , Xr with coefficients in Q. We write  H(X1 , . . . , Xr ) = Hβ (X1 , . . . , Xr ) β =0

where Hβ (X1 , . . . , Xr ), for any multi-index β = (β1 , . . . , βr ), denotes the component of H(X1 , . . . , Xr ) which is homogeneous of degree βi in the variable Xi , for any 1 ≤ i ≤ r. Proposition 32.7. i. For any β = 0 the coefficients of the associative (noncommutative) polynomial Hβ (X1 , . . . , Xr ) have a p-adic valuation ≥ − |β|−1 p−1 . ii. For any λ1 , . . . , λr ∈ Lω (G)

1 + p−1

we have the convergent expansion

log(exp(λ1 ) · · · exp(λr )) =



Hβ (λ1 , . . . , λr )

β =0

such that Hβ (λ1 , . . . , λr ) ∈ Lω (G)

1 + p−1

for any β = 0.

32

The Hausdorff Series

241

Proof. First of all we establish by induction with respect to r that each Hβ (X1 , . . . , Xr ) a) is a finite sum of associative monomials of the form cXj1 · · · Xj|β| , with c ∈ Q and 1 ≤ j1 , . . . , j|β| ≤ r, which are homogeneous of degree βi in Xi and such that v(c) ≥ − |β|−1 p−1 , and

b) lies in the free Lie algebra L{X1 ,...,Xr } with coefficients in Q (which, by Cor. 15.2, is naturally contained in the free associative algebra As{X1 ,...,Xr } ). For r = 2 this is (H1) and (H2) (the latter remains valid for Hβ —compare the proof of Prop. 17.6). The induction step is carried out by applying the following simple observation to the substitution H(H(X1 , . . . , Xr−1 ), Xr ) = H(X1 , . . . , Xr ). Let cZ 1 · · · Z m be an associative monomial in two variables X and Y (i. e., each Z i is equal to either X or Y ) with coefficient c ∈ Q such that v(c) ≥ − m−1 p−1 . Moreover, for any 1 ≤ i ≤ m, let Mi = bi Xji1 . . . Xjidi be an associative monomial in the variables X1 , . . . , Xr with coefficient in Q such i −1 . Substituting Mi for Z i then gives the monomial that v(bi ) ≥ − dp−1 cM1 · · · Mm = cb1 · · · bm Xj11 · · · Xjmdm of degree d := d1 + · · · + dm in the variables X1 , . . . , Xr with v(cb1 · · · bm ) = v(c) + v(b1 ) + · · · + v(bm ) 1 (m − 1 + d1 − 1 + · · · + dm − 1) ≥− p−1 d−1 =− . p−1 In a completely analogous way the substitution of Lie monomials for the variables in a given Lie monomial leads again to a Lie monomial. By definition, a Lie monomial is an element in L{X1 ,...,Xr } of the form cex with c ∈ Q and ex the image of any x ∈ M{X1 ,...,Xr } under the composite map ⊆

pr

M{X1 ,...,Xr } −−→ A{X1 ,...,Xr } −−→ L{X1 ,...,Xr } in Sect. 15. By construction, any element in L{X1 ,...,Xr } is (not uniquely) a finite sum of such Lie monomials.

242

VII

The Lie Algebra

The assertion i. follows from a), and b) implies that Hβ (λ1 , . . . , λr ) ∈ Lω (G) ⊆ ΛQp (G, ω) for any λ1 , . . . , λr ∈ Lω (G). But as a consequence of i. we have ω  (Hβ (λ1 , . . . , λr )) ≥ − If λ1 , . . . , λr ∈ Lω (G)

1 + p−1

|β| − 1  (λ1 ) + · · · + βr ω  (λr ). + β1 ω p−1

then the right hand side is >

1 p−1

and hence

Hβ (λ1 , . . . , λr ) ∈ Lω (G) 1 + . p−1 It remains to establish the expansion in ii. In case r = 2 this is a slightly more precise version of Thm. 32.1 which has the same proof. For general r we, in particular, have the convergent expansion  H(λ1 , . . . , λr ) = H(α1 ,α2 ) (H(λ1 , . . . , λr−1 ), λr ). (α1 ,α2 )

