LONOON MA1HEMA11CAL SOCIETY snJDENTlEXTS Managing editor: Dr C.M. Series, Mathematics Institute University of Warwick, Coventry CV4 7AL, United Kingdom
Introduction to combinators and A.-calculus, J.R. HINDLEY & J.P. SELDIN 2 Building models by games, WILFRID HODGES 3 Local fields, J.W.S. CASSELS 4 An inttoduction to twistor theory: Second edition, S.A. HUGGE'IT & K.P. TOO 5 Introduction to general relativity, L.P. HUGHSTON & K.P. TOO 6 Lectures on stochastic analysis: diffusion theory, DANIEL W. STROOCK 7 The theory of evolution and dynamical systems, J. HOFBAUER & K. SIGMUND 8 Summing and nuclear norms in Banach space theory, G.J.O. JAMESON 9 Automorphisms of surfaces after Nielsen and Thurston, A. CASSON & S. BLEILER 10 Nonstandard analysis and its applications, N. Cun.AND (ed) 11 Spacetime and singularities, G. NABER 12 Undergraduate algebraic geometry, Mn..ES REID 13 An introduction to Hankel operators, J.R. PARTINGTON 14 Combinatorial group theory: a topological approach, DANIEL E. COHEN 15 Presentations of groups, D.L. JOHNSON 16 An introduction to noncommutative Noetherian rings. K.R. GOODEARL & R.B. WARFIELD, JR. 17 Aspects of quantum field theory in curved spacetime, S.A. R.JLLING 18 Braids and coverings: selected topics, VAGN LUNDSGAARD HANSEN 19 Steps in commutative algebra, R.Y. SHARP 20 Communication theory, C.M. GOLDIE & R.G.E. PINCH 21 Representations of finite groups ofUe type, FRANCOIS DIGNE & JEAN MICHEL 22 Designs, graphs, codes, and their links, P.J. CAMERON & J.H. VAN UNT 23 Complex algebraic curves, FRANCES KIRWAN 24 Lectures on elliptic curves, J.W.S. CASSELS 25 Hyperbolic geometry, BIRGER IVERSEN 26 An Introduction to the theory of L-functions and Eisenstein series, H. HIDA 27 Hilbert Space: compact operators and the trace theorem, J.R. RETHERFORD 28 Potential Theory in the Complex Plane, T. RANSFORD 29 Undergraduate Commutative Algebra, M. REID 32 Lectures on Ue Groups and Ue Algebras, R. CARTER, G. SEGAL & I. MACDONALD
London Mathematical Society Student Texts 32
Lectures on Lie Groups and Lie Algebras
Roger Carter University of Warwick
Graeme Segal University of Cambridge
Ian Macdonald Queen Mary and Westfield College, London
CAMBRIDGE UNIVERSITY PRESS
Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 1RP 40 West 20th Street, New York, NY 10011-4211, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia © Cambridge University Press 1995
First published 1995 Printed in Great Britain at the University Press, Cambridge
A catalogue recordfor this book is available from the British li.brarJ Library of Congress cataloging in publication data available ISBN 0 521 49579 2 hardback ISBN 0 521 49922 4 paperback
Contents
Foreword
page vii
M. J. Taylor
Lie Algebras and Root Systems
R. W. Carter
Preface Introduction to Lie algebras 1 Basic concepts 1.1 1.2 Representations and modules 1.3 Special kinds of Lie algebra 1.4 The Lie algebras sln (Cl:) Simple Lie algebras over Cl: 2 2.1 Cartan subalgebras 2.2 The Cartan decomposition 2.3 The Killing form 2.4 The Weyl group 2.5 The Dynkin diagram 3 Representations of simple Lie algebras 3.1 The universal enveloping algebra 3.2 Verma modules 3.3 Finite dimensional irreducible modules 3.4 Weyl's character and dimension formulae 3.5 Fundamental representations 4 Simple groups of Lie type 4.1 A Chevalley basis of g 4.2 Chevalley groups over an arbitrary field 4.3 Finite Chevalley groups 4.4 Twisted groups 4.5 Suzuki and Ree groups 4.6 Classification of finite simple groups
v
1 3
5 5 7 8 10 12 12 13 15 16 18 25 25 26 27 29 32 36 36 38 39 41 43 44
Contents
VI
Lie Groups
Graeme Segal
Introduction 1 Examples l su2. S03, and SL2R. Homogeneous spaces 3 4 Some theorems about matrices Lie theory 5 Representation theory 6 Compaet groups and integration 7 Maximal compaet subgroups 8 9 The Peter-Weyl theorem 10 Functions on R.n and sn-1 Induced representations 11 1l 'Ibe complexification of a compaet group 'Ibe unitary and symmetric groups 13 14 The Borel-Weil theorem 15 Representations of non-compact groups Representations of S L 2R. 16 1be Heisenberg group 17
Linear Algebraic Groups
45 47 49 53 59 63
69 82 85 89 91
100 104
108 110 115
120 124
128
I. G. Macdonald
133 Preface 135 Introduction 137 1 Aftine algebraic varieties 139 l Linear algebraic groups: definition and elementary properties 146 Interlude 154 3 Projective algebraic varieties 157 4 Tangent spaces. Separability 162 5 1be Lie algebra of a linear algebraic group 166 6 Homogeneous spaces and quotients 172 7 Borel subgroups and maximal tori 177 8 The root structure of a linear algebraic group 182 Notes and references 186
Bibliography
187
Index
189
Foreword
This book consists of notes based on the three introductory lecture courses given at the LMS-SERC Instructional Conference on Lie theory and algebraic groups held at Lancaster University in September 1993: Lie Algebras by Roger Carter; Lie Groups by Graeme Segal; algebraic groups by Ian Macdonald. The aim of the course was to provide an introduction to this important area of mathematics for postgraduate students who had no previous specialised knowledge. Discussions with the students at the end of the meeting suggested that the conference had been extremely successful; it then seemed desirable to further impose on the lecturers by asking them to write-up their lectures, in order that future generations of students could also benefit from their efforts. All three lecturers adopted the same approach of providing a crisp, fast-moving, clear introduction, while at the same time taking care to indicate more advanced material, so as to give the full ftavour of the subject. It is clear, from both the lectures and the written account, that a substantial effort was made to ensure a coherent and well-harmonised presentation of these three highly interrelated themes. The general intention of the new series of LMS-SERC Instructional Conferences is to provide postgraduate students with the opportunity to learn important mainstream core mathematics, which they might not otherwise meet. Lie theory and algebraic groups seemed to be a natural first choice, since they are a central mathematical crossroads, which relate to a host of important areas such as group theory, number theory, algebraic geometry, differential geometry, topology, particle physics and strings; indeed, a knowledge of algebraic groups and Lie theory can be quite crucial in making significant progress in many aspects of these related areas.
Vlll
Foreword
Finally, on behalf of the LMS, I should like to express my deepest gratitude to the three authors for not just accepting to give their lectures and then write them up, but also for carrying out their alloted tasks with such infectious enthusiasm; extra special thanks go to Ian Macdonald for sage advice in the initial planning of the meeting. It is also a pleasure to thank both Roger Astley and David Tranah for their help and cooperation in producing these notes, which will be a very valuable contribution to the mathematical community. Martin Taylor UMIST Manchester
Lie Algebras and Root Systems R.W. Carter
Contents Lie Algebras and Root Systems
Preface 1 Introduction to Lie algebras 1.1 Basic concepts 1.2 Representations and modules 1.3 Special kinds of Lie algebra 1.4 The Lie algebras sln(q 2 Simple Lie algebras over cr: 2.1 Cartan subalgebras 2.2 The Cartan decomposition 2.3 The Killing form 2.4 The Weyl group 2.5 The Dynkin diagram Representations of simple Lie algebras 3 3.1 The universal enveloping algebra 3.2 Verma modules 3.3 Finite dimensional irreducible modules 3.4 Weyrs character and dimension formulae 3.5 Fundamental representations 4 Simple groups of Lie type 4.1 A Chevalley basis of g 4.2 Chevalley groups over an arbitrary field 4.3 Finite Chevalley groups 4.4 Twisted groups 4.5 Suzuki and Ree groups 4.6 Classification of finite simple groups
2
3
s s 7 8 10
12 12 13 IS
16 18 25 25 26 27 29 32 36 36 38 39 41 43 44
Preface
The following notes on Lie Algebras and Root Systems follow fairly closely the lectures I gave on this subject at the Lancaster meeting, although more detail has been included in a number of places. The aim has been to give an outline of the main ideas involved in the structure and representation theory of the simple Lie algebras over «:, and the construction of the corresponding groups of Lie type over an arbitrary field. It has not been possible to give aU the proofs in detail, and so interested readers are encouraged to consult books in which more complete information is given. The following books are particularly recommended. J. E. Humphreys. Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics 9 (1972) Springer. N. Jacobson, Lie Algebras. Interscience Publishers, J. Wiley, New York (1962). R. W. Carter, Simple Groups of Lie Type, Wiley Classics Library Edition (1989), J. Wiley, New York.
3
1 Introduction to Lie algebras
1.1 Basic concepts A Lie algebra is a vector space g over a field F on which a multiplication gxg
-+
g
x,y -+ [xy] is defined satisfying the axioms: (i) [xy] is linear in x and in y. (ii) [xx] = 0 for all x e g. (iii) [[xy]z] + [[yz]x] + [[zx]y] = 0 for all x,y,z E g. Property (iii) is called the Jacobi identity. We note that the multiplication is not associative, i.e., it is not true in general that [[xy]z] = [x[yz]]. It is therefore essential to include the Lie brackets in products of elements. For any pair of elements x, y E g we have [x + y,x + y] = [xx] + [xy] + [yx] + [yy]. We also know that [xx]
= 0,
[yy]
= 0,
[x + y, x + y]
= 0.
It follows that [yx] = -[xy] for all x,y E g. Thus multiplication in a Lie algebra is anticommutative. Lie algebras can be obtained from associative algebras by the following method. Let A be an associative algebra, i.e., a vector space with a bilinear associative multiplication xy. Then we may obtain a Lie algebra [A] by redefining the multiplication on A. We define [xy] = xy- yx. It is clear
s
6
I Lie Algebras
that [xy) is linear in x and in y and that [xx] == 0. We also have [[xy]z]
== (xy- yx)z - z(xy- yx) == xyz - yxz - zxy + zyx.
It follows that [[xy]z] + [[yz]x]
=
=
+ [[zx]y] xyz - yxz - zxy + zyx +yzx- zyx- xyz + xzy +zxy- xzy- yzx + yxz 0,
so that the Jacobi identity is satisfied. Let g1o g2 be Lie algebras over F. A homomorphism of Lie algebras is a linear map 8: g1 - g2 such that 8[xy] == [8x,8y] for all x,y e g 1• 8 is an isomorphism of Lie algebras if 8 is a bijective homomorphism. Let g be a Lie algebra and h,k be subspaces of g. We define the product [hk] to be the subspace spanned by all products [xy] for x e h, y e k. Each element of [hk] is thus a finite sum(XtYil + · · · +rxrYrlwith Xi e h, Yi e k. We note that [hk] == [kh], i.e., multiplication of subspai'ces is commutative. This follows from the fact that multiplication of elements is anticommutative. So if x e h, y e k we have [yx] == -[xy] e [hk]. A subalgebra of g is a subspace h of g such that [hh] c: h. An ideal of g is a subspace h of g such that [hg] c: h. We observe that, since [hg] == fgh], there is no distinction in the theory of Lie algebras between left ideals and right ideals. Every ideal is two-sided. Now let h be an ideal of the Lie algebra g. Let g/h be the vector space of cosets h + x for x e g. h + x consists of all elements of form y + x for y e h. We claim that g/h can be made into a Lie algebra, the factor algebra of g with respect to h, by introducing the Lie multiplication [h + x, h + y] = h + [xy]. We must take care to check that this operation is well defined, i.e., that if h + x == h + x and h + y == h + y' then h + [xy] == h + [x'y']. This follows from the fact that h is an ideal of g. We have
x' == a + x,
y' == b + y for a, b e h.
Thus [x'y'] == [ab] + [ay] + [xb] + [xy] E h + [xy]
1.2 Representations and modules
7
since [ab], [ay], [xb] all lie in h. This gives h+ [x'y'] = h+ [xy] as required. There is a natural homomorphism g.! g/h relating a Lie algebra with a factor algebra. (} is defined by lJ(x) = h + x. Conversely given any homomorphism (} : g1 - g2 of Lie algebras which is surjective, the kernel k of(} is an ideal of g1 and the factor algebra g,fk is isomorphic to g2. The set of all n x n matrices over the field F can be made into a Lie algebra under the Lie multiplication [A, B] = AB-BA. This Lie algebra is called gln(F), the general linear Lie algebra of degree n over the field
F. 1.2 Representations and modules
Let g be a Lie algebra over F. A representation of g is a homomorphism P: g- gln(F) for some n. Two representations p, p' of g of degree n are called equivalent if there is a non-singular n x n matrix T over F such that p'(x) =
r- 1p(x)T,
for all x e g.
There is a close connection between representations of g and g-modules. A left g-module is a vector space V over F with a multiplication gxV
-
V
X,V
-
XV
satisfying the axioms (i) xv is linear in x and in v (ii) [xy]v = x(yv)- y(xv) for all x,y
e g, v e V.
Every finite dimensional g-module gives a representation of g, as follows. Choose a basis e~o .. . ,en of V. Then xej is a linear combination of e~o .. . , en. Let n
xej =
:~:::>ij(x)ei. i=l
Let p(x) be the n x n matrix (Pij(X)). Then we have p[xy]
= p(x)p(y)- p(y)p(x) = [p(x) p(y)]
and so the map x - p(x) is a representation of g. If we choose a different basis for the g-module V we shall get an equivalent representation.
8
I Lie Algebras
Now let U be a subspace of V and h a subspace of g. Let hU be the subspace of V spanned by all elements xu for x E h, u e U. U is called a submodule of V if gU c: U. A g-module V is called irreducible if V has no submodules other than V and 0. Now g is itself a g-module under the multiplication g x g -+ g given by x,y-+ [xy]. To see this we must check [[xy]z] = [x[yz]]- [y[xz]] for x, y, z e g. This follows from the Jacobi identity using the anticommutative law. g is called the adjoint g-module, and it gives rise to the adjoint representation of g.
1.3 Special kinds of Lie algebra So far the theory of Lie algebras has been very analogous to the theory of rings, where one has subrings, ideals, factor rings, etc. However there is also a sense in which the theory of Lie algebras can be considered as analogous to the theory of groups, where the Lie product [xy] is regarded as analogous to the commutator x- 1y- 1xy of two elements in a group. This analogy motivates the following terminology. A Lie algebra g is called abelian if [gg] = 0. This means that all Lie products are zero. We shall now define a sequence of subspaces g 1, g2, g 3, .. · of g. We define them inductively by g'
= g,
gn+l
= (gng].
Now if h,k are ideals of g so is their product [hk]. For let x E h, y z e g. Then we have [[xy]z] = [x(yz]]
+ [[xz]y] E
e k,
[hk].
Thus the product oftwo ideals is an ideal. It follows that all the subspaces gi defined above are ideals of g. Thus we also have gn+l = (gng] c: gn and so we have a descending series g = g'
:::l
g2
:::l
g3
The Lie algebra g is called nilpotent if gi Lie algebra is nilpotent.
:::l ••••
=0
for some i. Every abelian
Example. The set of all n x n matrices (aij) over F with aij = 0 whenever i ~ j is a nilpotent Lie algebra under Lie multiplication [AB] = AB-BA.
1.3 Special kinds of Lie algebra
9
We now define a different sequence of subspaces g(01,g(ll,gC21, ... of g. We again define them inductively by g(OI =g,
The gCil are all ideals of g. Also we have g(n+ll = [g
and so we again have a descending series g = g(OI
::::1
g
::::1
g(2)
::::1 ••••
The Lie algebra g is called soluble if g
Proposition. Every nilpotent Lie algebra is soluble. Proof We show first that [gmgn] c: gm+n for all m, n. We proceed by induction on n, the result being clear if n = 1. Assuming inductively that [gmgn) c: gm+n, let X E gm, y E g 0 , z E g. Then we have [x[yz]] = [[xy]z]- [[xz]y] e gm+n+l by induction. Thus [gmgn+l] c: gm+n+l as required. We next observe that g(n) c: g2". This is clear for n = 0. Assuming it inductively we have g
as above. This completes the induction. We now assume that g is nilpotent. Then gm = 0 for some m. Hence there exists n with g2• = 0. It follows that g
0 Example. The set of all n x n matrices (ai;) over F with llij = 0 whenever i > j is a soluble Lie algebra. A Lie algebra g is called simple if g has no ideals other than g and 0. A Lie algebra g of dimension 1 is of course simple because g has no proper subspaces at all. We have g = Kx for some x e g. Since [xx] = 0 we have [gg] = 0. Such a !-dimensional Lie algebra will be called a trivial simple Lie algebra. We shall be mainly interested in non-trivial simple Lie algebras.
I Lie Algebras
10
1.4 The Lie algebras sln(CC) We shall now take F = CC. Let sln(CC) be the set of all n x n matrices of trace 0. sln(CC) is an ideal of gln(CC). For if A E sln(CC), B E gln(CC) we have trace[AB] = trace(AB- BA) = traceAB- traceBA
=0
since traceAB = traceBA for any two nxn matrices. Hence [AB] e sln(CC). Thus we see that gln(CC) is not simple. sln(CC) is, however, a non-trivial simple Lie algebra when n ~ 2. To see this suppose we have a non-zero ideal k and take a non-zero element in this ideal. By multiplying this element on the left or right by suitable elementary matrices Eij with i =I= j we may simplify its form, while remaining within the ideal k. Eij is the matrix with I in the i,j position and 0 elsewhere. Eventually we see that k contains some elementary matrix Eij, and by further multiplication we see readily that k is the whole of sln(CC). Thus sln(CC) is simple. We shall describle certain properties of sln(CC) in detail, because it is typical of simple Lie algebras in general. Let h be the set of diagonal n x n matrices of trace 0. Then h is a subalgebra of sln(CC) and dim h = n - I. Furthermore we have [hh] = 0, so h is abelian. We recall that g may be considered as a g-module, using [gg] c g. We thus have [hg] c g and so we may regard g as a left h-module. We may write down a decomposition of g as a direct sum of h-submodules: sln(CC) = h e
L C[Eij· i+j
We note that the 1-dimensional space C[Eij is an h-submodule since, for x e h, we have x = ( At
0 with At + · · · +An = 0 and
0 )
An
1.4 The Lie algebras sln(G::)
11
This h-module gives a 1-dimensional representation of h
X=
(A, ·.. A,) -.t;-.lj.
We note that there are n(n-1) 1-dimcnsional representations of h arising in this way. They are called the roots of sl0 (CC) with respect to h. Let ell be the set of roots. ell lies in h• = Hom (h, CC), the dual space of h. We note that if IX e ell then -IX e ell also since the map x - A.j - A.i is the negative of the map x - Ai - A.j. Thus the roots are certainly not linearly independent. The roots do however span h•. For define IXi e ell by IXj(X) = Aj - Ai+l·
Then 1Xt,IX2•····1Xn-1 are linearly independent and form a basis of h•. Let II= {1Xt,IX2,····1Xn-1}· II is called a set of fundamental roots, or simple roots. We consider the way in which the roots are expressed as linear combinations of the fundamental roots. The root x - Ai - A.j is equal to IXj
+ IXi+l + ... + IXj-1
if i < j
and to -(1Xj
+ IXj+l + ··· + IXi-1)
if i > j.