We write the associative formal power series  (α ,α ) H(α1 ,α2 ) (H(X1 , . . . , Xr−1 ), Xr ) = Hβ 1 2 (X1 , . . . , Xr ) (α1 ,α2 ) (α ,α )

as the sum of its homogeneous components Hβ 1 2 of degree βi in Xi . The induction step in the first part of this proof applies as well and gives that (α ,α ) each Hβ 1 2 (X1 , . . . , Xr ) satisfies a). It follows that (α1 ,α2 )

ω  (Hβ

|β| − 1  (λ1 ) + · · · + βr ω  (λr ) + β1 ω p−1 |β| − 1 + |β| min ω  (λi ) ≥− 1≤i≤r p−1   1 .  (λi ) − > |β| min ω 1≤i≤r p−1

(λ1 , . . . , λr )) ≥ −

Since the Hausdorff series has no constant term there are, for any β, at most (α ,α ) finitely many (α1 , α2 ) such that Hβ 1 2 (X1 , . . . , Xr ) = 0, and Hβ (X1 , . . . , Xr ) =



(α1 ,α2 )

Hβ

(X1 , . . . , Xr ).

(α1 ,α2 )

This, in particular, shows that (50)

lim α1 +α2 +|β|→∞

(α1 ,α2 )

Hβ

(λ1 , . . . , λr ) = 0.

33

Rational p-Valuations and Applications

243

Again by induction with respect to r we now assume that the asserted expansion holds for H(λ1 , . . . , λr−1 ). An appropriate version of Lemma 3.2 then implies that, for any (α1 , α2 ), we have the convergent expansion  (α ,α ) Hβ 1 2 (λ1 , . . . , λr ). H(α1 ,α2 ) (H(λ1 , . . . , λr−1 ), λr ) = (α1 ,α2 )

Because of (50) we finally may apply Lemma 3.3 and conclude that   (α ,α ) H(λ1 , . . . , λr ) = Hβ 1 2 (λ1 , . . . , λr ) (α1 ,α2 ) β

=

 

(α1 ,α2 )

Hβ

(λ1 , . . . , λr )

β (α1 ,α2 )

=



Hβ (λ1 , . . . , λr ).

β

33

Rational p-Valuations and Applications

In this section we will show that on any p-valuable group there exists a pvaluation with values in Q. The technique to do this actually is to construct rational valued functions w on Lω (G) 1 + and then to transfer them to p−1

σω (G) and hence by restriction to G. This important existence result will enable us to establish various fundamental ring theoretic properties of the completed group ring Λ(G). Lemma 33.1. There is a function w : Lω (G)

1 + p−1

1 \ {0} −→ ( p−1 , ∞) ∩ Q

p , ∞)) ⊆ which satisfies (b+), (c+), (d+) and has the property that w−1 (( p−1 pLω (G) 1 + . p−1

Proof. We fix a Zp -basis λ1 , . . . , λr of Lω (G) 1 + as constructed in Cor. 31.3. p−1 It necessarily satisfies p 1 <ω  (λi ) ≤ p−1 p−1

for 1 ≤ i ≤ r.

We let (51)

p 1 < τi ≤ p−1 p−1

for 1 ≤ i ≤ r

244

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The Lie Algebra

be arbitrary but fixed real numbers, and we introduce the function   1 w : Lω (G) 1 + \ {0} −→ ,∞ p−1 p−1 r  xi λi −→ min (v(xi ) + τi ). 1≤i≤r

i=1

It is straightforward to see that w satisfies (b+) and (d+) and has the additional property stated in the assertion. Let cijk ∈ Zp , for 1 ≤ i, j, k ≤ r, be such that r  cijk λk . [λi , λj ] = k=1

If w also satisfies (c+) then we, in particular, have   cijk λk = w([λi , λj ]) ≥ w(λi ) + w(λj ) = τi + τj . min(v(cijk ) + τk ) = w k

k

Hence the τi satisfy the further inequalities τi + τj ≤ v(cijk ) + τk

(52)

for any 1 ≤ i, j, k ≤ r.

Suppose, vice versa, that these inequalities hold. We then compute       =w w xi λi , yj λ j xi yj [λi , λj ] i

j

i,j

 =w



cijk xi yj λk

i,j,k

 

= min v k





+ τk

cijk xi yj

i,j

≥ min(v(cijk ) + v(xi ) + v(yj ) + τk ) i,j,k

≥ min(v(xi ) + v(yj ) + τi + τj ) i,j

= min(v(xi ) + τi ) + min(v(yj ) + τj ) i j     =w xi λi + w yj λ j i

j

33

Rational p-Valuations and Applications

245

which means that w satisfies (c+). In order to prove our assertion we therefore have to find rational numbers τ1 , . . . , τr which satisfy the inequalities (51) and (52), which both have rational coefficients. Note that these inequalities do have the real solutions τi := ω  (λi ). 1 < qi ≤ ω  (λi ) for 1 ≤ i ≤ r and consider We choose rational numbers p−1 the slightly more restrictive system of inequalities

(53)

−τi ≤ −qi p τi ≤ p−1 τi + τj − τk ≤ v(cijk )

for 1 ≤ i ≤ r, for 1 ≤ i ≤ r, for 1 ≤ i, j, k ≤ r.

The set S ⊆ Rr of real solutions of (53) still is nonempty and, of course, is convex. We want to show that it contains a rational solution. Let E denote the set of r3 + 2r inequalities in the system (53). For any e ∈ E and any s ∈ S we write s < e, resp. s = e, if s satisfies the strict inequality, resp. the equality, corresponding to e. We divide up E into the subsets E < := {e ∈ E : s < e for some s ∈ S} and E = := {e ∈ E : s = e for any s ∈ S}. The equations corresponding to the e ∈ E = define a rational affine subspace A ⊆ Rr which contains S. We claim that S as a subset of A has nonempty interior, and consequently contains a rational point. Consider the set S0 := {s ∈ A : s < e for any e ∈ E < } which obviously is open in A and is contained in S. It suffices to show that S0 is nonempty. If E < = ∅ then S0 = S(= A) is nonempty. Otherwise there is, for any e ∈ E < , a solution se ∈ S such that se < e. Then s :=

1  se |E < | < e∈E

is, by convexity, a solution in S as well and lies in S0 by construction. Theorem 33.2. There always is a p-valuation ω  on σω (G) with values in Q and such that (σω (G), ω  ) is saturated. Proof. Combine Lemma 33.1 and Prop. 32.6.

246

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Corollary 33.3. Any p-valuable group G carries a p-valuation ω with values in N1 Z for some N ∈ N. Proof. By the theorem we find an ω with values in Q. On theother hand we know that the set of values always is of the form ω(G)\{1} = 1≤j≤s νj +N0 . Since the νj have to be rational numbers we can take for N their smallest common denominator. To draw the conclusions for the completed group ring Λ(G) we from now on assume that the p-valuation ω on our p-valuable group G has values in some N1 Z. We also allow O to be again a general complete discrete valuation ring in which p is a prime element. Theorem 33.4. For any p-valuable group G the completed group ring Λ(G) is noetherian, regular of global dimension ≤ 1 + rank G, and is Auslander regular. Proof. According to Lemma 26.13.i. we may assume, by possibly replacing ω by ω − C for some sufficiently small rational number C > 0, that the Lie algebra gr G is abelian. We then know from Cor. 28.6 that the graded ring gr Λ(G) is a polynomial ring in 1+rank G variables over O/pO. In particular, gr Λ(G) is noetherian and regular of global dimension 1 + rank G. Let N ∈ N be a common denominator for the values of ω which then also is a common denominator for the values of ω  on Λ(G). We introduce the exhaustive and separated decreasing filtration Film Λ(G) := Λ(G) m N indexed by integers m ≥ 0 on Λ(G). It is a ring filtration in the sense that Fil Λ(G) · Film Λ(G) ⊆ Fil +m Λ(G) holds true for any , m ≥ 0. Moreover, Λ(G) is complete with respect to this filtration, i. e., Λ(G) = lim Λ(G)/ Film Λ(G). ←− m

m = +m Finally, the trivial reason that N + N N , the associated graded  for m m+1 ring m Fil Λ(G)/ Fil Λ(G) coincides, as an ungraded O/pO-algebra, with gr Λ(G). In such a situation it is a general fact that the three ring theoretic properties in our assertion lift from the associated graded ring to the complete filtered ring Λ(G). A detailed discussion of this lifting technique and its application to our three properties can be found in [LvO] (or [NkA]). The lifting of being noetherian also is treated in [B-CA] Chap. III §2.9 Cor. 2.