Thus each root in ell is a linear combination of fundamental roots with coefficients in Z which are either all non-negative or all non-positive. Thus we may write Cll = ell+ u .,- where cJ)+ consists of positive combinations of II and ell- negative combinations. We shall keep this example sin((:) in mind to illustrate the general theory of simple Lie algebras.
2 Simple Lie algebras over
l.l Cartan subalgebras Let g be a finite dimensional Lie algebra over of g we define I(h) by I(h) = {x e g; [yx]
cr:.
For any subalgebra h
e h for ally e h}.
It is readily checked that I(h) is a subalgebra of g containing h, and that his an ideal of I(h). Moreover if his an ideal of k then k is contained in I(h). Thus I(h) is the largest subalgebra of g in which his an ideal. I(h) is called the idealizer of h.
Definition A subalgebra h of g is called a Cartan subalgebra if h is nilpotent and h = I(h). Theorem Every finite dimensional Lie algebra g over cr: has a Cartan subalgebra. Moreover given any two Cartan subalgebras h,, h2 of g there exists an automorphism 8 of g (i.e. an isomorphism of g into itself) such that 9(h,) = h2.
Proof We shall not give the proof of this theorem, which is lengthy, but shall indicate briefly how a Cartan subalgebra can be obtained. For any element x e g we define the linear map adx : g- g by adx.y = [xy] Given any A. e C[: the eigenspace of adx with eigenvalue A. is {y e g; (adx- A.l)y = 0}. The generalized eigenspace of adx with eigenvalue A. is {y e g; (adx- A.l);y = 0 for some i}.
12
2.2 The Cartan decomposition
13
It is well known from linear algebra that g is the direct sum of its generalized eigenspaces for all A. E CC. (This is not true for the ordinary eigenspaces unless adx can be represented by a diagonal matrix). Let h(x) be the generalized eigenspace of g with respect to adx with eigenvalue A.= 0. We say that xis a regular element of g if the dimension of h(x) is as small as possible. It turns out that when xis regular h(x) is a Cartan 0 subalgebra of g. The fact that any two Cartan subalgebras are related by some automorphism of g is proved using a density argument and ideas from algebraic geometry. In fact one can use a special kind of automorphism of g, called an inner automorphism, to transform h1 to h2• Example Let g = sln(CC) and h be the subalgebra of diagonal matrices in g. Then his a Cartan subalgebra of g. Since [hh] = 0, h is clearly nilpotent. To show h = J(h) let EaijEij be any element of J(h). Choose p,q Epp - Eqq e h, hence
e
{1,· · ·,n} with p
:f=
i,j
q. Then
This gives
L aipEip - L OiqEiq - L apjEJi - L aq;Eci e h. i
i
Since this matrix is diagonal we deduce, by considering the coefficient of Epq, that apq = 0. Since this is true for all p, q with p :f= q we have EaijEij E h. Thus h = l(h). It turns out in fact that whenever g is a simple Lie algebra its Cartan subalgebras are abelian, as is the case in sl0 (CC).
2.2 The Cartan decomposition
Let g be a simple non-trivial Lie algebra over C[ and h be a Cartan subalgebra of g. Then we have [hh] = 0. Since we have [hg] c g we make regard g as a left h-module. h is then a submodule of g. In fact it is possible to express g as the direct sum of h with a number of 1dimensional h-submodules. Such a decomposition is uniquely determined
I Lie Algebras
14
by h. It is called the Cartan decomposition of g with respect to h. We write it as
G::e. is a 1-dimensional h-module. thus we have cx(x) E G:: for all x e h. ex lies in the dual space h• = Hom(h,G::) of h. The 1dimensional representations ex of h arising in the Cartan decomposition are called the roots of g with respect to h. The set of roots will be denoted by ~. Thus we have
and dim g = dim h+ I ~ I . The Cartan decomposition in the case g = sln(CC) was described in detail in §1.4. The root system ~ has the following properties. If ex e cJ» then -ex e Cl». Also the set ~ spans h•. However ~ is not linearly independent, so it is natural to choose a subset of ~ which will form a basis for h•. In fact such a basis can be chosen in rather a special way. There exists a subset n of Cl», called the set of fundamental roots, such that n is linearly independent and each « e cJ» can be expressed as a linear combination of roots in n with coefficients in Z which are either all ~ 0 or all =E; 0. Such a system n was given when g == sln(CC) in §1.4. The choice of the system n of fundamental roots is not unique. However once n is chosen we can define the sets ~+ and ~- of positive and negative roots. We have ~
== ~+ u ~-
and~-
== -C()+.
We shall denote by hit the set of elements of h• which are linear combinations of elements of n with coefficients in R.. The definition of hit is in fact independent of the choice of n since it consists of all real combinations of elements of~- We have dimll hit ==dime: h.
=dime h = l.
l is called the rank of the Lie algebra g.
2.3 The Killing form
15
l.3 1be Killing fom
We consider the map g x g -+ ([ defined by < x, y >= x,y -+ <x,y > trace(adxady). adx, ady and adxady are linear maps of g into itself. trace(adxady) is the trace of any matrix representing adxady, and is independent of the choice of such a matrix. Since traceAB = traceBA for any two square matrices A, B we have< x,y >=< y,x >.Thus we have a symmetric bilinear form on g. This is called the Killing form. We now assume that g is a non-trivial simple Lie algebra over «:. Then the Killing form on g is non-degenerate in the sense that < x,y >= 0 for ally E g implies x = 0. We may restrict the Killing form on g to h, to give a map h x h -+ Cl:. It can be shown that this map remains non-degenerate on h. Thus
x E h and < x, y >= 0 for all y E h implies x = 0. We may thus define a map h-+ h• given by x-+ fx where fx(Y) =< x,y > for ally E h.
This is a linear map from h to h•. Since the Killing form is non-degenerate on h this map is bijective. Thus each element of h• has form f x for just one x E h. We may thus define a map h. x h• -+ ([ by < fx,/y >=< x,y > for x,y E h. We may restrict this bilinear form to the real vector space hit. It can be shown that its values then lie in R.. Thus we have a map
This map has the property that < A., A. >~ 0 for all A. E hit. Moreover < A., A. >= 0 implies A. = 0. Thus the scalar product on hit is positive definite. hit is therefore a Euclidean space. This Euclidean space hit contains the set of roots cD. The properties of the configuration formed by the roots in hit is important in the classification of the simple Lie algebras g.
16
I Lie Algebras
Examples. Let g = sl2(CC). Then dimh = 1. Let ll = {ext}. Then cD = {cx~o-cxt}. The configuration formed by cD in the 1-dimensional is Euclidean space
hit
0 Now let g = sl3(CC). Then dimh = 2. Let n = {cx 1,cx2}. Then, as shown in §1.4, we have cD = {ext. cx2, txt + cx2, -cx~o -«2, -«t - «2}. The configuration formed by cD in the 2-dimension Euclidean space is
hit
2.4 The Weyl group The configuration formed by the root system cD is best understood by introducing a certain group of non-singular linear transformations of called the Weyl group. For each ex E cD let s11 : -+ be the map defined by
hit
hit
hit
1 2
(X, ( 1)-A<ex.«>
S11A
Note that s11(cx) = -ex and s11(l) =A. whenever < cx,.A. >= 0. Thus s11 is the reflection in the hyperplane orthogonal to ex. Let W be the group generated by the maps s11 for all ex e cD. W is called the Weyl group. W has some favourable properties. In the first place it permutes the roots, i.e. w(cx) E cD for all ex E cD and all w e W. It follows that W is finite, since there are only finitely many permutations of cD, and each such permutation comes from at most one linear transformation since cD spans Also we have cD = W(ll), i.e., given any ex e cD there exists cx1 e n and w e W such that ex = w(cx;). Moreover W is generated by the S11; for CXj E ll. The importance of the Weyl group is that it enables us to reconstruct
hit.
2.4 The Weyl group
17
the full root system cJ) given only the set 11 of fundamental roots. For given 11 the Weyl group is determined, being the group generated by the reflections s~~; for CXi e 11. The root system cJ) is then determined, since cJ) = W(l1). Hence, given 11, the root system (J) is obtained by successive reflections s11; until no further vectors can be obtained. An example when g = sl3(CC) is shown in the figure.
Given 1X1,cx2 the remaining roots are obtained by reflecting successively by s111 , S111 • We note that S11;(1Xj)
= IXj -
+
2 < IXj,IXj >
< IXj,IXi >
IXj.
If CXj, CXj E 11 with i j s11;(«j) is a root, so is a Z-combination of «i and tXj. Since the coefficient of CXj is 1 the coefficient of CXi must be a non-negative integer, since the given root lies in (J)+. It follows that
2 < IXj,IXj > _ _;;;,.....:'--- E < IXj, IXj >
Z' 2 <
IXj,IXj > < IXj, IXj >
~
'"""
O•
2 < lXI, lX) > We define A,1 = < ex,, «i 5 . The numbers Aij are called the Cartan integers and the matrix A = (Aij) which they form is called the Cartan matrix. We have Aij E Z, Aii = 2 and Aij <. 0 if i j. Let 8ij be the angle between CXj, CXj· This angle can be determined by the cosine formula
+
< IXj,IXj >=< IXj,IXi > 1' 2 < IXj,IXj >l/l COS8ij·
Thus we have
Hence 4 cor 8ij =
AijAji·
18
I Lie Algebras
We shall write nij = AijAji· Then nij E Z and nij -1
~
cosOij
~
~
0. Moreover, since
1
we have 0 ~ 4cos2 6ij ~ 4. In fact when i =I= j we have Oij =I= 0 and so 0 ~ 4cos2 Oij < 4. Hence the only possible values for Rij are nij = 0, 1, 2, 3. We shall now encode this information about the system mental roots in terms of a graph.
n of funda-
2.5 The Dynkin diagram The Dynkin diagram A of g is the graph with nodes labelled 1, · · · , I in bijective correspendence with the set n of fundamental roots, such that nodes i, j with i =I= j are joined by Rij bonds. Example Let g Sar2 (1XJ)
= IXJ + tX2.
= s/3(CC}.
Thus A12 = -1, A21
Then
= -1
n = {«t.«2}
and s111 (a2)
= a 1 + a 2,
and so n12 = 1. Thus A is the graph
0---___,0 2
The Dynkin diagram is uniquely determined by g. The choice of Cartan subalgebra does not matter since any two Cartan subalgebras are related by some automorphism of g. The choice of fundamental system n does not matter, since it can be shown that any two fundamental systems 0 1, D2 have the property that 0 2 = w(D 1) for some we W. The Dynkin diagram of g has the following properties. A is a connected graph provided g is a non-trivial simple Lie algebra. Any two nodes are joined by at most 3 bonds. Also let Q(x 1, • • • ,XJ) be the quadratic form I
Q(xJ, · · ·, x,)
= 2 L x~ i-1
L .jnijXiXj. ij
i+J
This quadratic form is determined by the Dynkin diagram. For example if A is 12 then we have Q(XJ.X2) =
2xt + 2x~- 2XJX2.
2.5 The Dynkin diagram
19
Now the quadratic form Q(x~o · · ·, XJ) is positive definite, since we have
We shall consider the problem of determining all graphs 11 with the above properties.
'lbeorem Consider graphs 11 with the following properties: (a) 11 is connected.
(b) The number of bonds joining any two nodes is 0, 1, 2, 3. (c) The quadratic form Q determined by 11 is positive definite.
Then 11 must be one of the graphs on the following list, A1 0
A2
o--o
As
o--o--o
B2
Bs
~
~
A4
o--o--o--o
--------- -·
B4 o---o--<:t:::D
~--------· Es
~~~ Fig. 2.1. list I.
The graphs on this list will be called Dynkin diagrams. Proof A subgraph of a graph 11 is one obtained from 11 by removing certain nodes or decreasing certain bond strengths or both. For example a:=o is a subgraph of~ . The list of graphs given in the theorem will be called the standard list We note that each subgraph of a graph on the standard list is also on the standard list It is not difficult to show that if the quadratic form of a graph 11 is positive definite, then the quadratic form of any subgraph of 11 is positive definite also.
20
I Lie Algebras
Now the quadratic form of a graph ll is represented by a symmetric matrix
We recall from linear algebra that Q(xt, · · · , Xt) is positive definite if and only if all the leading minors of M have positive determinant. However the leading minors of M are simply matrices M corresponding to certain subgraphs of fl. In order to show that Q(x" · · · ,xt) is positive definite it is therefore sufficient to check that det M > 0 for each graph ll on the standard list. This is readily verified. We now wish to prove conversely that the graphs on the standard list are the only ones satisfying the given conditions. In order to do this we introduce a second list.
c6 A3 q OA4 =--------· ~~~ A2
n
62 u
D
~
~
>-zr-< - - - -·
r~~ o---o--a:::=o--
i'4
Fig. 2.2. list 2.
It may be readily checked that each graph ll on list 2 has a quadratic form Q(xt, · · · , Xt) with symmetric matrix M satisfying det M = 0. Thus Q(xt. · · · , Xt) is not positive definite. Hence any graph ll satisfying our given conditions can contain no subgraph on list 2. Let ll be a graph satisfying our conditions (a), (b), (c). Then ll has no cycles, otherwise ll would contain a subgraph of type At. ll has at
2.5 The Dynkin diagram
21
one multiple bond, otherwise ll would contain a subgraph of type cannot have both a m~ltiple bond and a branch point, otherwise ll would contain a subgraph B,. Also ll cannot have more than one branch point, otherwise ll would contain a subgraph i51• Suppose ll has a triple bond. Then ll must be G2, as otherwise ll would contain a subgraph G2. We may therefore assume that ll contains no triple bond. Suppose ll has a double bond. Then ll contains no branch point, so is a chain. If the double bond is at one end of the chain then ll = B,. If not ll must be F4, since otherwise ll would contain a subgraph F4. Thus we may assume ll contains only single bonds. If ll has no branch point then ll = A,. Thus we suppose that ll contains a branch point. This branch point has only 3 branches, otherwise ll would contain a subgraph D4• Let the lengths of the branches be lt. h, l3 with I = It + l2 + l3 + 1 and 11 ~ l2 ~ l3. Then l3 = 1, otherwise ll would contain a subgraph E6. Also l2 ~ 2 otherwise ll would contain a subgraph E7. If l2 = 1 then ll = D,. So we may suppose h = 2. We then have It ~ 4, otherwise ll would contain a subgraph Es. If It = 2 then ll = E6. If It = 3 then ll = E,. If It = 4 then ll = Es. Thus ll must be one of the graphs on the standard list. D
1110st 1• ll
c
We now consider to what extent the Dynkin diagram determines the matrix of Cartan integers. We recall that nij
=
if j
AijAji
and that Aij, Aji are integers ~ 0. Moreover Aij = 0 if and only if Aji = 0. If nij = 0 we must therefore have Aij = 0 and Aji = 0. If nij = 1 we must have Aij = -1 and Aji = -1. If nij = 2 however, there are two possible factorisations of nij· Either we have Aij = -1, A11 = -2 or we have Aij = -2, A11 = -1. Since k _ 2< IJ-
<
> >
IXj, IXj
IXj,IXj
we have Aij Aji =
< IXj, IXj > < IXj,IXj >.
Thus in the first case above we have < txj, txi > > < IXj, OCj > and in the second case < txi, txi > < < IXj, IXj > . We distinguish between these two cases by putting an arrow on the Dynkin diagram pointing towards the long root. (The arrow can be interpreted as an inequality of root lengths.) In the first case we have the diagram
22
I Lie Algebras io
;>
oj
and in the second case the diagram
Similarly if nij = 3 we get two possible factorisations nij = Ai;A;i which are distinguished by putting an arrow on the given triple bond. In the cases when A is B2, F4, G2, it does not matter in which direction the arrow is inserted, since the graphs are symmetric. However when A is Bt for I ~ 3 we can obtain two different diagrams by inserting an arrow. Those diagrams will be labelled Bt, C1 as shown:
Bt o--o----o----o-----------~
Thus in type Bt the last fundamental root is shorter than the others, whereas in type Ct it is longer than the others. The main theorem on the classification of the finite dimensional simple Lie algebras over CC is as follows.
Theorem. Let g be a finite dimensional simple non-trivial Lie algebra over G::. Then the Cartan matrix of g is one of those on the standard list Bt I ~ 2, E6, E,, Es, F4, G2.
At I
~
1,
Ct
1~3,
Moreover for any Cartan matrix on the standard list there is just one simple Lie algebra, up to isomorphism, giving rise to it. The classification of the simple Lie algebras was achieved by W. Killing, in a series of papers in Mathematische Annalen between 1888 and 1890, and independently by E. Cartan in his thesis in Paris in 1894. The dimensions of the simple Lie algebras may be calculated as follows. The Dynkin diagram determines the configuration formed by the set n of fundamental roots, i.e., the angles between the fundamental roots and their relative lengths. We may then obtain the full root system ci» by successive reflection by elements of the Weyl group, as explained earlier. Finally, since we have
2.5 The Dynkin diagram
23
it follows that dimg
= dimh + 1~1.
The dimensions of the simple Lie algebras are given in the following table. dim A1 = 1(1 + 2) dim B1 = 1(21 + 1) dim q = 1(21 + 1) dim D1 = 1(21-1) dim G2 = 14 dim F4 =52 dim E6 = 78 dim E1 = 133 dimEs= 248. The algebras of classical type A1, B1, C1, D1 can be described conveniently in terms of matrices. The simple Lie algebra A1 is isomorphic to the Lie algebra sl1+1(C£:) of all (I+ 1) x (I+ 1) matrices of trace 0. The simple Lie algebra D1 is isomorphic to the Lie algebra so21(C£:)of all 21 x 21 skewsymmetric matrices. Although this is the simplest description of this Lie algebra, another is more convenient. D1 is isomophic to the Lie algebra of all 21 x 21 matrices T satisfying the condition
TA+AT 1 =0 where A = (
~ ~
) . The advantage of this description of g is that
the diagonal matrices in g form a Cartan subalgebra h, and the Cartan decomposition can be readily obtained. The simple Lie algebra B1 is isomorphic to the Lie algebra S021+t(C£:) of all (21 + 1) x (21 + 1) skew-symmetric matrices T. It is also isomorphic to the Lie algebra of all (21 + 1) x (21 + 1) matrices satisfying T A +AT1 = 0 where
A=(i:, :) The simple Lie algebra cl is isomorphic to the Lie algebra of all 21 X 21
24
I Lie Algebras
matrices T satisfying T A +AT' = 0 where
A= (
0 It). -It 0
Each of these Lie algebras is the Lie algebra Lie G of some Lie group G, i.e., the tangent space of G at the identity element with suitable Lie multiplication. (G is not uniquely determined up to isomorphism by its
Lie algebra). For the classical types we have Type At
slt+t(CC)
=
Lie SLt+t(CC)
S02l+ I ( (;)
=
Lie S02t+t(G::)
SP2t(CC)
=
Lie Sp21(CC)
Type Ct
S02t(CC) =
Lie S02t(CC)
Type Dt
Type Bt
The exceptional simple Lie algebras G2, F4, E6, E,, E8 can be constructed in terms of the Cayley algebra or algebra of octonians a. Given any algebra A a derivation D : A ...... A is a linear map such that D(ab) = Da · b+a ·Db.
If Dt.D2 are derivations of A so is [Dt D2] = DtD2- D2Dt. The derivations of A form a Lie algebra DerA. The Lie algebra of derivations of the Cayley algebra a over CC is the simple Lie algebra G2. The vector space of all 3 x 3 hermitian matrices over the Cayley algebra a forms a Jordan algebra J under the operation
A·B= AB+BA. 2 We have dim J = 27. The Lie algebra of all derivations of J is the simple Lie algebra F4 • The algebras E6, E7, Es can also be constructed by making use of a and J in different ways.
3 Representations of simple Lie algebras
In the present section we shall discuss the finite dimensional irreducible g-modules, where g is a simple non-trivial Lie algebra.
3.1 The universal enveloping algebra Let g be any finite dimensional Lie algebra over tensor algebra of g.
cr:.
Let T{g) be the
T(g) = C[;l $ g $ (g ®g) E9 (g ® g ®g) E9 • • • T(g) is a vector space over cr: on which a multiplication is defined in a natural way. Let I be the 2-sided ideal of T(g) generated by all elements of the form X
® y - y ®X - (xy) for
X, y
E g.
Let U(g) == T(g)/1. U(g) is an associative algebra called the universal enveloping algebra of g. A basis of U(g) can be obtained as follows. If Xt, • · ·, Xn is a basis of g then the set of elements irEZ,ir~O
forms a basis of U(g). This is called the Poincare-Birkhoff-Witt basis theorem. In the special case when [gg] = 0, i.e., g is abelian, we have XiXj == XjXi in U(g) and so U(g) is isomorphic to the polynomial ring CC[x 1, ••• x 0 ]. In general, however, Xi, Xj do not commute and we have instead XjXj - XjXi
==
(XiXj).
Thus U(g) is a kind of non-commutative polynomial ring.
25
26
I Lie Algebras
The importance of the enveloping algebra U(g) is that it has the same representation theory as g. If V is a g-module then V can be regarded as a T(g)-module in a natural way. Since [xy]v = x(yv)- y(xv)
for x,y E g, v E V we see that (x®y- y®x- [xy])v =0
for all v E V. Thus elements x ® y- y ® x- [xy] lie in the kernel of V. This kernel is a 2-sided ideal of T(g), so contains I. Thus V may be regarded as a U(g)-module since U(g) = T(g)/1. Conversely every U(g)-module may be regarded as a g-module using the map g - T(g) - U(g). This map is injective by the PBW-basis theorem, and so g may be regarded as a subspace of U(g).
3.2 Verma modules
We now suppose that g is a non-trivial simple Lie algebra. Let h be a Cartan subalgebra of g and
be the Cartan decomposition of g with respect to h. We recall that the Killing form gives a bijection h +-+ h". Let h, e h be the element corresponding to < oc~~J1 5 E h" under this bijection. Then «j(hi)
OCj,OCj > = 2< = Aij E Z.
Thus all the fundamental roots «1,. · ·, «1 take integer values at hi. h1, · · ·, h1 form a basis of h. They are called the fundamental coroots. Let A. e h" and J(A.) be the left ideal of U(g) generated by the elements e«, oc E C()+, and hi- A.(hi)l fori= 1, ···,I. Thus J(l) =
L
I
U(g)e« +
L U(g)(hi- A.(hi)l).
J(A.) is a U(g)-submodule of U(g). Let M(A.) = U(g)/J(A.). Then M(A.) is
3.3 Finite dimensional i"educible modules
27
also a U(g)-module, called the Verma module determined by ..t We have a natural homomorphism U(g).! M(l) of left U(g)-modules. Let m;. = 6(1). Then we have e«m;.
him;.
= =
0 for allac e e~»+ l(hi)m;. for i = 1, · · ·,I.
Since each element u e U(g) satisfies u = ul, each element of M(l) has the form um;. for some u E U(g). Thus M(l) = U(g)m~ is a cyclic U(g)-module. We may regard M(l) as an h-module. M(l) decomposes into the direct sum of !-dimensional h-submodules. The !-dimensional representations Jl e h• obtained form these submodules are called the weights of M(l). l is a weight of M(A.) since him;.= l(hi)m;.. All the weights of M(.A.) tum out to have the form
l - mt!Xt- • • · - mt!Xt where ac 1, • • • , IXJ are the fundamental roots and m1, • • ·, m1 are non-negative integers. lis thus in a natural sense the highest weight of M(l). M(.A.) is called the Verma module with highest weight l. It can be shown that M(l) has a unique maximal submodule K(.A.). Let L(.A.) = M(l)/K(.A.). Then L(.A.) is an irreducible U(g)-module. We thus have a procedure for constructing irreducible g-modules. For each .A. e h• we have obtained an irreducible g-module L(l) as the top quotient of the Verma module M(.A.). L(l) is not necessarily finite dimensional - this depends on the choice of l.
3.3 Finite dimensional irreducible modules Theorem dim L(A.) is finite if and only if .A.(hi) E Z, l(hi) ~ 0 for all i = 1, ... 'l. A. e h• is called integral if l(M e Z for all i and dominant integral if in addition l(hi) ~ 0 for all i.
28
I Lie Algebras
Theorem Every finite dimensional irreducible g-module has the form L(A.) for some dominant integral A. E h•. Thus we have a bijective correspondence between finite dimensional irreducible g-modules, up to isomorphism, and dominant integral weights A.. This classification of the irreducible g-modules goes back to E. Cartan's thesis of 1894. The dominant integral weights can be described conveniently in the following way. Let Wi E h. satisfy
Wj(hi) Wi(hj)
1
=
0 for j =I= i.
The elements w1, • • • ,Wt of h• uniquely determined in this way are called the fundamental weights. They form a basis of h•. Let A. e h. and
We see from this that the dominant integral weights are precisely the non-negative integral combinations of the fundamental weights. We consider the relation between the fundamental weights w,, · · · , Wt and the fundamental roots ex 1, ···,ex,. Let I
exi
= L:mijWj. j=l
Then we have exi(hj)
= mijWj(hj) = mii· By definition of hi we have
- ( ""IJ mIJ.. -- N·(h·) ""I 1 N•
2ex·1
< exj, exj >
)
--
2 < ex·1• ex·I > < exj, exj >
-
-
A··Jl•
Thus I
exi
= LAjiWj. j=l
Thus the transpose of the Cartan matrix transforms the fundamental weights into the fundamental roots.
Examples Suppose g has type A,. Then A= (2) and so ex, = 2w,. Thus w 1 =!ex,.
3.4 Weyl's character and dimension formulae
29
Now suppose g has type A2. Then we have
A= (
2 -1 ) -1 2
and so «t
=
2rot
-w2
«2
=
-WI
+ 2w2.
The fundamental roots and weights are shown in the following figure. a2
Wt
We note that ro2 is orthogonal to «t. Wt is orthogonal to oc2, and + W2 = «t + «2. 3.4 Weyl's character and dimension formulae
Suppose A. e h• is dominant and integral, so that L(A.) is a finite dimensional irreduable g-module. For each Jl e h• we define L(A.)11 = {v
e L(A.);xv =
Jl(x)v for all x
e h}
The Jl E h• for which L(A.)11 =/= 0 are called the weights of L(A.). L(A.)11 is called the Jl-weight space of L(A.). Its dimension dim L(A.)11 is called the multiplicity of the weight Jl in L(A.). We would like a formula which will enable us to find dim L(A.)11 for all Jl. Now all weights Jl of L(A.) are integral, although not necessarily dominant Let X be the set of aU integral weights. Then x~ze
... ez
is a free abelian group with basis w1, ••• ,w, of the fundamental weights. Let ZX be the integral group ring of X. Its elements are finite sums E RiAi where ni e Z and Ai e X. To give the dimensions of the weight spaces L(A.)11 is equivalent to giving an element E dimL(A.)11 Jl of ZX. peX
Now we have a problem in working in the group ring ZX since we have two types of addition, viz addition in X and addition in ZX. In order to
I Lie Algebras
30
eliminate the confusion arising because of this we define a multiplicative group e(X) isomorphic to the additive group X. e(X) consists of elements e(l) for l e X where
= e(lt + l2).
e(lt)e(l2)
We then work in the group ring Ze(X). We define the character of L(l) by char L(l) =
L dim L(l) e(#l) e Ze(X). 11
peX
Now Ze(X) is an integral domain, so can be embedded in its field of fractions. H. Weyl determined a formula giving the character of L(l). Let p = (1) 1 + ... + (1)1• (One can show that pis also given by p = 1/2 E ex). liE~
1beorem (Weyl's character formula).
E
charL(..l)=
detwe(w(..t + p))
weE detwe(w(p))
.
weW
This is an equality in the field offractions of Ze(X). We note that det w = ±1 for each we W. This is because W is generated by reflections s11 and det s11 = -1 for each such reflection. Example. Suppose that g has type At. Then the dominant integral weights are those of form mco1, where m e Z. m ~ 0. We consider the character of L(mcot). We have p = (/)J. Thus Weyl's character formula gives E detwe(w(m+ 1)(/)t) charL(mrot) =
E
wew
detwe(w(wt))
weW
Now the Weyl group W is generated by the single reflection s111 . Thus W = {1,s41 } has order 2. We have det 1 = 1 and dets11 1 = -1. Hence e((m + 1}wt)- e(-(m + 1)(/)t) char L(mwt) = e(wt)- e(-wt) e(wt)m+l _ e((l)t)-(m+t) = e((/)1) - e((/)1 )-I --•- ...._., -2 H Now ... ·z-!-• = zm + zm + · · · + z-. ence char L(mcot) = =
e((l)t)m + e((l)t)m-2 + · · · + e((l)t)-m e(mcot) + e((m- 2)(/)t) + · · · + e(-mcot).
3.4 Weyl's character and dimension formulae
31
Thus mw1, (m-2)wt, ... , -mw1 are the weights of L(mwt) each occuring with multiplicity 1. In particular we see that dimL(mwt) = m+ 1. -mw -(m-2)w
(m-2)w mw
By specialising Weyl's character formula one can obtain a formula for the dimension of L(A.).
Theorem. (Weyl's dimension formula)
n
dim L(A.) = -'ore;;.;.•. . ,.+- - - 0 ore~
A slightly different version of this formula is useful for calculating the dimensions in practice. For each positive root IX E ~+ we may express IX as a linear combinations of the fundamental roots by I
IX= LkjiXj, i=l
The dominant integral weight A. can be expressed as a combination of the fundamental weights by I
A.= 2:miWi, i=l
Let IX E n be a fundamental root of minimal length. Then we know that for each fundamental root IXi e n we have
< IXj, IXj >= Wj < IX, IX > where Wi = 1, 2 or 3. Wi = 1 if IXi is a short root and Wi = 2 or 3 if IXi is a long root. Wi = 3 only in type G2• With this notation we have the following corollary of Weyl's dimension formula I
Ekiw;(mi + 1) 1 1---:dimL(A.) = 0 '--..;... 1 --ore~ "k LJ jWj i=l
Examples. Let g have type A1. Let A.= m1w1. Then we have a single positive root 1Xt and so dim L(A.) = mt
+ 1.
32
I Lie Algebras
Now let g have type A2. Let A.= m1ro1 + m2ro2. This time we have
4>+ =
{tXt. tX2, IXJ
+ tX2}.
We have Wt = 1, w2 = J since rxt, rx2 have the same length. Thus dimL(A.)
= (mt +
l)(m2 + ~)(mt +m2 +2).
The dimensions of some of these irreducible g-modules are shown in the figure
24
3.5 Fundamental representations The modules L(roi) i = l, ... ,l are called the fundamental irreducible gmodules. If these are known all the others can be obtained as submodules of their tensor products. Thus if A. = mt rot + · · · + mtCOJ then L(l.) is a submodule of L(rot) ® · · · ® L(rot) ® · · · ® L(COJ) ® · · · ® L(ro,)
.,._---m•------+
~---ma---~
We shall describe the dimensions of the fundamental representations of the simple Lie algebras of classical type A,, B1, c,, D1• It is convenient to
3.5 Fundamental representations
33
write these dimensions on the appropriate node of the Dynkin diagram. The dimensions of the fundamental modules are as follows I+ 1 ( 1 ~ 1 ) At
('t 1)
I+ 1
C>--0---0----
21+1 ( 2 3 Bt C>--0---0----
Dt
(2'+1) 1-1
2'
--a::;:m
(1-1 21 ) - (1-3 21 )·• (2') 21 ) I - (1-2
~-----------~
2~----------~ 21-t
~21-1
The representation coming from the fundamental module L(w 1) is called the basic representation. This representation gives the description of gas an algebra ofnxn matrices where n = 1+1,21+1,21,21 respectively for types At. Bt, c., Dt. Most of the other fundamental representations can be obtained by considering exterior powers of the basic representation. Let V be the module giving the basic representation. Let T(V) be the tensor algebra of V, given by T(V) = (:1 $ V E9 (V ® V) E9 (V ® V ® V) e
· ··.
Let I be the 2-sided ideal of T(V) generated by elements v ® v for all v e V. Let /\(V) = T(V)/1. 1\(V) is called the exterior algebra of V. We have /\(V) = /\0(V) $/\ 1(V) $/\2(V) e
· ··
where /\1(V) is the image of T 1(V). We have dim /\1(V) = ( dimV) i . Let v1 /\ ···I\ Vi E /\1(V) be the image of Vt ® · · · ® v; E T 1(V). /\1(V) can be made into a g-module by the rule
I Lie Algebras
34 for all x
e g. We have
The /\;V are called the exterior powers of the g-module V. If g is of type A, then the exterior powers /\;V fori= 1,2, ... ,1 give all the fundamental g-modules. If g is of type B1 the exterior powers /\;V fori= 1,2, ... ,1- 1 give fundamental g-modules. There is one remaining fundamental module not given in this way. L(c.o1) is called the spin module. It has dimension 2' and can be constructed from an algebra called the Clifford algebra of
v. If g is of type C1 the exterior powers /\;V of the basic module are not in general irreducible. There exist g-module homomorphisms
(expansion)
(contraction)
for 1 ~ i ~ I - 1. 8 is injective, t/J is surjective and we have
1\i+l V
= ker 4> E9 imfJ.
The modules ker 4> for i = 1, ... , I - 1 give the fundamental modules in addition to V. Now let g have type D,. Then the exterior powers /\;V for i = 1, ... , 1-2 give fundamental g-modules. This time there are two remaining fundamental modules L(C.OJ-t), L(c.oJ). They both have dimension 2'-1• They are called the spin modules and are constructed using the Clifford algebra of V. We shall also give the dimensions of some of the fundamental modules for the simple Lie algebras of exceptional type.
7
G2
3.5 Fundamental representations 14
26
273
1274
52
F.~
21
e2'1) e;) e2'1)
21
78 56
133 E1 o----o--~r---o---~---o
248
Es
912
35
4 Simple groups of Lie type
It is possible to use the theory of simple Lie algebras over CC to construct simple groups of matrices over any field. This was discovered by C. Chevalley. We shall outline Chevalley's theory in the present section.
4.1 A Chevalley basis of g Let g be a non-trivial simple Lie algebra over CC and h be a Cartan subalgebra of g. Let
be the Cartan decomposition of g with respect to h. We aim to choose a basis of g adapted to this Cartan decomposition with favourable properties. We first choose a basis of h consisting of the fundamental coroots h1, • • ·, ht. It is useful to define the coroot hrt e h corresponding to any root a e C!l. This is the element of h corresponding to e h• under our isomorphism h -+ h•. It can be shown that any coroot hrt is a linear combination of the fundamental coroots ht. · · ·, ht with coefficients in Z. We then choose non-zero vectors ert e grt. Since dimgrt = 1, ert is determined up to a scalar. One can show that [erte-rt] is a non-zero multiple of hrt. Thus we may choose ert e grt, e-rt e g-rt such that [erte-rt] = hrt. We next consider the product [ertep] where a + P =I= 0. We have, for
<;;>
x eh, [x[ertep]]
= =
+ [ert[xep]] ~Z(x)[ertep] + P(x)[ertep] [lxert]ep]
(a+ p)(x)[ertep] 36
4.1 A Chevalley basis of g Thus if ex + P fl. we have [e11ep] = 0 and if ex + [e11 ep} E gm+P· We suppose ex+ P E and let
37
pE
we have
[e11ep] = N11,/Je11+P We also have
One can show that
where p is the non-negative integer such that p, -cx + p, -2cx + p, · · ·, -pex + P are roots but -(p + 1)ex + P is not a root Such an integer p certainly exists since the set of roots is finite. It is in fact possible to choose the vectors e11 E g11 such
whenever ex + P E <1>. The multiplication of basis elements is then given by
[hihj]
=
[hi ell]
=
[e~~e-~~l
=
0 2 < CXj,IX > e11 < exj,IXi > h11, a Z- combination of h1. · · ·, h1
[e11ep]
=
±(p + l)e11+/l if IX+ p E
[e11ep]
=
0 if IX +
p ~ , IX + p :i= 0
Thus the Lie product of any two basis elements is a Z-combination of basis elements. This kind of basis is called a Chevalley basis. The choice of Chevalley basis is not in general unique. Now the Chevalley basis described above has an even more favourable property than the fact that the multiplication constants lie in Z. For any IX E , A. E C[, we consider the map ad(A.e~~)
given by ad(A.e11 )x = nilpotent, i.e.
A.[e~~x].
:g - g
It is not difficult to see that this map is
(ad(A.e11 ))k = 0 for some k.
I Lie Algebras
38
We can then form the linear map expad(..te01) = 1 + ad(A.e01)
+T
ad(.te ) 2
+ ···
One can show that this map is an automorphism of g. (The sum on the right is finite since ad(..te.) is nilpotent). The Chevalley basis has the following very favourable property. expad(..te11 ) transforms every element of the Chevalley basis into a linear combination of basis elements with coefficients which are polynomials in A. with coefficients in Z. This is in spite of the denominators appearing in the formula for the exponential! Let A 01(A.) be the matrix representing expad(~) with respect to the Chevalley basis. Then the entries of the matrix A 01 (A.) are polynomials in A. with coefficients in Z. Let Gac~(G:) be the subgroup of the group of automorphisms of g generated by the elements expad(..te01 ) for all ex e cJ) and all A. e ct. This is called the adjoint algebraic group with Lie algebra g. It is a simple group. We shall now show how one can consider analogous groups over any field. 4.2 Chevalley groups over an arbitrary field
Now let k be any field. Then for each ex e cJ) and each J.t e k we have a non-singular matrix A01(J.t) obtained by replacing the indeterminate A. by the element J.t e k. Let Gac~(k) be the group of non-singular matrices over k generated by the matrices A01 (J.t) for all ex e cJ) and all J.t e k. Gac~(k) is called the adjoint Chevalley group of type g over k. It turns out that the group Gac~(k) is simple, apart from a small finite number of exceptions when k is finite. Examples (i) Suppose g is of type At. Then Gad(k) is isomorphic to PSLt+t(k). the projective special linear group of degree I + 1 over k. We have PSLt+t(k) = SLt+t(k)/Z where Z is the centre of SLt+t(k). (ii) Suppose g is of type Ct. Then Gac~(k) is isomorphic to PSP2t(k). the projective symplectic group of degree 21 over k. We have PSP2t(k) = SP2t(k)/Z where Z is the centre.