34

Coordinates of the First and of the Second Kind

34

247

Coordinates of the First and of the Second Kind

Throughout this section we assume that the pair (G, ω) is saturated. We view G as a p-adic Lie group as constructed in Cor. 29.6. We recall that this means the following. Given any fixed ordered basis (g1 , . . . , gr ) of (G, ω) the coordinate functions ϕi : G −→ Zp ⊆ Qp g −→ xi

if g = g1x1 · . . . · grxr

for 1 ≤ i ≤ r are locally analytic. These (ϕ1 , . . . , ϕr ) are called coordinates of the second kind . On the other hand it follows from Cor. 31.3 (more precisely, its proof) that the elements log(g1 ), . . . , log(gr ) form a Zp -basis of Lω (G) 1 + satisfying p−1

ω 

 r  i=1

yi log(gi )

= min (v(yi ) + ω(gi )) = ω(g1y1 · · · gryr ) 1≤i≤r

for any y1 , . . . , yr ∈ Zp . we therefore obtain functions ψi : G −→ Zp ⊆ Qp g −→ yi

if log(g) =

r 

yi log(gi ).

i=1

These (ψ1 , . . . , ψr ) are called coordinates of the first kind . We have seen at the end of Sect. 31 that the maps exp

Lω (G)

1 + p−1

G log

are isomorphisms of manifolds inverse to each other. This immediately implies that the functions ψi are locally analytic. But there is a more precise statement which we want to establish.

Proposition 34.1. There are formal power series Fi (Y ) = β ai,β Y β and

Ei (Y ) = β bi,β Y β , for any 1 ≤ i ≤ r, in r variables Y = (Y1 , . . . , Yr ) with coefficients in Zp such that we have: i. lim|β|→∞ v(ai,β ) = ∞ and lim|β|→∞ v(bi,β ) = ∞;

248

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ii. ψi (g) = Fi (ϕ1 (g), . . . , ϕr (g)) for any g ∈ G; iii. ϕi (g) = Ei (ψ1 (g), . . . , ψr (g)) for any g ∈ G. Proof. The assertion i. guarantees that the Fi and Ei converge on Zrp so that the assertions ii. and iii. make sense. There is nothing to prove for r = 1, so we assume in the following that r ≥ 2. Let g ∈ G and put xi := ϕi (g) and yi := ψ(gi ). For the construction of the power series F1 (Y ), . . . , Fr (Y ) we start from the identity g = g1x1 · · · grxr = exp(log(g1 ))x1 · · · exp(log(gr ))xr = exp(x1 log(g1 )) · · · exp(xr log(gr )), where the last line follows by continuity from the case where the xi are integers. We obtain r 

 yi log(gi ) = log(g) = log exp(x1 log(g1 )) · · · exp(xr log(gr )) .

i=1

Applying Prop. 32.7 to the right hand side and using the homogeneity property of the Hβ give rise to the expansion r 

yi log(gi ) =

i=1



xβ1 1 · · · xβr r Hβ (log(g1 ), . . . , log(gr ))

β =0

in Lω (G) 1 + . In particular, lim|β|→∞ ω  (Hβ (log(g1 ), . . . , log(gr ))) = ∞. We p−1 let r  Hβ (log(g1 ), . . . , log(gr )) = ai,β log(gi ) i=1

with ai,β ∈ Zp and lim|β|→∞ v(ai,β ) = ∞, and we obtain r  i=1

yi log(gi ) =

r   i=1



xβ1 1 · · · xβr r ai,β log(gi ).

β

It remains to define Fi (Y ) := β ai,β Y β . The construction of the power series E1 (Y ), . . . , Er (Y ) proceeds along the same lines as the proof of Prop. 29.2. We have the expansion   x1   xr  (54) g = g1x1 · · · grxr = ··· bα α α 1 r α

34

Coordinates of the First and of the Second Kind

249

in Zp [[G]]. On the other hand the identity g = exp(y1 log(g1 ) + · · · + yr log(gr )) leads to the expansion  r n  1   β 1 g= yi log(gi ) = y1 1 · · · yrβr Mβ (log(g1 ), . . . , log(gr )) n! |β|! n≥0

i=1

β

in ΛQp (G, ω) where Mβ (Z1 , . . . , Zr ) denotes the sum of all noncommutative (associative) monomials in the variables Zi which, for any 1 ≤ i ≤ r, have exactly βi factors Zi . We have   r  1 ω  βi ω  (log(gi )) Mβ (log(g1 ), . . . , log(gr )) ≥ −v(|β|!) + |β|! i=1