4.3 Finite Chevalley groups
39
(iii) Suppose g is of type Dt. Then Gad(k) is isomorphic to P02t(k,f0 ). Here
and 02t(k,fo) is the commutator subgroup of the orthogonal group 02t(k,fo). This is the group of all non-singular linear transformations of 21-dimensional vector space over k leaving invariant the quadratic form fo with symmetric matrix
(iv) Now suppose g is of type Bt. Then Gad(k) is isomorphic to P02l+t(k,fo). This time 02t+t(k,fo) is the commutator subgroup of 02t+t(k,fo), the orthogonal group of all non-singular linear transformations of (21 + 1)-dimensional vector space over k leaving invariant the quadratic form f 8 with symmetric matrix
4.3 Finite Chevalley groups We now consider the special case when k is a finite field. We recall that the number of elements in any finite field is a prime power, and that for each prime power q = pc there is just one field Fq, up to isomorphism, with q elements. When k = Fq we shall write Gad(k) = Gad(q). The number of elements in Gad(q) turns out to be given by an order formula of the following type: IGad(q)l = !qlcz,+l(qd• - 1)(qd2- 1) ... (qd• -1). d Here dis a small number bounded by an integer independent of q. The numbers d1,- • ·, dt are certain positive integers which can be obtained from the root system cJ) as follows. For each IX E cJ)+ we can write
We define the height of IX by ht
IX
= kt + ··· + kJ.
I Lie Algebras
40
Thus the fundamental roots are the roots of height 1. Suppose there are r 1 roots of height 1, r2 of height 2, etc. One can show that
and r1 +r2 +· · · = lq,+l. Thus we obtain a partition of lq,+l. This partition can be represented by a diagram with rt squares in row 1, r 2 squares in row 2, etc. For example the diagram of the partition 3221 is
I
The dual partition is the partition whose parts are the lengths of the columns of this diagram. For example the dual of 3221 is 431. Since r 1 = 1 the dual of the partition r 1, r2, r3 · · · will have 1 parts. Let them be m1,m2•··,mt. Then the numbers d1,···,d1 we require are given by di =mi + 1. The group Gad(q) turns out to be a finite simple group, except in the cases A1(2), A1(3), B2(2), G2(2). These are called the simple Chevalley groups. Their orders are given in the following table. 1(1+1)/2 IAt(q)l =
q
(I+
(q 2 -1)(q3 -1) ... (ql+l - 1)
t,q- 1)
J2
q
IBt(q)l = ICt(q)l =
(2,q -1)
1(1-1) IDt(q)l =
q
(4,q•- 1)
IG2(q)l = q6(q 2
-
(q 1
-
(q 2 - 1)(q4 - 1) · · · (q 21 - 1)
1)(q4 - 1) ... (q 21 - 1)
l)(q6 - 1)
IF4(q)l =q24(q2 -l)(q6 -1)(qs -l)(qt2 -1) 36 12 2 5 6 8 9 IE6(q)l = 1)(q -l)(q -l)(q -1)(q -l)(q -1)(q -1) 63 IE7 (q)l = q (q2_ 1)(q6-l)(qs_ 1)(q1o_ 1)(q12_ 1)(q14_ 1)(qts_ 1) (2,q-1) IEs(q)l = qt20(q2-t)(qs_ 1)(qt2_t)(qt4 _ 1)(q1s -l)(q2o _ 1)(q24 _ 1)(q30 _ 1)
(J.: _
4.4 Twisted groups
41
4.4 Twisted groups The finite Chevalley groups are not the only finite simple groups obtainable from the Lie theory. There are also twisted groups, obtained independently by R. Steinberg and J. Tits. One can obtain twisted groups whenever the Dynkin diagram (including arrows) has a symmetry. The possible symmetries are
Ar
Dr
Suppose the Dynkin diagram of g has a symmetry IX
-+
li. This symmetry
I Lie Algebras
42
has order 2 or 3. We suppose the field k has an automorphism A - I of the same order. We note that a finite field with an automorphism of order 2 must be Fq2 with I = Aq and a finite field with an automorphism of order 3 must be Fql with I = Aq (or I = Aq\ Let U be the subgroup of Gad(k) generated by the matrices A11(A) for a e II, A e k and V be the subgroup generated by the A11(A) for -a e II, A e k. Then Gad(k) is generated by U and V. Now there is an automorphism q : U - U uniquely determined by IX
Similarly there is an automorphism
q :
V - V uniquely determined by -
Let
ua = {u E U;q(u) =
E II, A E k.
IX
E II, A E k.
u}
ya = {v E V;q(v) = v}.
Let G1(k) be the subgroup of Gad(k) generated by ua and va. Then G1(k) turns out to be a simple group, again with few exceptions. In fact there is only one exception, when g = A2 and k = F 2z. G1(k) is called a twisted simple group. We obtain in particular finite simple groups
(The top suffix gives the order of the symmetry). Their orders are given by a formula IGl(k)l =
~~ ql•+l(qd• -
Et )(qdz- E2) ... (qd•- Et)
where Et. · · · , EJ are certain roots of unity. They are the eigenvalues of the symmetry of the Dynkin diagram. In particular we have:
where e
= e2Jt/il
4.5 Suzuki and Ree groups
43
4.5 Suzuki and Ree groups There are still further ways of finding finite simple groups from the Lie theory. These arise in those cases where the Dynkin diagram bas a symmetry without arrows, but not when the arrows arc included. These are the following cases:
In these cases the symmetry of the diagram does not extend to a symmetry of the root system because the simple roots which correspond under the symmetry do not have the same length. In spite of this, it is still possible to obtain simple groups over certain special fields. In type B2 this is possible only in characteristic 2. In the finite case one must take a field with 22e+ 1 clements, ic an odd power of 2. In type G2 this is possible only in characteristic 3. In the finite case one must take a field with 32e+l elements. In type F4 one bas characteristic 2 and a finite field with 22e+l elements. In these cases the subgroups U and V have an automorphism uniquely determined by
a(A.(.il.)) =
{
A«(.il.p~+•)
A«(.il.l")
if if if if
ex e n is short ex E n is long -ex E 0 is short -ex e n is long
(Here p = 2 or 3 as appropriate). Let Uti= {u E U;a(u) = u} and Vtl = {v E V;a(v) = v}. Let G1(k) be the subgroup of Glld(k) generated by Uti and Vtl. Then G1(k) is a simple group. It is called a Suzuki group when g = B2 and a Rec group when g = G2 or F4. The finite simple groups obtained in this way are 2B2(22e+l) 2G2(32e+l) 2F4(32e+l)
e;;?:: 1 e;;?:: 1 e;;?:: 1.
44
I Lie Algebras
It is convenient to write q2 = 22c+l or 32c+l as appropriate. (Thus q is irrational). With this choice of q we have an order formula
IGI(q2)1 = q141+1(qd• - E!)(qdl- E2) ... (qd• - Et) as before. To be specific we have 12B2 (q2)1 12G2(q2)1 12F4(q2)1
=
= =
q4(q2 _ 1)(q4 + 1) q2 = 22c+l q6(q2 _ l)(q6 + 1) q2 = 32c+l q24(q2 -1)(q6 + 1)(q8 _ 1)(q12 + 1)
q2 = 22e+l
The Chevalley groups, twisted groups, Suzuki and Ree groups over finite fields are called the finite groups of Lie type.
4.6 Classification of finite simple groups The classification of finite simple groups was completed in 1981, after many years of intense effort by a number of mathematicians. Every finite simple group is isomorphic to one on the following list: A cyclic group of prime order. An alternating group of degree n ~ 5. A finite simple group of Lie type. A sporadic simple group. There are 26 sporadic simple groups, of which the largest is the MONSTER. Most of them are subgroups of the MONSTER. It is interesting to consider to what extent the MONSTER is related to the Lie theory. It is known that the MONSTER is the automorphism group of an infinite dimensional algebra called a vertex operator algebra. Vertex operators appear in the representation theory of the infinite dimensional Lie algebras known as affine Kac-Moody algebras. These are Lie algebras corresponding to the extended Dynkin diagrams on list 2 of §2.5. Thus the MONSTER can be related to the theory of Kac-Moody algebras. Vertex operators are also important in string theory, the branch of theoretical physics which attempts to unify the possible particles and forces. It is intriguing to speculate whether the MONSTER will have a part to play in string theory.
Lie Groups Graeme Segal
Contents Lie Groups Introduetion 1 Examples l SU2, S03, and SL2R 3 Homogeneous spaces 4 Some theorems about matrices S Lie theory 6 Representation theory 7 Compact groups and integration 8 Muimal eompaet subgroups 9 Tbe Peter-Weyl theorem 10 Fanetions on R• and s•-• 11 Induced representations 1l The eomplexiftcation or a eompaet group 13 The unitary and symmetric groups 14 1be Borei-Weil theorem IS Representations or non-compact groups 16 Representations or SL2R 17 1be Heisenberg group A list or the groups that will be mentioned The circle group, T, pages 51, 82. The general and special linear groups: GL,.R, GL.G:, SL.R, SL.{;, pages SO, Sl, 108. Othogonal groups: o., SO., page 49. sot. the Lorentz group, page 54. o.. (t:), page 109. Unitary groups: U,., SU., pages 53, 73. su•.~. page 56. Symplectic and metaplectic groups: Sp,.(R), Mpl:~oo(R), page 131. The Euclidean group, E., page 49. The Heisenberg group, pages SO, 128, 131.
46
47
49 53 59 63
69 82 85 89 91 100 104
108 110 115
120 124 128
Introduction
These notes are an expanded version of the seven hours of lectures I gave at Lancaster. I have kept to the original plan and policy, which perhaps need some explanation. Roughly speaking, the contents are what I should like my own graduate students to know about Lie groups, and my general idea was to show how the theory is a natural continuation of basic linear algebra. As root systems and the classification of semisimple Lie algebras were treated in the companion lecture courses I felt I had an excuse for concentrating firmly on the general linear groups. But in any case I believe that is the right way to approach the subject: the taxonomic side of the theory is not to my taste. I tried to make my lectures useful to people with rather different amounts of mathematical knowledge and sophistication. That means the level is uneven: remarks aimed at the more advanced readers are scattered throughout, and are meant to be ignored by others. I hope the chapters can be read in almost any order: I tried to make them fairly independent. The first four are devoted to a survey of concrete examples of the theory to be developed. This is mainly "undergraduate.. material, and so I put it before the formal definition of a Lie group in Chapter S. But it does not need to be read in advance, and sometimes it uses terminology which is defined only later. More than half of the book - nearly everything from Chapter 6 on - is concerned with representation theory. I did not at first envisage that this would bulk so large, but in retrospect it does reflect my judgement of what is important. I feel sad that there is nothing about the differential geometry or algebraic topology of Lie groups: I should especially have liked to include the Chern-Weil theory of characteristic classes. I decided,
47
48
II Lie Groups
however, that I could not give a worthwhile elementary treatment of these things in the prescribed time. I strongly recommend Milnor's books on Morse Theory and Characteristic Classes to fill the gap. The text is now at least twice as long as what I actually said, although I have only added ..details", and some more proofs. I am not sure the expansion was a good idea: I may well have spoiled the overall perspective by over-egging, while I have certainly not produced a comprehensive treatise.
1 Examples
A good example of a Lie group is the group El of all isometrics of euclidean space .Rl. Euclidean geometry is the study of those properties of subsets of Rl which are preserved when the subset is transformed by an element of El, so to know what El is is the same thing as to know what is meant by Euclidean geometry. In general, Lie groups are the basic tools of geometry. Besides being a group a crucial property of El is that it has a topology, i.e. it makes sense to say that one clement is "near" another, or to speak of a "continuous path" in El. Thus El consists of two connected components, one formed by the elements which preserve orientation and the other by those which reverse it, and there is no continuous path from one of the former to one of the latter. A simpler example is the subgroup Ol of El consisting of isometrics of Rl which leave the origin fixed. This can, of course, be identified with the group of 3 x 3 real orthogonal matrices A. Again it consists of two connected components, the subgroup of matrices A with determinant +1, which is called SOl, and the coset of matrices with determinant -1. The group S0 3 consists of all rotations about axes through the origin in Rl. A rotation is determined by its axis and the angle of rotation, which is taken between 0 and n. Representing a rotation by a vector along its axis whose length is the angle of rotation, and observing that the direction of the axis becomes ambiguous when the angle of rotation is n, one sees that, as a topological space, SOl can be made from a solid ball in R 3 of radius n by identifying antipodal points on the boundary sphere. This produces a non-simply-connected space which is not easy to visualize.
49
so
II Lie Groups
Matrix groups The orthogonal group is an example of a matrix group, i.e. a closed subgroup of the group GL,.R. of invertible real n x n matrices (the composition being, of course, matrix multiplication). All matrix groups are Lie groups. The converse is almost, but not quite, true: all Lie groups are locally isomorphic to matrix groups, as will be explained. For the most part the groups we are interested in are matrix groups: the essential reason for preferring the more general concept is that the same group can be realized as a matrix group in many ways, and to make a particular choice often obscures its simplicity and introduces irrelevant features. For example, the additive group R can be identified with the positive 1 x 1 matrices (r), or with the 2 x 2 matrices of the form
( ~ ~)' or with the 2 x 2 matrices of the form ( cosh x sinh x
sinh x ) cosh x ·
The Euclidean group E3 is a matrix group because it can be identified with the 4 x 4 matrices of the form
with A E 0 3 and b E R 3. Another reason for considering Lie groups rather than matrix groups is that some groups closely related to matrix groups are not matrix groups. For example, in the group N of 3 x 3 real upper-triangular matrices with 1's on the diagonal the matrices of the form
n)
1 0 ( 0 1 0 0 0 1
with n e Z form a normal subgroup Z. But N/Z is not a matrix group, as we shall prove in on page 83. This group N /Z can be described in a quite different way. It is called the Heisenberg group, and is very important in quantum mechanics. It arises as a group of operators in Hilbert space, i.e., roughly speaking, as
51
1 Examples
a group of oo x oo matrices. On the Hilbert space L2(R.) let Ta be the operation of translation by a, i.e. (TJ)(x) = f(x- a).
Let Mb be the operation of multiplication by the function fil•ibx, and let Uc be multiplication by the constant fil'dc. Then the transformations of L2(R) of the form TaMbUc form a group which is isomorphic to N/Z.
Low dimensional examples We can list all the connected Lie groups of dimension
~
3.
There are just two connected 1-dimensional groups, R. and 'I' = { z e CC : I z I = 1} :!!! R./2nZ. They are locally isomorphic, in the sense I shall define in a moment There is only one connected 2-dimensional group which is not abelian, namely the group of affine transformations x ....__ ax + b of the real line, with a> 0. Up to local isomorphism, four new groups appear in dimension 3, namely
S03, SL2R., E2, and N, where N is the group of matrices
0i ;) already mentioned, and SL2R is the group of real 2 x 2 matrices with determinant 1. Of course there are also the 3-dimensional groups which are products of 1- and 2-dimensional groups. Most of the theory of Lie groups is exemplified by the groups just listed, and to begin with it may be best not to think about any others. The classification of so-called ..semisimple" Lie groups by Dynkin diagrams is enormously important in many areas of mathematics, but it is not very relevant to the kind of questions I shall be concerned with. For our purposes, it tells us that any semisimple group is an interlocking system of copies of SL2R and S03. The way the copies interlock can be described purely combinatorially by the techniques of root systems, which are treated in the accompanying lectures on Lie algebras.
52
II Lie Groups
Local isomorphism Two groups G1 and G2 are locally isomorphic if there is a homeomorphism f : u, ----+ u2 between neighbourhoods of the identity elements in the respective groups which preserves the composition law in the sense that f(xy) = f(x)f(y) whenever xy belongs to u,. The most obvious locally isomorphic groups are R and T: we can take f(x) = eh' for 1x I< 11:. A much more interesting example is the local isomorphism between SU2 and S03 which I shall describe in the next section.
2 SU2, S03, and SL2R
The group of 2 x 2 unitary matrices with determinant 1 is denoted by SU2. There is a homomorphism SU2 -+ S03 which is 2-to-1 and onto. It is of enormous importance in particle physics, because, while S03 can be regarded as the set of possible positions of a rigid body whose centre of mass is fixed at the origin, the set of states of an electron which is at rest at the origin is SU2 • The electron has two states for each way of orienting it in space, and one can change it from one to the other by rotating it through 2n about any axis. (See page 76.) The elements of the group S U2 can be written (2.1) where a, b are complex numbers such that I a 12 + I b 12 = 1. This is the same as the group of unit quaternions, i.e. quaternions q = t +xi+ yj +zk with t, x, y, z real and c2 + x 2 + y2 + z2 = 1. For quaternions can be identified with 2 x 2 complex matrices of the form (2.1) by
i-
(~~i)i- (~~~)k- (~~).
Thus SU2 is topologically a 3-dimensional sphere, easier to visualize than S03. Quaternions .were invented to describe rotations. Thinking of a quaternion as a real part plus a vector part, i.e. q = t + v with t e R and v e R 3, quatemion multiplication is defined by (tt
+ Vt)(t2 + V2) =
(ttf2- (Vt,V2)) + (ttV2
+ t2VJ + VJ
X V2),
where v1 x V2 denotes the usual vector product in three dimensions.
53
II Lie Groups
54 In terms of matrices,
R.l ++{skew hermitian matrices with trace 0} For any non-zero quaternion g V E R.3 => gvg-l E R.3,
so each g E SU2 defines a linear transformation T1 of R.3 by v t-+ gvg-1. If u e R.3 is a unit vector then g=cos6+usin6 is a unit quaternion, and T1 : R.3 -+ R.3 is rotation about the axis u through the angle 26. So g ...... T1 is a surjective homomorphism T :SU2
-soJ.
It is easy to check that the kernel of T consists of the two elements ±1: a rotation is represented not by one quaternion but by a pair ±g. Thus SU2 is a two-sheeted covering of S03, and is non-trivial in the sense that there is no continuous choice of a single quaternion representative for each rotation. For if {Ru.B }0<6Ei2x is the closed path in S03 consisting of rotations through 6 about u, and we choose 1 e SU2 to represent R,p, then we must choose go = cos + u to represent Ru.B· But then Ru.lx is represented by cos 1r: + u sin 1r: = -1 =/= 1, and the path {g6} in su2 does not close.
!
sin!
There are a number of closely related double-covering homomorphisms worth mentioning at this point (i) Thinking of the quaternions as R.4, an arbitary element of S04 can be written v 1-+ 8tVB2 1, where 81 and 82 are unit quaternions. This gives a double covering
(ii) The homomorphism T: SU2-+ S03 extends to a double covering
T : SL2CC-+ SOt.J• where sot.J is the Lorentz group (i.e. the elements of S~(R.) which preserve the quadratic form r- x 2 - y 2 - z 2 and also preserve the direction of time, ie. do not interchange the two sheets of the hyperboloid
55.
t2- x 2- y 2- z2 = 1). To see this, we identify .R4 with the 2 x 2 hermitian matrices by (t,x,y,z)- ( t ++ ~ yt- iz)' y IZ - X and let g E SL2d"; act on them by A~---+ gAg- 1• The quadratic form is preserved because
t+xy-iz) 2 2 = t - x -l- z 2 • det ( . y+1z t-x
(iii) Restricting T : SL2d"; covering
SOt3 to real matrices, we get a double
su2 Xsu2 - so4 defines a double covering n : so4 - so3 x so3 ,
(iv) Finally, the covering
for the two elements of SU2 X SU2 above g E S04 have the same image in S03 x S03. The homomorphism ll describes the action of S04 on A21R4 , which splits into two 3-dimensional pieces A~ e A'l_, the self-dual and anti-self-dual parts, under the action of S0 4 • The relation between SU2 and S03, and between SL2d"; and SOt.J is so important that it is worth giving an alternative description of it. Think of the unit sphere S 2 in 1R3 as the Riemann sphere E = d";u{oo} by stereographic projection, i.e.
(0, 0, I)
(x,y,z) E S2 ~ ~
+ 11f. =
x+iy E E. -z
-1 -
56
II Lie Groups
To g=
(a_~) -ba
in S U2 we associate the Mobius transformation Zl-+
az+b • -bz+a
This is a bijection l: -1:, which, when regarded as a map S2 precisely the rotation Tg.
-
S2, is
Any holomorphic bijection l: - l: is a Mobius transformation z 1-+ (az + b)j(cz +d), and one may as well assume that ad- be = 1. Changing the sign of a,b,c,d is immaterial, so the group of Mobius transformations is the quotient group of SL2«:: by its centre, which consists of the matrices ±1. This group is denoted by PSL2«::. The sphere S2 can be regarded as the "celestial sphere", i.e. the set of light rays through the origin in Minkowski space. Thus the Lorentz group acts as a group of transformations of S2• The surjection
SL2«::- sotJ tells us that the Mobius transformations of l: are precisely the Lorentz transformations of the celestial sphere S2• One way of looking at this, emphasised by Roger Penrose, is to say that the celestial sphere is naturally a !-dimensional complex manifold, and the Lorentz group is the group of all holomorphic bijections S2 - S 2•
A picture of SL2R. To visualize SL2R. it helps to notice that it is isomorphic to the group of complex matrices of the form
such that I a 12 -I b 12 = 1. This group is called SU1,1· In fact SL2R. and SU1,1 are conjugate subgroups in GL2«::, for g(SL2R)g-1= SU1,1. where g=
G~i)·
The group SL2R corresponds to the Mobius transformations z 1-+ (az + b)j(cz +d) of the Riemann sphere (; U {oo} which preserve the
2 SU2, S03, and SL2R.
57
upper half-plane H = {z e ([ : lm(z) > 0}, while SU1,1 corresponds to those which preserve the disc D = {z e ([: 1z I< 1}. The transformation z H (z - i)/(z + i) defined by g takes H to D. The group SU1,1 is homeomorphic to an open solid torus S 1 x D by
(b !) -(a/lal,b/a)eS
1
xD.
Whether regarded as SL2R. or as SU1,1, the group has three kinds of elements apart from the two elements ±1 which form the centre: (i) those with !trace I > 2, called hyperbolic, which in SL2 ([ are conjugate to ( ~ J.~l) for some A. e R.; (ii) those with !trace I < 2, called elliptic, which are for some a e R.;
~njugate
to
( ~ e
(iii) the interface, with Itrace I = 2, called parabolic, conjugate to
±(~
!)·
The elliptic elements, which form the sausage-like region B, are the union of all subgoups of SL2R which are isomorphic to the circle-group T. The closure of the region A consists of the elements with trace ~ 2. It is the union of all subgroups isomorphic to R. The region C is
58
11 Lie Groups
the elements with trace < -2. These do not belong to any 1-parameter subgroup. (See page 74.) The two kinds of 1-parameter subgroups in SL2R are related to the positive and negative values of the Killing form (see page 15). A neighbourhood of the identity element in S L2 R can be identified with a neighbourhood of 0 in the vector space 9 of 2 x 2 matrices of the form
a b+c) ( b-e -a by the exponential map (see page 73). In this notation, the Killing form on 9 is 8(al + b2 - cl), and the regions A and B correspond to the parts of 9 where the Killing form is positive and negative. We see here the simplest case of a general fact. A semisimple group is one for which the Killing form is non-degenerate. In a matrix group of this type the Lie algebra of a maximal compact subgroup is a maximal subspace of 9 on which the Killing form is negative-definite. The simply connected covering group of S L2R. is the infinite open cylinder got by unwrapping the torus S1 x D.
It is homeomorphic to R 3, and is not a matrix group (see page 130). The elements in the regions Ao and B belong to subgroups isomorphic to R, while those in Aic for k =I= 0 do not belong to any 1-parameter subgroup.
3 Homogeneous spaces
Lie groups arise as transformation groups. Spaces on which a Lie group acts transitively are known as homogeneous spaces. For example, (i) the sphere sn-l
= {x ERn :II X II= 1} is homogeneous under On;
(ii) the upper half-plane H = {z e CC : lm(z) > 0} is homogeneous under SL21R, acting by z 1-+ (az + b)f(cz +d); (iii) the space ~ of positive-definite real symmetric n x n matrices is homogeneous under GLnR. acting by (A, P) 1-+ AP At; (iv) the Grassmannian Gr,(R.n), defined as the set of all k-dimensional vector subspaces of Rn, is homogeneous under the action of GLnlR. but also homogeneous under the subgroup On. because any point of Gr,(R.n) has an orthonormal basis; (v) the space !l' of lattices in R.2 - a lattice is a subgroup isomorphic to Z E9 Z generated by a basis {v1, 1'2} of R.2 - is homogeneous under GL2R; and the subspace !l' 1 of unimodular lattices (those where the basic parallelogram {Vt. v2} has unit area) is homogeneous under SL2R..
When a group G acts transitively on a set X we can identify X with the set GI H of left-cosets of the isotropy group H of a point Xo e X. (To be "" is gH ~-+ gxo.) precise, H = {g e G : g.xo = .xo}, and the map G/H ..=..X In the five examples above we get 59
60
II Lie Groups
(i) sn-l ~ 0,/0,_ 1 by taking (ii) H (iii)
~
f!J ~
xo = e,, the n'11 basis vector of R";
SL21l/S02 by taking xo = i; GL,R/0,, by taking
xo =
1;
(iv) Grk(JR.") a!! GL,R./GLk,n-k ~ 0,/(0k x On-k), where GLk,n-k is the group of echelon matrices (~:) ;
So far in this section we have ignored topology. But in fact each of the spaces S"- 1,H,B', and Grk(R") has a natural topology (for Grk(R") see page 71), and the isomorphisms just listed are homeomorphisms between the natural topology and the topology which the set of cosets GI H acquires as a quotient space of G. The proof is in each case an easy exercise. The spaces of lattices !£ and !£ 1 have several remarkable and unobvious descriptions. It turns out that .f£1 is homeomorphic to the complement of a trefoil knot in R 3
(see [Milnor][2] page 84), while !£ is homeomorphic to the space of unordered triples of distinct points in the plane R 2, with centre of mass at the origin. Symmetric spaces
If X is a Riemannian manifold the group of isometrics of X is always a Lie group. For, because an isometry f : X - X preserves geodesics, it
3 Homogeneous spaces
61
is completely determined by /(Xo) and D/(Xo). where Xo is a base-point in X. A connected Riemannian manifold X is called a symmetric space if for each x e X there is an isometry f x : X - X which reverses geodesics through x, i.e. is such that Dfx(x) = -1. A symmetric space is always homogeneous, for any two points x and y can always be joined by a geodesic y, and fz(x) = y if z is the mid-point of y. In fact X ~ G/H, where H is the subgroup of G left fixed by an automorphism « of G such that «2 = 1. Symmetric spaces can be completely classified, and are of great importance in geometry. (See the book by Helgason.) The spaces in examples (i) to (iv) above are symmetric. For future reference (see page 119) we shall mention another important symmetric space.
Complex structures on R.ln Let J n be the space of complex structures on R 2n which are compatible with the inner product, i.e. .fn
= {J E 02n: J 2 = -1}.
For any J e .In it is clear that we can find an orthonormal basis {vi} of R ln such that JVUc-1 = V2k and JVUc = -V2k-l·
So any two Js are conjugate in 02n, and .In is the homogeneous space 02n/Un. The space J n can also be identified with the isotropic Grassmannian of CC2n, i.e. the set of n-dimensional complex subspaces W of ~n such that W = W J. with respect to the CC-bilinear extension of the innerproduct of R 2n. For a complex structure J is the same thing as a splitting CC2n = W e W into isotropic subspaces, where W and W are the (±i)-eigenspaces of J. This description of J n shows that it is a complex manifold. In fact it is a homogeous space of 02n(CC). To see this we choose a basis of 4::2n of the form {u~o ... , un; UJ. ... iln}. where (u1,u1} (ui,Uj}
=
=
(u1, u1} = {Jij·
0 and
II Lie Groups
62
(For example, take Utc = 2-!(e21c-1 + ie21c), where {e,} is the usual basis.) Then 02,.(«::) consists of the complex matrices g such that g'Ag =A, where A is the block matrix
(~ ~). and 0 211 is the subgroup of matrices of the form
Let P be the subgroup of 0211(~) consisting of matrices of the form
a
where denotes (a')- 1 and a- 1b is skew. P n 02,. = U,., and 02,./U,. ~ 02,.(~)/P .
It is easy to check that
Finally, it should be mentioned that alongside ,/11 there is another symmetric space Jlf,. = Sp2,.(R)/U,. formed by the complex structures on R 211 which preserve a skew rather than a symmetric bilinear form. The space Jlf,. is called the Siegel generalized upper half-plane.
4 Some theorems about matrices
In this section I shall recall four well-known theorems about matrices which have important generalizations as theorems about Lie groups. The first three describe canonical ways of factorizing a general invertible matrix. A The polar decomposition Theorem 4.1 Any invertible real n x n matrix g has a unique factorization g = pu, where p is a positive-definite symmetric matrix, and u is orthogonal.
Proof. One defines p = (gg1)i and u = p- 1g, observing that the positivedefinite symmetric matrix gg1 bas a unique positive-definite square-root. The positive-definite symmetric matrices form a convex open subset fJ' in the vector space p of symmetric matrices, so fJ' is homeomorphic to Rin
II Lie Groups
64
a linear combination of u,, ... ,U/c-l from v1c to obtain a vector D~c which is orthogonal to u., ... , Uk-l· Then the vectors u1c = D~c/ II iJ~c II form an orthonormal basis of C', and we have Vl
V2 VJ
= = =
AllUl
A12Ul AlJUl
+ +
A22U2 A2JU2
+
AJJUJ,
and so on.
If the vectors {v1} are the columns of an element g of GL,.CC then the {u1} are the columns of a unitary matrix u, and g = ub, where b is the upper-triangular matrix with entries (A.ij). Thus we have proved Theorem 4.2 Any g E GL,.CC can be factorized uniquely g = ub, where
u E U,. and b is an upper-triangular matrix with positive real diagonal entries. If B is the group of all n x n complex upper-triangular matrices then n B = T, where T ~ T" is the subgroup of diagonal matrices in U,.. So (4.2) implies U,.
Theorem 4.3 The natural map of homogeneous spaces
U,./T--+ GL,.CC/B is a homeomorphism. The homogeneous space U,./T ~ GL,.CC/B is very important in the representation theory of U,. and GL,.CC. It is the space of flags in CC": a flag is a sequence of subspaces
E, c E2 c: ... c E,. = C' with
dim(E~:)
= k. See page 174.
There is a corresponding theorem for any linear algebraic group : we shall meet it in Chapter 14. C Reduced echelon form: the Bruhat decomposition
v., ... ,
Once again, let { v,.} be the basis of C' formed by the columns of an invertible matrix g, and let us perform the same kinds of column operations as were used in the Gram-Schmidt process, i.e. multiplying a
4 Some theorems about matrices
65
column by a scalar, and subtracting from it a multiple of any column to its left. Thus once again we are really trying to find a basis for a flag. But this time, instead of trying to make the basis orthonormal, we construct the unique basis w = { Wt. ... , w,.} which is in reduced echelon form, i.e. like
. 100) ( w= • 0 • 1 1 0 0 0 0 0 1 0
'
in which (i) each column w, ends in a 1, say in the nfh row, and (ii) the entries to the right of each 1 vanish, i.e. W~r,J = 0 if j > i. The sequence
is necessarily a permutation of (1,2, ... ,n), and the matrix w is got by permuting the columns of an upper-triangular matrix
n=
( 10. 0) 0 1 • • 0 0 1 0 0 0 0 1
by n. In fact w = nn, where the permutation n is identified with the corresponding n x n permutation-matrix. We have proved
Theorem 4A Any element g E GL,.CC can be factorized g = nnb, where n belongs to the subgroup N of upper-triangular matrices with l's on the diagonal, 1t is a permutation matrix, and b belongs to the subgroup B of upper-triangular matrices. Equivalently, the permutation matrices n form a set of representatives for the orbits of the action of N on the homogeneous space GL,.f£1B. As expressed in (4.4) the factorization g = nnb is not unique, though the permutation n is uniquely determined by g. In fact the conditions (i) and (ii) characterizing w above can be reformulated as (i)' wn- 1 e N (ii)'pr 1w eN where N is the group of lower-triangular matrices with 1's on the diagonal. So we have
66
II Lie Groups
Theorem 4.5 The decomposition g = nnb is unique if n is chosen in N 11 = N nniln-1• Equivalently, the orbit of nB e GLnf:-1 B under N is isomorphic to N11 • The number of variable entries in the matrix w above is lx, the length of n, which is defined as the number of pairs i < j such that n1 > n1, i.e. as the number of crossings when n is depicted in the form
2
2
1t
4
3
3
= {3, 1,4,2} ~
4
111
= 3
This means that Nx is homeomorphic to the vector space C•, giving us Corollary 4.6 The orbits of N on the flag-manifold GLnf:-1 B decompose it into n! cells C11 , with C11 ~ N11 ~ cr'•. For future use let us notice that for almost all g e GLnf:- we shall have 1t = {n,n -l,n- 2, ... , 1} and lx = !n(n -1). In this case nne nN, and, replacing g by ng, we have Corollary 4.7 Almost all g and be B.
e GLnf:- have a unique factorization
g = fib with ii
eN
Everything in this section can be generalized from GLnf:- to any reductive Lie group over any field, even a finite field. In the general case the decomposition is called the Bruhat decomposition. The role of the permutation group is taken over by the Weyl group of G. (see page 16) It is instructive to consider the case G = GLnFq, where F q is a finite field with q elements. Then G has order
(qn _ l)(qn _ q) ... (qn _ qn-1) ,
4 Some theorems about matrices
67
the upper-triangular subgroup B has order (q - 1)11qin(n-ll, and N. has order q1•. The Bruhat decomposition gives us the identity
I G/B I==
L
I N. I •
•eS.
i.e.
D Diagonalization and maximal tori In the unitary group Un each element g is conjugate to a diagonal matrix. The diagonal matrices in Un form a torus T ~ 'r, and Tis a maximal abelian subgroup of U11 • In fact any abelian subgroup of U11 is conjugate to a subgroup of T. These facts are proved by elementary . linear algebra. No such simple statements can be made about GL11£. But for any compact Lie group G the picture is much as for U11 • We can always choose a maximal torus T in G, i.e. a maximal subgroup of the form 'r.
1beorem 4.8
If G is a compact connected Lie group then
(i) any element of G is conjugate to an element of T, and more gener-
ally, (ii) any connected abelian subgroup of G is conjugate to a subgroup of
T. In particular, any two maximal tori are conjugate. The word "connected" cannot be omitted in (ii): not every maximal abelian subgroup is a torus. Example. S01 is a maximal torus of SOl, and the statement (i) amounts to the fact that any element of SOl is a rotation about some axis. The diagonal matrices in SOl form a maximal abelian subgroup A with four elements, and A is clearly not isomorphic to a subgroup of the circle-group so1. There are various ways to prove Theorem (4.8). The crucial part is (i), for (ii) follows easily from (i) because any compact connected abelian
68
11 Lie Groups
group A contains an element g whose powers are dense in A (see [Adams] page 79), and then x- 1gx E T ===> x- 1Ax cT. There is a very attractive proof of (i) by means of algebraic topology. Again I shall refer to [Adams] (page 90) for more details, but I shall describe the idea. We first reformulate the assertion as a fixed-point theorem: to find x e G such that x- 1gx e T is the same as to find a fixed-point of the map fr : GIT - GIT defined by f 1 (xT) = gxT . The map fr depends continuously on g E G, and G is connected, so fr is homotopic to the identity-map f 1 of GIT. We now use a well-known theorem of topology. Theorem 4.9 If X is a compact space with non-zero Euler number then any map f : X - X which is homotopic to the identity has a fixed point. The Euler number is an integer-valued topological invariant x(X) defined for compact spaces X which possess a decomposition into cells. It is characterized by three properties. (i) x(X) = 0 if X is empty. (ii) x(X) = 1 if X is contractible . (iii) x(Xt U X2) = x(Xt) + x(X2)- x(Xt n X2). By decomposing the n-dimensional sphere S" as the union of two hemispheres we find, by induction on n, that (S") = { 2 if n is even X 0 if n isodd.
(The fact that any map S 2 - S 2 which is homotopic to the identity has a fixed point is the "hairy ball" theorem, probably the most famous result of elementary topology.) A similar argument shows that if X has a cellular decomposition with Ct cells of dimension k then x(X) = :E(-1)/cct.
To prove Theorem (4.8) we need to know that x(GIT) is nbn-zero. The Bruhat decomposition of GIT into even-dimensional cells tells us that x(GIT) is the order of the Weyl group of G, but there are easier proofs.
5 Lie theory
Smooth manifolds To define a Lie group we need the concept of a smooth manifold. A manifold is simply a topological space X which is locally homeomorphic to some Euclidean space R", i.e. each point of X has a neighbourhood U which is homeomorphic to an open subset V of R". Such homeomorphisms tp : U __. V are called charts for the manifold. A smooth manifold is a manifold X together with a preferred collection of charts tp11 : U11 __. Y11 which cover all of X and are smoothly related, i.e. for any a.,p the transition map tp11p = tpp o tp; 1 from tp"(U11 n Up) to tpp(U11 n Up) is smooth. ( I shall use "smooth" to mean C 00 , i.e infinitely differentiable.) The preferred collection of charts is called the atlas of the manifold X, or simply the "smooth charts". It is best to assume that the atlas is maximal, i.e. that any chart which is smoothly related to all the charts of the atlas belongs to the atlas. Example. One can cover the sphere
S2 = {(x,y,z) E R 3 : x2 + y2 + z2 = 1} by six open sets U1, ... , U6, where U1 consists of the points where x > 0, U2 consists of the points where x < 0, U3 consists of the points where y > 0,
and so on. There are obvious charts tp1 : U; __. J.j c R.2 ; for example, tp 1(x, y, z) = (y, z ). These charts are smoothly related -e.g. the transition map '1'13 is given by tpu(y,z) = (+(1- y2- z2)i,z).