= −v(|β|!) +

r 

βi ω(gi )

i=1

 ≥ |β| min(ω(g1 ), . . . , ω(gr )) −

 1 . p−1

Next we use the “standard” expansions  1 cα,β bα Mβ (log(g1 ), . . . , log(gr )) = |β|! α with cα,β ∈ Qp and lim|α|→∞ v(cα,β )+

r

i=1 αi ω(gi )

= ∞ for any β. We have



 1 v(cα,β ) + αi ω(gi ) ≥ ω  Mβ (log(g1 ), . . . , log(gr )) |β|! i=1   1 . ≥ |β| min(ω(g1 ), . . . , ω(gr )) − p−1 r 

It follows in particular that lim

|α|+|β|→∞

v(cα,β ) +

r 

αi ω(gi ) = ∞.

i=1

Hence we obtain the expansion   (55) g= cα bα with cα := cα,β y1β1 · · · yrβr . α

β

250

VII

The Lie Algebra

By comparing (54) and (55) we deduce      xr x1 cα,β y1β1 · · · yrβr ··· = α1 αr β

and, in particular, xi =



ci,β y1β1 · · · yrβr

β

β for the multi-index α = i. We define Ei (Y ) := β ci,β Y . The coeffi1 cients satisfy v(ci,β )+ω(gi ) ≥ |β|(min(ω(g1 ), . . . , ω(gr ))− p−1 ) which implies lim|β|→∞ v(ci,β ) = ∞. It remains to show that ci,β ∈ Zp or, equivalently, that v(ci,β ) > −1. By construction ci,0 = 0. For β = 0 we have the more precise estimate v(ci,β ) + ω(gi ) ≥ |β| min(ω(g1 ), . . . , ω(gr )) − Since (G, ω) is saturated we have ω(gi ) ≤

p p−1

|β| − 1 . p−1

and hence

 v(ci,β ) + 1 ≥ |β| min(ω(g1 ), . . . , ω(gr )) −

1 p−1

 > 0.

Since a change of basis of Lω (G) is a linear and hence locally analytic map we deduce that, for any Qp -basis λ1 , . . . , λr of Lω (G), the functions ψi : G −→ Qp g −→ yi

if log(g) =

r 

yi λ i

i=1

are locally analytic. Moreover, this remains true even if (G, ω) is not saturated (since it is open in its saturation). It gives us a more general kind of coordinates of the first kind .

References [BGR]

Bosch S., G¨ untzer U., Remmert R.: Non-Archimedean Analysis. Berlin - Heidelberg - New York: Springer 1984

[B-CA] Bourbaki N.: Commutative Algebra. Berlin - Heidelberg - New York: Springer 1989 [B-GT] Bourbaki N.: General Topology. Berlin - Heidelberg - New York: Springer 1989 [B-LL]

Bourbaki N.: Lie Groups and Lie Algebras. Berlin - Heidelberg New York: Springer 1998

[B-VA]

Bourbaki N.: Vari´et´es diff´erentielles et analytiques. Fascicule de r´esultats. Berlin - Heidelberg - New York: Springer 2007, reprint of 1982 printing

[DDMS] Dixon J.D., du Sautoy M.P.F., Mann A., Segal D.: Analytic Prop-Groups. Cambridge Univ. Press 1999 [Fea]

F´eaux de Lacroix C.T.: Einige Resultate u ¨ber die topologischen Darstellungen p-adischer Liegruppen auf unendlich dimensionalen Vektorr¨ aumen u ¨ber einem p-adischen K¨ orper. Thesis, K¨oln 1997. Schriftenreihe Math. Inst. Univ. M¨ unster, 3. Serie, Heft 23, pp. 1– 111 (1999)

[Gab]

Gabriel P.: Des Cat´egories Ab´eliennes. Bull. Soc. Math. France 90, 323–448 (1962)

[Haz]

Hazewinkel M.: Formal Groups and Applications. New York: Academic Press 1978

[Hum]