69
II Lie Groups
70
They define a smooth structure on S 2 • Another chart belonging to the same atlas is the one given by stereographic projection (see page 55) from the north pole N = (0, 0, 1) to the equatorial plane z = 0. This is the homeomorphism 1p from U = S 2 - {N} to R.2 defined by 1p(x,y,z) = (x/(1- z),y/(1- z)). At first sight one would say that a manifold is a very natural concept, while a smooth manifold seems a cumbersome and inconvenient thing. But that turns out to be quite wrong: experience shows that smooth manifolds are very practical, while manifolds in general are intractable. It is worth emphasizing that a smooth manifold is completely described by giving the set of points and prescribing which real-valued functions on it are smooth. Manifolds sometimes arise embedded as submanifolds of Euclidean space, but sometimes they do not. The orthogonal group 03 is naturally a 3-dimensional submanifold of the space R.9 of all 3 x 3 matrices, defined by the six equations 03 = {A :A' A = 1}.
(There are six equations here, because A'A is symmetric.) Charts for 03 can be given in many ways. One elegant way is the Cayley parametrization: if U = {A E 03 : det(A + 1) =,': 0},
then a bijection 1p : U --+ V = {skew 3 x 3 matrices}
a:: R.3
is defined by 1p(A) =(A -1)(A + 1)-1• The group is covered by the open sets {gU} for all g e 0 3 - actually it is enough to let g run through the eight diagonal elements of the group- and a chart 1p1 : gU - V is defined by 1p1 (A) = tp(g- 1A.) A good example of a manifold which does not arise naturally as a subset of Euclidean space is the projective space Pl- 1 = P(R.n), which consists of all lines through the origin in R.n. A point of P".t1 is represented by n homogeneous coordinates (xt. ... ,xn). not all zero, and (xt. ...• xn) represents the same point as (A.xt. ... ,A.xn) if A. 0. If Un is
+
5 Lie theory
71
the part of P~t_- 1 consisting of points with Xn =I= 0 then we have a bijection V'n : Un - Rn- 1 given by V'n(XJ, ... ,Xn) = (x1x;1, ... ,Xn-1x;1). Obviously P~t_- 1 is covered by n such sets U1, ... , Un, with bijections VJ; : U; - Rn- 1. One readily checks that they define a smooth atlas. Notice that in situations like this we do not need to define a topology on ~-t explicitly: the atlas provides it with a topology which makes it a manifold. Only slightly more general is the case of the Grassmannian Grk(Rn), which is the set of all k-dimensional vector subspaces ofRn. A point W of Grk(Rn) is represented by an n x k matrix x of rank k, whose columns form a basis for W. In this case x and xA. represent the same point if A. is an invertible k x k matrix. For each k-element subset S of {1, ... , n} let Us be the part of Grk(Rn) represented by matrices x whose S'11 k x k submatrix xs is invertible. As with projective space, x ~--+ xx:s1 defines a bijection between Us and the vector space of (n- k) x k matrices. (For 1 is an n X k matrix whose S'" block is the k X k identity matrix.) The reader may like to check that the transition map between Us and Uris
xxs
x ...._ (a+ bx)(c + dx)- 1, where
is the permutation matrix corresponding to the shuffle which takes S to
T. If X and Y are smooth manifolds then by using the charts we can say what it means for a map f : X - Y to be smooth. To be precise, f is smooth ifi]}ofoVJ-1 is a smooth map from VJ(Unf- 1U) to V whenever 1p : U - Y and q, : fJ - Y are charts for X and Y. We can now give the long-postponed definition of a Lie group.
Definition 5.1 A Lie group is a smooth manifold G together with a smooth map G x G - G which makes it a group.
Any closed subgroup of GLnR is a Lie group, but I shall omit the proof. (A short elegant proof can be found in [Adams], pages 17-19.)
II Lie Groups
72
By applying the implicit function theorem to solve the equation xy = 1 for y in terms of x one finds that in any Lie group the map x ~--+ x- 1 is a smooth map G - G. Finally, the smoothness requirement in the definition of a Lie group is actually superfluous. But that is difficult and tedious to prove, and, as far as I know, it is a theorem without any applications. (It was proposed by Hilbert in 1900 as the fifth of his celebrated problems for the 20th century, and was proved by Gleason, Montgomery, and Zippin in 1953.) A closely related fact, but much more useful and quite easy to prove, is that any continuous homomorphism of Lie groups is smooth. Tangent spaces A smooth n-dimensional manifold X has a tangent space TxX at each point x. It is ann-dimensional real vector space. If X is a submanifold of R.N one can think of TxX as a vector subspace of R.N. We consider aU smooth curves y : (-e,e)- X such that y(O) = x. Then TxX is the set of all the velocity vectors y'(O) E R.N. Equivalently, if q, : V- X c R.N is a local parametrization such that ,P(y) = x, then TxX is the image of the linear map D,P(y) : Rn -R.N. (Here a local parametrization q, : V - X means a map which is the inverse of a chart tp : U- V, where Vis an open subset of Rn.) But we can define TxX without invoking the ambient space RN: an element of TxX is defined by a triple (x, tp, e), where tp : U - V is a chart such that X E u, and is a vector in Rn which we think of as the representative of the element of TxX with respect to the chart tp. A triple (x, tp, e> is regarded as defining the same tangent vector as (x, ip, ~) if and only if = DO(y}e, where 6 = ip o tp- 1 in a neighbourhood of y.
e
e
Example. If G = On. regarded as a submanifold of the n x n matrices, then T 1G is the }n(n-1)-dimensional vector spaceS of all skew matrices, and T1 G = gS = Sg.
Proof. For any skew matrix A the matrix e-A is orthogonal, so y(t) = ge-A defines apathy : R.- G such that y(O) = g and y'(O) = gA. Conversely, if y : (-e,e) - G is a path such that y(O) = g then by differentiating yry = 1 we find y'(Otg + g'y'(O) = 0, which shows that g- 1y'(O) is skew, i.e. that T1 G c gS.
5 Lie theory
73
Exercise. If G = Un then T1 G is the n2-dimensional real vector space of skew hermitian matrices.
Notation (i) A smooth map f
:X
- Y obviously induces a linear map TxX -
Tf(x)Y for any x EX, and it is natural to denote this map by Df(x).
(ii) If G is a Lie group, and g E G, there is a smooth map L1 : G - G given by left-translation, i.e. L 1 (x) = gx. This induces an isomorphism TxG- T1xG which I shall write simply as thinking of the matrixgroup example above. The corresponding isomorphism TxG - T xgG given by right-translation will be written
e....,. ge,
e...... eg.
One-parameter subgroups and the exponential map A homomorphism f : R- GLn(R) -a so called one-parameter subgroup - is necessarily of the form f(t) = f!A, where A is the matrix /'(0). For f'(t)
= =
=
lim h- 1{/(t +h) - f(t)}
h-oO
lim h-1{f(h) - 1}/(t)
h.....O
Af(t),
and the unique solution of the differential equation f'(t) = Af(t) such that /(0) = 1 is f(t) = f!A. Furthermore, the map exp : MnR- GLnR is bijective in a neighbourhood of zero, its inverse being the smooth map g ....,.logg defined, when II g - 1 II< 1, by log(1-A) = -'£Akfk. Theorem S.l In any Lie group G there is a 1-1 co"espondence between the tangent space T1G and the homomorphisms f: R- G.
Proof. The argument is essentially the same as for GLnR. A homomorphism f gives us a tangent vector /'(0) e T1 G. Conversely, A e T1 G defines a tangent vector field eA on G by eA(g) = Ag, and we have only to show that eA has a unique solution curve with /(0) = 1. The theory of differential equations gives us a solution
f: (-e,e)- G for some e > 0. It is a homomorphism where defined because both t .._. f(t + u) and t .._. /(t)f(u) are solution curves Of eA Which take the
II Lie Groups
74
value f(u) when t = 0. But then for any t e R the element f(t/n)" is defined for all sufficiently large n, and is independent of n, because
f(t/n)"
= f(t/nm)"m = f(t/m)m.
So we can define f(t) = f(t/n)" for any large n. We have, therefore, a map exp: TtG-G whose derivative at 0 is the identity. In general it is neither 1-to-1 nor onto, but by the Inverse Function Theorem there is a smooth inverse map which we call 'log' defined in a neighbourhood of 1 in G.
Examples. (i) If G = SLnR then Tt G consists of the n x n matrices with trace 0, because det(etA) = etrace(rAI. (ii) If G = SU2 then Tt G is the skew-hermitian matrices with trace 0, i.e. the pure vector quaternions R 3• If u e R 3 is a unit vector, regarded as a quaternion, then u2 = -1, so exp(tu) =cost+ u sin t . This is the 1-parameter subgroup of rotations about u. In particular, exp is surjective.
Remark. In fact exp is surjective in any compact group, for such a group has a Riemannian metric for which the geodesics emanating from 1 are precisely the 1-parameter subgroups, and in a complete Riemannian manifold any two points can be joined by a geodesic. In non-compact groups exp is usually not surjective. The next four examples can serve as exercises for the reader. (iii) (See Chapter 2) In SL2R the elements with trace > -2 are on 1-parameter subgroups. Those with trace~ -2 (i.e. those in the region C of the diagram on page 57) are not, with the exception of -1.
(iv) In SL2(; the matrix (
~2 ~!)
is on a 1-parameter subgroup,
but not on any !-parameter subgroup of SL2R. (v) In GL2(; the matrix (
~1 ~ 1 )
is on a 1-parameter subgroup,
but not on any 1-parameter subgroup of SL2C (vi) In GLn«= the map exp is surjective, as one can see by using the Jordan normal form.
15
5 Lie theory
Lie's theorems In a Lie group G with T1 G = g the map log: U- 9 inverse to the exponential map is a canonical chart defined in a neighbourhood U of the identity element. It is natural to ask what the composition law G x G - G looks like in this chart, i.e. how to express
C(A,B) = log(exp(A)exp(B)) in terms of A and B. atA=B=O:
We can expand C in a Taylor series
C(A, B) =A+ B + !b (A, B)+ (terms of order~ 3),
(5.3)
where !b (A, B) is the second order term. Because
C(A, 0) =A and C(O, B) = B the map b is necessarily a bilinear map
b:9x9--9. and it is skew because C( -B, -A) = -C(A, B). One way of stating the basic miracle of Lie theory is that (i) the infinite series (5.3) can be expressed entirely in terms of the bilinear map b, and (ii) the series converges in a neighbourhood of the origin. For example, the third order terms are
nb
b(A, B) = [A, B]
= AB -
BA,
so one writes [ , ] : 9 x 9 - - 9 for the skew bilinear map b in general. It is easily seen to satisfy the Jacobi identity [[A,B],C] + [[B,C],A] + [[C,A],B] = 0. In other words, it makes 9 into a Lie algebra. Example. If G = S03 then 9 is the 3 x 3 real skew matrices, and can
76
II Lie Groups
be identified with R 3• The Lie bracket R 3 x R 3 - R 3 is the "vector product" of elementary geometry. The Jacobi identity follows from the well-known formula
(ax b) x c = (a,c}b- (b,c}a. But the miracle of Lie theory is even better than I have said. The picture which Lie worked out is stated in modern language as Theorem 5.4 The functor taking G to T1 G is an equivalence of categories
between the category of connected simply connected Lie groups and the category of Lie algebras. This means that every Lie algebra g arises from a simply connected Lie group G, and that G is determined up to isomorphism by g. Futhermore, group homomorphisms Gt - G2 are in 1-1 correspondence with Lie algebra homomorphisms Tt Gt - Tt G2. The theorem reduces the study of Lie groups to questions in the vastly simpler realm of linear algebra. After 120 years there is still no altogether easy proof of Lie's theorem. I shall try to sketch the main ideas in the remainder of this section. Three preliminary remarks may be helpful. (i) Groups which are locally isomorphic clearly have the same Lie algebra. The theorem tells us that there is precisely one connected and simply connected group locally isomorphic to a given group. One half of this is easy: any group G is locally isomorphic to its simply connected covering group G, whose elements are pairs (g, y), where g e G and y is a homotopy class of paths in G from 1 to g. When G is S03 the definition of(; is appealingly illustrated by the party trick called "Dirac's spanner". An element g eGis represented by a rigid body such as a spanner or undergraduate, whose centre of mass is fixed. The path y is represented by a collection of strings which run from g's hands and feet to fixed points in space. Experiment shows that by rotating g one can get the strings y into exactly two states - "tangled" and "untangled" - for each position of g, and that rotating g through 360° interchanges the two states of y. (ii) When a homomorphism 8 : Tt G1 - Tt G2 of Lie algebras is given it is obvious that there is at most one group homomorphism I : Gt - G2 which induces it. For I is determined by its restriction to a neighbourhood of 1, and hence by its values l(expe) fore e TtGJ. But
5 Lie theory
77
f(exp~) =expO(~).
because t 1-+ f(expt~) and t 1-+ exptO(~) are both 1-parameter subgroups of G2 with the same derivative at t = 0. One must remember, however, that exp : T1 G - G is usually not surjective. (See page 74.)
(iii) If ~ is a sub-Lie-algebra of T1 G there is a Lie group H with T1H = ~ and a homomorphism H - G inducing the inclusion ~ - T1G. But the image of H in G need not be closed, and H need not be homeomorphic to any topological subgroup of G. The classical example is when G is a torus T x T and ~ c: R E9 R = T1G is a line of irrational slope. If~ has rational slope pfq the corresponding !-parameter subgroup f : R - G closes up and forms a circle in the torus after winding p times round the left-hand T and q times round the right-hand T. But when the slope is irrational f : R - G is injective, and its image is a curve which winds densely round the torus.
Turning now to the proof of Lie's theorem, I think the best place to start is with the problem of constructing a homomorphism f : G1 - G2 of Lie groups when one is given a homomorphism 0 : 91 - 92 between their Lie algebras. In a neighbourhood of the identity f can be defined by f(exp~) = One way to prove the theorem is to show that this is a homomorphism (where it is defined) by constructing the Campbell-BakerHausdorff series (5.3) explicitly, and proving that it converges. This is arduous; but it is elegantly described in [Serre]. Even then one has still to extend f to the whole group, which involves using the simpleconnectedness of G1• expO(~).
The method essentially used by Lie seems much more illuminating to me; in particular, it makes clear where the simple-connectedness is used. To define f(g) we choose a smooth path y : [0, 1] - G from 1 to g. Then we consider the path t ~--+ y'(t)y(t)- 1 = ~(t) , say,
in the Lie algebra 91• We transfer this to a path
t ~--+ O(~(t)) = ~(t) , say,
78
II Lie Groups
in 92 , then we solvet the ordinary differential equation
ql(t) = ~(t)rp(t) in G2, with initial condition rp(O) = 1. Finally, we define f(g) = rp(1). The main point is to show that rp(1) does not depend on the choice of the path "I from 1 to g. If we have two different paths then, using the simple-connectedness, they form part of a family of paths
{tl---+ .,,(t)}~ ..
all from 1 to g. Let
~(t,s)
=
:t"'s(t)."'s(t)- 1 E 91•
'l(t, s)
=
:s
"'s(t)."',(t)- 1 E g1•
We calculate that (5.5) This is called the Maurer-Cartanequation. Define~= Ooe and f1 = Oo,, so that - because 8 is a homomorphism of Lie algebras (5.6) This is precisely the compatibility condition which enables us to solve* the pair of equations
(5.7) to obtain rp: [0,1] x [0, 1]- G. (In modem language, the Maurer-Cartan equations (5.5) and (5.6) express the fact that the Lie-algebra-valued 1-forms A= edt+,ds and A= ~dt+flds arejlat connections on R 2• The equation (5.5) can be written dA = [A, A], and (5.6) is dA = HA,A].)
l
t If one thinb of "' as the path of an aircraft ftyina through G1, then
~ is the record in its ftight recorder, and rp is the path of an aircraft which fties in G2 accordina to the programme ~ = 9 o ~. :1: If (5.7) holds then (5.6) follows by equating iJ/iJs(iJrp/iJt) to iJ/iJt(iJrp/iJs). Conversely, if (5.6) holds one can fint define rp(t,O) by integratina iJrp/iJt = ~rp alona the lines • o, and then define rp(t,s) by integratina iJrp/iJs ~rp holding t constant Then (5.6) tells us that iJ/iJs{iJrp/iJt- ~rp} = 0, which implies that rp satisfies both equations (5.7).
=
5 Lie theory
79
Now 'I and if vanish when t = 1 by definition, so OfPios = 0 when t = 1, and qJ(1,s) is independent of s, as we want The most difficult part of Lie's theorem is the proof that any (finite dimensional) Lie algebra arises from a Lie group. The easiest route is first to prove Ado's theorem that g is isomorphic to a subalgebra of the Lie algebra of matrices Mn.R. Then we consider all smooth maps ~
: (0, 1) - + g c MnR.
which vanish together with aU their derivatives at 0 and 1. For each such ~ we solve the differential equation
(5.8) to obtain q>~ : [0, 1] -+ GLnR such that q>~(O) = 1. The elements GLnR. obtained by this process form a subgroup of GLnR.. for
q>~(l)
of
(5.9) where 'I • ~ : [0, 1]-+ g is the concatenation of~ and,, i.e. ('I. ~)(t) =
if { 2ec2t) 2'1(2t- 1) if
o~ t ~ t
t ~ t ~ 1.
(To prove (5.9) we observe that, if q>(
then
(/J
satisfies
!
t) _ { qJ~(2t) for 0 ~ t ~ qJ11 (2t -l)qJ~(1) for ~ t ~ 1,
(/J 1
!
= (, • e)qJ.)
The subgroup of GLnR so defined is almost, but not quite, the group we want to associate to g: unfortunately it may not be a closed subgroup of GLnR.. as the dense winding on the torus illustrated. Instead, we consider the vector space ~ of all maps ~ as above, and introduce the equivalence relation (5.10)
The quotient space ~I - is a topological group under the operation of concatenation. It is the group we want We must show it is locally homeomorphic to the Lie algebra g. But if e ~ is small then fp~ = log qJ~ is a well-defined path in MnR. We shall show that it is actually contained in g. This means that locally ~I- is the same as the space
e
II Lie Groups
80
of smooth paths 1/J : [0, 1] - g with q,(O) = 1, modulo the equivalence relation
In other words, locally &'I- looks like g. One must check that the composition law in &'I- is smooth, but that is easy. It remains to give the proof that the path 1/J~ lies in g. Its velocity is related by the derivative of the exponential map to q>~(t), and q>~(t)q>c(t)- 1 belongs to g. There is an elegant formula for the derivative of the exponential mapt : 1/J~(t)
cS(eA)e-A = F(ad A)cSA,
(5.11)
where F: End(MnR)- End(MnR) is defined by
= (~ -1)1x =Ex~' l(k + 1)!.
F(x)
k~
Here ad A e End(MnR) is given by ad A(B) =[A, B). The formula (5.11) shows that A(t) = 1/Jdt) satisfies the differential equation F(ad A)A' = ~(t)
(5.12)
for a function A : [0, 1] - g, and this completes the proof. I should, however, say a word about the derivation of (5.11 ). It is got by combining two results whose proofs can be left as straightforward exercises, namely
~(eA) = dt
t
}0
f!A dA e
for any smooth function A : R - MnR. and
Finally, it is easy to see how we could have avoided invoking Ado's theorem. It was used only to define the equivalence relation (5.1 0) on &'. We could have replaced that by prescribing ~o- ~~ if ~o and~~ are joined by a path {es} in&' such that the Maurer-Cartan equation (5.5) is satisfied for some, : [0, 1] x [0, 1] -g. This gives us a topological group &'I - as before. To prove it is locally like g we associate to each small t It holds in any Lie group.
5 Lie theory
81
ee &' a path 1/J, in g by solving the equation (5.12). The hardest step is to check that
eo- er <=> 1/J,o(l) = 1/J,J(l). We shall not do that here. The discussion of Lie's theorem I have given has the advantage that it can mostly be applied to infinite dimensional groups. though I should emphasize that the theorem is not true without restrictions for infinite dimensional groups. A very attractive account of the subject can be found in [Milnor][3], which influenced my treatment here.
6 Fourier series and representation theory
From now on these lectures will mostly be concerned with representations of Lie groups G. This means that we have a topological vector space V, and (i) each g E G defines a linear isomorphism V- V, written v 1-+ gv, (il) (g1g2)v = g1(g2v) for g1,g2 E G and v E V, and
(iii) (g,v) ~-+ gv is a continuous map G x V- V. We shall always assume that the vector space V is complex, and also locally convex and complete. The significance of the last two conditions is that they permit us to integrate any continuous function I : [a, b) - V to obtain an element l(x)dx of V.
J:
One of the most important theorems is mathematics is Fourier's theorem, which asserts that a smooth function I : 1l' - CC on the circle can be expanded in a Fourier series 1(8)
= I>nt"9 ,
where
nEZ
an= { I(O)e-inOd(J }y
2x
.
(6.1)
This can be viewed as a theorem about group representations. If V is a representation - perhaps infinite-dimensional - of the group T, then any ~ E V can be expanded (6.2)
and (6.3)
Here
Ro : V - V denotes the action of the group-element 8 on V. 82
6 Representation theory
83
The proof of (6.2) is exactly the same as that of (6.1): substituting the definition of a,. or en into the corresponding series one sees that each result is equivalent to the fact that the functions SN defined by
SN(O) =
_!_ "" einO 27t LJ
lnlllltN
tend to the delta-function cS(O) as N-+ ex:>. Furthermore, (6.2) really does contain (6.1). For if Vis the space C 00(T) of all smooth functions T-+ ([; then (6.3) implies that
(Rtxen)(O) = en(O -a)= e-tnmen(O) , and hence that en(a) = an~1111 , where an= en(O). We can state the result as Theorem 6.4
If Vis a representation ofT then V =
EB V,., where IIEZ
V,. = {e E V : Rtze = e-i•e for alia E T} .
EB
The notation is meant to imply that each Vn is a closed subspace of v' and that each E v has a unique convergent expansion = E en with en E V,.. In other words, V is a completion of the algebraic direct sum E9 V11 • But there are many possible completions: V cannot be reconstructed from the V,. without more information.
e
e
As a simple application of Fourier's theorem let us prove Theorem 6.5 The Heisenberg group N /Z of page 50 is not a matrix group.
Proof. N /Z has a circle subgroup T formed by the matrices
g,= for t
0! D
e .R. It is trivial to check that
(i) Tis contained in the centre of N/Z, and (ii) each element ofT is a commutator uvu-•v- 1 in N/Z. We must show that whenever we have a finite dimensional representation V of N/Z, ie. a homomorphism p: N/Z-+ Aut(V)
II Lie Groups
84
into the group of automorphisms of V, then p is not injective. But V can be decomposed V = E9 Vn under the action of the subgroup T. As Tis in the centre of N/Z each Vn is an invariant subspace for N/Z. Now g, e T acts on Vn by multiplication by e-lainr. But because it is a commutator it acts with determinant 1. This is a contradiction unless n = 0. So V = Vo, and T acts trivially on V, and p is not injective.
General remarks about representations A representation V is called i"educible if it has no closed G-invariant subspaces, except 0 and V. When a representation V is reducible, with a closed invariant subspace W, we can ask whether it decomposes, i.e. whether we can find an invariant subspace W' such that V = WED W'. In general we cannot. For example, the group of matrices of the form ( the subspace W = (
~
~
; ) acts on CC2 , and
) is invariant, but there is no other invariant
subspace. But if the representation V is unitary, i.e. if V is a Hilbert space and the inner product ( , ) is invariant, i.e.
(ge,g,) =<e.,)
e,,
for all g E G and E V, then W is invariant if and only if W.l is invariant, and V = WED W.l. In particular, a finite dimensional unitary representation is always a direct sum of irreducibles. We shall constantly make use of a trivial but crucial remark about representations, which is always called Schur's lemma.
If V1 and V2 are finite dimensional i"educible representations ofG then any G-map f: V1 - V2 (i.e. linear map such that f(ge) = gf(e)) is either zero or an isomorphism. Furthermore, if V1 = V2 then f = .:U for some A. e CC.
Lemma 6.6
The first half of the lemma holds because ker(f) and im(f) are invariant subspaces, the second half because any eigenspace of f is an invariant subspace.
7 Compact groups and integration
We should like to generalize the Fourier theory from T to other Lie groups. The only tool used was the ability to average functions over the group T. The same can be done for any compact group. If G is a compact group, V is a topological vector space (locally convex and complete), and f : G-+ Vis continuous, then we can define
Lf(g)dg
=fatE v.
More precisely, there is a continuous linear map
L:
C(G; V)-+
v.
where C(G; V) ={continuous maps G-+ V}, such that (i) fa f(g)dg = v if f(g) = v for all g, and (ii) fa f(hg)dg = fa f(gh)dg = fa f(g)dg for any h E G. In addition, the measure on G is positive. This means that (i) can be generalized to (iii) iff takes its values in a convex subset C of V, then fa f E C. From the general theory of integration on an orientable manifold we know that to define the integral it is enough to give a volume element at each point g of G, i.e. an alternating n-fold multilinear form w, on the tangent space T1G, where n = dim( G). We can choose Wt arbitrarily, and then define w, at other points g by identifying T1 G with Tt G by lefttranslation. This gives us a left-invariant integral J, obviously unique up to a scalar multiple, which can be chosen so that f 1 = 1. Performing a right-translation by h could only multiply the linear map f by a positive 85
86
II Lie Groups
real number #l(h), which would give us a homomorphism ll: G- R~. But if G is compact ll must be trivial, for R~ has no compact subgroups. So the integral is right-invariant as well. If G = SU2, regarded as the unit sphere S 3 , then f6 has the obvious meaning. It may be helpful to mention that if G = GL,.R, which is not compact, the formula
1f ! =
/(A) det(A)-"dA ,
where dA denotes ordinary Lebesgue measure on the vector space R 112 of matrices, defines a left- and right-invariant integral for functions f with compact support on G; but of course then the function f = 1 is not integrable.
A formula for integration on U,. It is not practical to give an explicit formula for integrating a general function on a group such as U,., for there are no convenient coordinates to use. But if the function f to be integrated is a class-function, i.e. if f(gxg- 1) = f(x) for all g and x, then there is a very elegant result Its form and its derivation illustrate some aspects of Lie group theory so well that I shall include it, even though I shall make no use of it A class-function on U,. is a symmetric function of the eigenvalues (ut. ... , u11 ) of the group element. Theorem 7.1Jf f
f f lu. where Ilk =
=
~ n.
: U,.- C[ is a class-function, then {2" ... {2" /(u~o ... ,u,.)
lo
lo
II lu;- ui~Ot ... ~6,. ' i<J n n
e«Jt.
The formula tells us, essentially, that the volume of the n(n- I)dimensional conjugacy class of the diagonal matrix with entries (ut. ... , u,.) is
IJ!u1 -ui. i<J
(This vanishes if two of the u; are equal, which is right, since then the conjugacy class has dimension less than n(n-1).) The 1/n! in the formula corresponds to the fact that each conjugacy class is counted n! times on
7 Compact groups and integration
87
the right-hand side: it could be removed if one restricted the integration to the region
Proof. Let G = U"' and let T denote the diagonal matrices. We consider the map T
X
(G/T) --+ (t,gT) ~--+
G gtg- 1 .
Let the Jacobian of this map at (t,gT), with respect to invariant measures on T,G/T, and G, be J(t): it is clear that it depends only on t and not on gT. Then for any function f on G we have
ff
la
=
~ f
n. lrx(G/TI
so if f is a class-function then
1
f = K- 1
f(gtg- 1)J(t)dtd(gT),
i
f(t)J(t)dt ,
where K- 1 = volume (G/T)/n!. To calculate J(t), let t and g denote the Lie algebras of T and G. We can identify the tangent space toG/Tat its base-point with tL, the orthogonal complement oft in g. If t changes infinitesimally to t(l + with e t, and g changes from 1 to 1 + ,, where '7 e t.L, then the change in gtg- 1, to first order, is
e>.
e
p
= =
(1 + '7)t(1 + e)(l- '7)- t te + ,t - t, .
So
and
J(t) = det(A(c 1) -1), where A(t- 1) denotes the adjoint action of t- 1 on t.L. The complexification of t.L has a basis consisting of the matrices EJk for j ::/= k, where EJk has 1 in the U,k) place and 0 elsewhere. Evidently Elk is an eigenvector of A(t-1) with eigenvalue ukuj 1, if t has diagonal elements (uJ. ... ,un). So
J(t) = IT
=II luij<.k
Ukl 2 •
88
II Lie Groups
We have now proved (7.1) apart from the calculation of the constant K = JT J(t)dt. But J(t) = l!\(t)l 2, where !\(t) =
IJ
is the Vandermonde determinant 1
1
uT-1 Expanding the determinant !\(t) =
E± u~•u~ · · ·u::'",
where (m~o ... , m11) runs through the permutations of {0, 1, ... , n -1 }, and integrating each of the (n!) 2 terms separately, we find that K = (2n)"n!. The preceding calculation has been arranged so that it applies essentially without change to any compact Lie group G with maximal torus T. The matrices Ejk are replaced in general by the root vectors, and J(t) becomes, in the notation of page 30, the function
IJ<e(cx)- 1) II
on T, where ex runs through the roots of G. As before we have J(t) = l!\(t)l 2, where !\(t) = IJ<e0
=L
detw e(w(p))
weW
is the denominator of the Weyl character formula. The equality of the last two expressions, which is often called the "Weyl denominator formula", generalizes the product formula for the Vandermonde determinant.
8 Maximal compact subgroups
In the following sections I shall discuss the representation theory of compact groups in some detail. One of the main reasons for being interested in it is to use it as a tool for studying non-compact groups. The following basic structural theorem is therefore important.
Theorem 8.1 In a Lie group G with a finite number of connected components there always exist maximal compact subgroups. If K is one of them, then any compact subgroup of G is conjugate to a subgroup of K, and in particular any two maximal compact subgroups are conjugate. Furthermore, G is homeomorphic to K x Rm for some m. I shall prove this only for GLnR. But first let us consider the five 3-dimensional groups described in Chapter 1 :
E2, SL2R., SU2, N, N /Z . (i) The group E 2 acts on R 2, and a compact subgroup H must leave a point fixed, for
eo=
L
hedh
e
is fixed under H for any e R 2 • SoH is contained in the isotropy group G~0 of some point of R.2, and G~0 is conjugate to the isotropy group 0 2 of the origin. So 02 is a maximal compact subgroup, and E2 is clearly homeomorphic to 0 2 x R 2 •
eo
(ii) SL2R. acts on the upper half-plane H, which is homeomorphic to R 2• Again, though it is not quite so obvious, a compact subgroup must leave fixed some point of H, and so the maximal compact subgroups are the isotropy groups, such as the isotropy group S02 of i E H. We saw in Chapter 4A that SL2R is homeomorphic to S02 x R 2•
89
II Lie Groups
90
(iii)
sul
is compact.
(iv) N is homeomorphic to R.l, and its maximal compact subgroup is
{1}. (v) N /Z has the circle T of elements g, (see page 83) as its unique maximal compact subgroup. It is a normal subgroup, and N /Z is homeomorphic to T x R 2• Now let us return to GLnR.. To prove that any compact subgroup K is conjugate to a subgroup of On it is enough to prove that the action of K on Rn preserves an inner product on R.". For we can find an orthonormal basis with respect to any inner product by the Gram-Schmidt process (see Chapter 4B), and after changing to the ortbornormal basis K will be represented by orthogonal matrices. We find a K-invariant inner product ((, )) by averaging the standard inner product (, ), i.e. we define
<<e.,))=
k
{ke,k,)dk.
We saw in Chapter 4A that, as a topological space,
GLnR
~
~
On x {positive definite symmetric matrices} On X Rln(n+ll.
A proof of Theorem (8.1) can be found in [Hochschild].
9 The Peter-Weyl theorem
We shall now describe the analogue of Fourier series for a compact Lie group G. Let V be a representation of G. We can assume V is unitary, as we can find an invariant inner product by averaging an arbitary one, as was described at the end of Chapter 8. If P is a finite dimensional irreducible representation of G we shall define a subspace Vp of V, called its P-isotypical part, consisting of the "vectors which transform according toP". This is analogous to the space
V,.
= {~ E V: ~~ = e-inlll~}
in the Fourier case, for the irreducible representations of T are the !-dimensional ones (} 1-+ ei"' We define Vp as the sum of all the copies of P contained in V. Equivalently, it is the image of the natural inclusion t
P®HomG(P;V)- V
(9.1)
where HomG(P; V) is the vector space of G-equivariant linear maps P-+ V. One version of the Peter-Weyl theorem is
Theorem 9.2 The isotypical part Vp is a closed subspace of V, and A
V=EB Vp, p
t
The map (9.1) is injective because it is injective when V = P by Schur's lemma, and hence injective when V is a sum of copies of P. But the left-hand side of (9.1) does not change if V is replaced by the sum of all the copies of P contained in V.
91
II Lie Groups
92
where P runs through the finite dimensioMl irreducible representations of
G. 1\
The meaning of the notation
E9 is explained after Theorem (6.4).
Corollary 9.3 All irreducible representations of G are finite dimensional. There are many equivalent but rather different-looking versions of the Peter-Weyl theorem. The second of the three I shall mention is
Theorem 9A Any compact Lie group is isomorphic to a subgroup of a unitary group U,. In particular, it is a matrix group.
Of course it is a matter of tradition, perspective, and taste w~ich theorems are regarded as "equivalent" to which, and it may seem odd to claim that Theorem (9.2) is equivalent to a theorem which is vacuous if G is U,. The argument in favour of the tradition will be given presently. To state the third version of the Peter-Weyl theorem we need a definition. A representative function on G is a function G -
1M;~•., =
(,,
C[
of the form
ge).
e,
where M is a finite dimensional unitary representation of G, and '1 E M. (In other words, I M:~,.,(g) is a matrix element of the action of g on M with respect to a suitable basis.) The representative functions form a subalgebra Catg(G) of the algebra C(G) of continuous functions on G, for
I M.;~I.IJI + I Mz;~z.IJz IMI;~I,IJI
•
IMz;~loiJZ
I M•GlMz;~le(MIEiliJz• =
I MI®Mz;~.®~z,IJI®IJz•
and A./M;(,IJ
=
IM;A(,IJ•
In fact Catg( G) is the coordinate ring (in the sense of Macdonald's lectures) of a linear algebra group Gc which I shall return to in Chapter
12. The third version of the Peter-Weyl theorem is
Theorem 9.5 Cata(G) is a dense subring ofC(G).for the topology of uniform convergence.
9 The Peter-Weyl theorem
93
Proof that (9.4) <==> (9.5 ). First suppose G c MnG::. Then the matrix entries and their complex conjugates belong to C1 t1( G). But by the Stone-Weierstrass theorem polynomials in these coordinate functions are dense in the continuous functions on the compact subset G of MnG::. Conversely, if Cat1(G) is dense in C(G) then the representative functions separate the points of G, so any g e G acts non-trivially on some finite dimensional representation V1 of G. Let us consider the kernels of G--+ Aut(V11 E9 ... ,E9V1.)
for larger and larger finite subsets g~. ... ,gn of G. Eventually the kernel must be {1}, for compact Lie groups obey the descending chain condition : in any decreasing chain either the dimension or the number of components must fall at each step.
Proof that (9.2) => (9.5). For any representation V of G we define the sub-vector-space G-finite vectors by yfin
=
yfin
of
{ ~ e V : ~ is contained in some finite dimensional G-invariant subspace of V}.
Because any finite dimensional representation is a sum of irreducible representations Theorem (9.2) is equivalent to the assertion that y&n is dense in V. So the implication we want follows from Theorem 9.6
If we let G act on C( G) by left-translation, then C( G)6n = Cata(G).
Proof If f belongs to a finite dimensional G-invariant subspace W of C( G) then f is a matrix clement of the action of G on W. For if {fl, ... ,fn} is an orthonormal basis of W, with f =ft. and g acts on W by then f(g)
E Mjl(g- )/j(l)
=
(g-l ft)(l) =
=
EMu(g}fJ(l),
1
so f is a linear combination of the matrix elements MIJ.
II Lie Groups
94
Proof that (9.5) ===> (9.2). We must introduce a new construction. For each f e C(G) we can define an operator T1 : V - V as a weighted average of the action of the elements of G: r1e =
fa
f(g)gedg.
(Physicists call this "smearing the G-action with f".) This makes V into a module over C(G) when the usual product in C(G) is replaced by the convolution product (f•• /2)(h) = 1ft(g)f2(g-1h)dg. Iff e C1Ja(G) then Tfe e V60 for any e e V, as one can easily check that gTfe = T,fe, and the functions gf, for all g, are linear combinations of the matrix elements of a single finite dimensional representation. There is no identity element in the convolution algebra C( G): the identity element "would be" the delta-function at 1 E G, which is not continuous. But the delta-function can be approximated arbitrarily closely by continuous functions f satisfying f = 1 which are supported in a small neighbourhood of 1 E G. So for any E V we can find f E C(G) such that Tfe is very close to If (9.5) holds then we can find fP in Cats very close to f. Thus 41 is very close to But E yftn, so V60 is dense in
J
Te
e.
e
e.
T.,e
V.
We have now shown that the various versions of the Peter-Weyl theorem are equivalent. If we are only interested in matrix groups there is no more to do. For general groups the most convenient version to prove is probably that C( G) 60 is dense in C( G). I shall give only a very brief sketch of the proof, referring to [Adams], or [Chevalley] for more details. For any f E C( G) the operator Tf : C( G) operator with a continuous kernel, for
C( G) is an integral
(Tpp)(x) =
Ia f(g) fP (g- x)dg
=
[ f(xg- 1)fJ (g)dg.