Humphreys J.: Introduction to Lie Algebras and Representation Theory. Berlin - Heidelberg - New York: Springer 1972

[Lam]

Lam T.Y.: A First Course in Noncommutative Rings. Berlin - Heidelberg - New York: Springer 2001

[La1]

Lazard M.: Quelques calculs concernant la formule de Hausdorff. Bull. Soc. Math. France 91, 435–451 (1963)

[La2]

Lazard M.: Groupes analytiques p-Adiques. Publ. Math. IHES 26, 389–603 (1965)

P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8, © Springer-Verlag Berlin Heidelberg 2011

251

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References

[LvO]

Li Huishi, van Oystaeyen F.: Zariskian Filtrations. Dordrecht: Kluwer 1996

[NkA]

Schneider P.: Ausgew¨ ahlte Kapitel aus der nichtkommutativen Algebra. Vorlesungsskriptum, M¨ unster 2000

[NFA]

Schneider P.: Nonarchimedean Functional Analysis. Berlin - Heidelberg - New York: Springer 2002

[Se1]

Serre J-P.: Local Fields. Berlin - Heidelberg - New York: Springer 1979

[Se2]

Serre J-P.: Lie Algebras and Lie Groups. Lect. Notes Math. 1500. Berlin - Heidelberg - New York: Springer 1992

Index a-derivation, 67 algebra free A-, 107 Lie, 71 Magnus, 111 tensor, 102 universal enveloping, 102 atlas, 46 equivalent, 46 maximal, 46 n-dimensional, 47 ball, 4 closed, 4 open, 4 Banach space, 11 Campbell-Hausdorff theorem, 114 Lie group germ, 125 chain rule, 18, 58, 66 chart, 45 around x, 45 compatible, 45 dimension, 45 domain of definition, 45 completed group ring, 157 comultiplication, 106, 133 coordinates of the first kind, 247, 250 of the second kind, 247 covering locally finite, 51 refinement, 51 derivation, 67 derivative, 18, 59 diagonal, 106

diameter, 6 distribution, 165 dual space, 13 Dynkin’s formula, 117 ε-convergent, 25 expansion, 31 exponential map, 153 exponential power series, 219 finite rank, 181 formal group law, 132 formal homomorphism, 143 group-like element, 219 Hausdorff series, 115, 232 homomorphism formal, 143 into associative algebra, 101 local, 136 of Lie algebras, 101 of Lie groups, 101 identity theorem, 33 index for M , 76 invertibility for power series, 33 local, 22, 42, 58 Iwasawa algebra, 157 Jacobi identity, 71 Lie algebra, 71 graded, 174 of G, 100, 219, 222 Lie group, 89 p-adic, 192

P. Schneider, p-Adic Lie Groups, Grundlehren der mathematischen Wissenschaften 344, DOI 10.1007/978-3-642-21147-8, © Springer-Verlag Berlin Heidelberg 2011

253

254

local homomorphism, 136 ring, 136 local invertibility, 22, 42, 58 locally analytic function, 38, 49 manifold, 47 map, 50 locally convex final topology, 82 inductive limit, 83 topology, 81 vector space, 81 logarithm power series, 219 Magnus algebra, 111, 175, 232 manifold, 47 n-dimensional, 47 open sub-, 48 product, 49 map differentiable, 17 exponential, 153 locally analytic, 50 locally constant, 25 polynomial, 118 strictly differentiable, 20 tangent, 57 nonarchimedean absolute value, 8 field, 8 norm, 10 seminorm, 79 norm, 10 open mapping theorem, 23 operator norm, 12 ordered basis, 182

Index

p-adic field, 9 p-adic valuation, 182 paracompact, 51 partial derivative, 20 Poincar´e-Birkhoff-Witt theorem, 104 primitive element, 219 product rule, 19, 59 pseudocompact module, 165 ring, 165 p-valuable, 192 p-valuation, 169 rational, 243 rank, 181 residue class field, 9 ring of integers, 9 saturated, 187 saturation, 208 seminorm, 79 strictly paracompact, 51 tangent bundle, 64 map, 57 space, 57 vector, 56 Taylor expansion, 33 ultrametric space, 3 complete, 6 spherically complete, 6 universal enveloping algebra, 102 vector field, 66 left, right invariant, 96 vector space dual, 13 locally convex, 81 normed, 10