1
Such an integral operator acting on the functions on a compact space is a compact operator. It is self-adjoint iff is hermitian, i.e. f(g) = f(g- 1). The eigenspaces of a compact self-adjoint operator are finite dimensional. and any function in the range can be expanded in a uniformly convergent
95
9 The Peter- Weyl theorem
series of eigenfunctions. If we choose I to be conjugation-invariant (i.e. l(xyx- 1) = l(y)) then TJ commutes with the left-action of G on C(G), and so the eigenspaces are finite dimensional subspaces, and the eigenfunctions belong to C( G) 6n. So if I is a conjugation-invariant hermitian approximation to the delta-function at 1 e G its range is nearly the whole of C(G), and Ca~g(G) is dense in its range. That completes the proof. The structure of C8~g( G) The group G acts on the representative functions C81g( G) by lefttranslation and also by right-translation, so altogther we have an action of G x G on it The diagonal copy of G in G x G acts by conjugation. The whole situation can be described very elegantly.
Theorem 9.7 (i) There is an isomorphism of representations ofG x G
Ej1 P ® P - + Calg( G)
(9.8)
given by 'I® e......_. IP;~~,C• where P runs through the i"educible unitary representations of G, and the left and right copies of G act on P and P respectively.t (ii) Each P ® P is an i"educible representation of G x G. (iii) The isomorphism (9.8) takes the inner-product defined on the left as the orthogonal direct sum of the inner products ('It 0
on
1 -e•. ,2 0 e2> = dim(P) (, •• '12He •• e2>
P ® P to the usual L 2 inner product on C8 111(G).
Before giving the proof of this theorem let us notice some of its many useful corollaries. First, concerning characters. The character of a finite dimensional representation of V of G is the function xv : G - «: defined by xv(g) = trace(gy ), where gy : V - V is the action of g on V. We have t The complex conjugate representations P is identical with P as a set, but the scalar field ~
acts on it in the complex-conjugate way.
96
II Lie Groups
Corollary 9.9 ( i) A finite dimensional representation of G is determined up
to isomorphism by its character. ( ii) If P and Q are i"educible representations, then P~Q 1 if (xp, XQ} = { 0 if not
(iii) The characters of the i"educible representations form an ch"thonormal basis for the Hilbert space of class-functions on G, i.e. functions f such that f(xyx- 1) = f(y). To deduce the corollary from the theorem we notice that if {e;} is an orthonormal basis of an irreducible representation P then XP is the image of Ee; ® e; under the map (9.8). So (9.9)(ii), and hence (9.9)(i), follows from (9.7)(iii). But (9.9)(iii) also follows, as the conjugation-invariant elements of Catg( G) correspond to the part of ~ 1t ® P left fixed by the diagonal action of G , and this part is spanned by the single element E e, ® e;, because (P ® P)G ~ HomG(P; P) ~ CC
by Schur's lemma. Another corollary, or really reformulation, whose justification I shall leave to the reader, is Proposition 9.10 The map
Catg(G) -
E9 End(P)
which takes rp to the collection of "smeared'' operators rpp : P - P is an isomorphism of algebras when Cata( G) is given the convolution product. There is also an analytical aspect to Theorem (9.7). It tells us that any
f
E Catg(G) can be expanded as a finite series
f =LIP
(9.11)
where f p is in the image of P ® P, and that 2"" 1 2 II f II = L., dim(P) II fp II ,
(9.12)
where II f II is the L 2 norm, and II f p II is the natural norm on P ® P. When functions on the circle are expanded as Fourier series /(8) = 2.,a.~nB
9 The Peter-Weyl theorem
97
we know that there are many correspondences of the type L2 functions f C 00functions f real-analytic functions f
......-
square-summable sequences {an}, rapidly decreasing sequences {an}. exponentially decreasing sequences {an}.
Here "square-summable" means that E lanl 2 converges, "rapidly decreasing" means that {n"an} is bounded for all k, and "exponentially decreasing" means that {Kinlan} is bounded for some K > 1. These facts generalize to any compact group. The result for L 2 functions follows directly from (9.12). To state the others, which I shall not prove, one needs to know that the irreducible representations P of G are classified by their highest weights lp, and that lp has a norm lllp 11. (See Chapter 14.)
Proposition 9.13 Under the correspondence (9.11) L 2functions C 00functions real-analytic functions
f ...- square-summable sequences {fp }, f ...- rapidly decreasing sequences {fp }, f ...- exponentially decreasing sequences {fp }.
In this case "square-summable" means that the right-hand side of (9.13) converges, "rapidly decreasing" means that {lllp Hie II f p II} is bounded for each k, and "exponentially decreasing" means that {KIArl II fp II} is bounded for some K > 1. Finally, (9. 7) gives us a description of the functions on a homogeneous space GI H, for C(GI H) is just the part of C(G) which is invariant under the right-hand action of H. We have
Corollary 9.14 1\
C(GIH) ~ $P ®P 8 , p
compatibly with the G-actions on C(GIH) and the representations P. Proof of (9.7). We begin with (ii). If P1 and P2 are irreducible representations of compact groups G1 and G2 then P1 ® P2 is an irreducible representation of G1 x G2. To see this it is enough to show that (9.15)
98
II Lie Groups
because representations of any compact group are sums of irreducibles. But the space of (Gt x G2)-equivariant maps Pt ® P2 - Pt ® P2 is the (G 1 x ~)-invariant part of the matrix algebra End (Pt ® P2)
~
End (Pt) ®End (P2),
and so (9.15) follows at once from Schur's lemma. (i) When G acts on C(G) by left-translation we know that C(G)fin Cat1 ( G), and so that
c•••(G) =
=
E9P ®Hom G(P;C(G)).
But there is a map (9.16)
P - - Hom G(P; C(G)),
equivariant with respect to the right-hand action of G on C(G), defined by ~-+ F~;, where
e
Ft:<'1> =
('J,ge).
The map (9.16) is injective by Schur's lemma, and is surjective because if f: P- C(G) we can find E P such that
e
('7. e) = 1<'1 > for all '7 e P, and then
F~;(,)(g) = (,,ge)
F~; =
f because
= (g-•,,e) = f<1> = g- 1!<,><1> = t<,>·
(iii) We first observe that the L 2 inner product on C(G) induces a (G x G)-invariant inner product on E9 P ® P. As the spaces P ® P are non-isomorphic irreducible representations of G x G, Schur's lemma tells us that the summands P ® P arc orthogonal for any invariant inner product, and that, up to a scalar multiple, there is only one invariant inner product on each P ® P. We therefore have (fp;,,..~;, ./P:rrz.t:2) = Kp('Jt.'72Het.e2)
for some number Kp which depends only on choose an orthonormal basis {e,} for P. Then
P.
To determine
Kp
we
fp;e,#J(g) = M;j(g), where Mij(g) is the unitary matrix representing the action of g on P. Taking {'7t.et.'72.e2} = {l,ebet.el} we find
L
Mfj(g)Mk/(g)d(g) = Kp~IJc~jl·
(9.17)
9 The Peter-Weyl theorem
99
But Mjj(g) = M1,(g- 1). Putting i = k in (9.17), and summing over i. we get
Ia MJI(l)d(g) = Kp dim(P)cS11• But MJI(l) = cSJ'• and
fG ld(g) = 1, so Kp =
1/ dim(P), as we want.
10 Functions on Rn and sn-l
The spaces of functions on Euclidean spaceR" and on the unit sphere sn-l provide simple concrete illustrations of the representation theory we have been developing. Let Pk denote the homogeneous polynomials of degree k on R", with complex coefficients, and let Fk denote their restrictions to S"-1• Thus d:[x 1, ••• ,x,.] = E9Pk, k;iiiO
while
because
E.xf =II X 11 2= 1 on sn-l.
The space Pk is a representation of 0 11 , and we can introduce an invariant inner product. Let Hk be the orthogonal complement of Fk- 2 in Fk, so that Fk
=
Hk
E9 Fk-2 =
Hk
E9 Hk-2 E9 Hk-4 E9 · • ••
The space Hk is called the space of spherical harmonics of degree k on S"- 1• We shall see in a moment that it is an irreducible representation of 0,., and the reason for the name "harmonic" will also appear. If n = 3 it 1). has dimension 2k + 1, since in general dim(Pk) =
<"+!-
The polynomials on R" are dense in the space C(S"-1) of continuous functions on sn-l by the Stone-Weierstrass theorem. This implies C(s"- 1)fiD =
UFk k;iiiO
= E9Hk.
(10.1)
k;;.G
Let us compare this decomposition with the assertion of Theorem (9.14), 100
10 Functions on Rn and
101
sn-l
writing qsn-l) as C(On/On-1). If Htc is irreducible, then (10.1) and (9.14) are compatible if Hf-• is one-dimensional for each k (for clearly we have Htc == Htc in this case). But, conversely, (9.14) implies that dim(:;;D-•) ~ 1 for each irreducible representation P which occurs in C(sn- 1). So if we show that dim(Hf-•) == 1 it will follow that Htc is irreducible. Suppose that On- I is the subgroup of On which leaves the x 1-axis fixed. Then f e Pte belongs to Ptc0 -• only if it is a linear combination of the polynomials It _i-2 -"-4 2 X"j, X"j p, X"j p '
... '
where p == ~ + ~ + ... + x~. So dim(P~c0-') dim(Hf-•) == 1. This proves that Htc is irreducible.
==
[k/2], and hence
If(/) e Htc then r"(f) is a homogeneous polynomial of degree k on Rn, where r == II x II • Let Htc denote the polynomials of this form, so that Htc .! Hte under restriction to sn-l. We have proved that A
2A
4A
Pte== nte E9 r nte-2 E9 r Hte-4 E9 ••• , or, alternatively, that any polynomial f on Rn can be decomposed
/(ru) == where
u
e sn-l, (/)k e Hte
L r"/te(r )(f)tc(u), 2
and ftc is a polynomial in r 2• In other words (10.2)
CC[xi•···•Xn] ==$Etc ®Htc te;a.Q where Etc is the space of radial functions of the form The Laplacian t::.
r" /tc(r2).
== E
It commutes with the action of On. and maps Pte into Pte-2· By Schur's lemma it must map the irreducible subspace lite of Pte to zero, as lite does not occur in Pte-2· In fact Proposition 10.3 The space Hte is exactly the kernel oft::. restricted to Pte. i.e. it is the space of harmonic polynomials of degree k on Rn. Proof. It is easy to check that
(10.4)
II Lie Groups
102 for any (/)
e P1c, and hence, inductively, that A(r21 qJ) = 2s(n + 2k- 2s- 2)r21-
2qJ
+ r 2' A(/)
for (/) E P1c-2.r· So A maps r 21 filc-2.r isomorphically to r2s-2flle-2s if s > 0. It is well-known that the Laplacian A on R.n can be written A=
2+ (.£.) or
n- 1
.£. +.!.As, rl
ror
(10.5)
where As is the Laplacian on sn-l. As As commutes with the action of On it must preserve each subspace H~e, and must act on H1e by multiplication by a scalar. Because A(r"qJ) = 0 when (/) E H1e we can substitute in (10.5) to obtain Proposition 10.6 (i) The Laplacian As acts as -k(k + n- 2) on H~e.
(ii) In terms of the decomposition (10.2) we have A=EA~e®1, le~O
where A1c : E1e - E1e is given by A1e = ( .£.)2 + n -1 .£. _ k(k + n- 2).
or
ror
r2
There is a striking curiosity to be noticed here. On the polynomial ring G::[x1. ... , Xn] we have three operators
1
a
2A,r or
n
+ 2'
12
2r •
(10.7)
where !r2 denotes the multiplication operator by !r2. They all commute with the On-action, and can be regarded as operators on the spaces E~e. If one calls them {e,h,f} one can rapidly check, using (10.4), that they satisfy [h,e] = -2e, [h,/] =
2/, [e,/] = h,
i.e. they define a representation of the Lie algebra sl2R. of SL2R. on CC[x~o ... , Xn]. with the harmonic polynomials as lowest-weight vectors. Each E1e is an irreducible representation of sl2.IR, as it is generated by the lowest-weight vector r", of weight k + J. This Lie algebra action does not come from an action of the group SL2R on G::[x1. ... , Xn], and there is no prima facie geometrical reason for sl2R to appear. I shall not pursue this any further, but I shall give a fuller account of a closely related situation in Chapter 17.
10 Functions on R" and s•- 1
103
1be Radon transform A
An amusing application of the decomposition qs•-1) = ffiH~c is to the Radon transform. This is the linear map dl : qsn-1)-+ qsn-1)
defined by (dlf)(x) = (average off over the great (n- 2)-sphere with pole x). As fJtj is automatically an even function on sn-l, i.e. fJtj( -x) = atj(x), and atj = 0 iff is odd, it is best to think of at as a map
at : C+(s"-1) -+ C+(s•- 1), A
where C+ denotes the even functions. We have C+(S"- 1) = EI)H21c. One would like to know whether at is bijective, i.e. whether f can be reconstituted from its averages. Because at commutes with On it must map each H21c to itself by multiplication by some scalar An/c. To calculate An/c we observe that it is the value of atj at (1,0, ... ,0) when f = (x1 + ix2)21c e H21c. By explicit integration we get
+ 1) A _ (-1)" r(!!::l)r(k 2 2 "" - .Ji r(k + "2') · If n = 3 this is ~ (~). In any case, it is non-zero, and decays like 1/ki-l as k -+ oo. We conclude that the Radon transform is injective, and, by (9.13), bijective on coo functions.
11 Induced representations
The Peter-Weyl theorem describes the G-action on C(X) when X = GI H is a homogeneous space of a compact group G. It works equally well if we want to study, say, the space Vect(X) of tangent vector fields on X. Let T = TXoX be the tangent space to X at its base-point Xo = [H]. The subgroup H acts on T, for each h E H defines a map X - X which leaves Xo fixed and therefore induces h: T- T. Proposition 11.1 Tangent vector fields ' on GI H can be identified with maps~ : G- T which are H-equivariant in the sense that
The G-action on Vect(GIH) co"esponds to the action (g.e)(x) = e(g- 1x). In symbols Vect(GIH) ~ Map8 (G; T) as representations of G. Proof. The tangent vector '(gH) lies in TgH(GIH), so e(g) = g- 1,(gH) belongs to T. It depends, however, on g and not just on the coset gH: in fact e(gh- 1) = he(g). Conversely, given G - T we can define '(gH) = ge(g) E TgH·
e:
Now we can apply the Peter-Weyl theorem. As we have not discussed real representations we had better consider the complexification Vect(GIH) ® G::
~
104
Map8 (G; Tt:),
11 Induced representations
105
where Tc:: = T ® cr;, We have Map8 (G; Ta;) = {C(G) ® Ta;} 8 = EBP®(P®Ta;)8 , p
where P runs through the irreducible representations of G. Because each P is unitary we have P ~ P*, and so (P ® Ta;)8 can be identified with the space HomH(P; Ta;) of H-equivariant linear maps P - Ta;. Example. If X= S2 = 03/02 then T ~ R 2 with the obvious action of 02. The irreducible representations of 03 are the spaces Hk of spherical harmonics, and (Hk ®~2 )02 is zero if k = 0, and is 2-dimensional if k > 0. So Vect(S 2 ) ® ~ ~ E9
e Hk).
k>O
There is no 0 3-invariant vector field on S 2, so Ho does not occur.
Representations of G of the form Map8 ( G; M), where M is a representation of H, are called induced representations; more precisely, Map8 (G;M) is called the representation of G induced by the representation H of M. They are natural generalizations of the spaces C(G/H) - i.e. the case M = ~ - and they can always be interpreted as the spaces of sections of G-vector-bundles on GI H analogous to the tangent bundle. A G-vector-bundle on a space X on which G acts is a family of vector spaces {Ex}xex together with linear maps g :Ex- E1x for each g e G and x eX. The family is required to be locally trivial in a sense I shall not discuss. Example. Let X be complex projective space J'C- 1 = P(C). This is a homogeneous space of Un. and also one of G = GLnC[;· Let us think of it as G/H, where H = GLt,n-t(C[;) is the group of echelon matrices
(
h~t :~:
:::
:~)
.~.
~~~
::: ~~~
0
hn2
"'
hnn
Let Vk denote the space (SkC)* of homogeneous polynomials on ([;" of degree k. A polynomial p e Vk is not a function on ~-• : it assigns to each point L of ra:-t (where L is a line in cr;") a point p(L) of the
106
II Lie Groups
!-dimensional space EL of homogeneous functions of degree k on L, i.e. it is a section of the !-dimensional complex vector bundle E = {EL}Le~· on At first sight this may not seem a very fruitful way of regarding the polynomial p, but in the long term it is the right point of view. (See Chapter 14.)
r.:-•.
Alternatively, p can be regarded as a map p : G -+ CC which satisfies f}(gh)
= h't.P
(11.2)
when h e H is as above. For applying (11.2) when h is of the form
teUs us that p(g) depends only on the first column ofg, i.e. that pis a function C- {0}-+ CC; and then applying it when h=
teUs us that
(~ ~)
p is homogeneous of degree k.
In fact VA: consists precisely of aU holomorphic functions p : G -+ CC satisfying (11.2). Whenever G and H are complex Lie groups it makes sense to speak of the representation Maprf1(G;M) of G holomorphically induced from a holomorphic representation M of H (or, equivalently, of the holomorphic sections of the vector bundle E on GI H). So we can state what we have proved as Proposition 11.3 The representation (SA:([:")* of GL,CC is holomorphically induced from the }-dimensional representation h~---+ htf' ofGL 1 ,~~_ 1 C For future use let me point out that if p : GL,CC -+ CC is a holomorphic function then to prove that p(g) is a homogeneous polynomial function of the first column of g we need to assume (11.2) only for h in the subgroup B of upper-triangular matrices. One can prove this by an elementary explicit argument, but the point is that the homogeneous space GL1.n-d B is a compact complex manifold, and so any holomorpbic function on it is constant In any case, beside (11.3) we have Proposition 11.4 The representation (SA:C')* of GL,.CC is holomorphically induced from the }-dimensional representation h ~---+ htf' of B.
11 Induced representations
107
We shall prove in Chapter 14 that every holomorphic representation of GLnCC is holomorphically induced from a 1-dimensional representation
h ......_ h"•II of B for some k == (k., ... ,kn) E Z".
J)cz ••• "22
'-"• rrm.
12 The complexification of a compact group
The unitary group U,. is a maximal compact subgroup of the complex Lie group GL,.([;. There are three other important aspects of the relationship between these two groups. The first is very obvious. The Lie algebra of U,. is the space of n x n skew hermitian matrices, and any matrix can be expressed uniquely in the form A + iB with A and B skew hermitian. This gives us Propositioa ll.l The Lie algebra M,.f:. of GL,.f:. is the complexification of
the Lie algebra of U,.. This is what is usually meant by saying that the group GL,.(; is the
complexification of the group U,.. Propositioa 12.2 The algebra Calg(U,.) of representative functions on U,. is precisely the algebra f:.[a 11, !l- 1] of polynomial functions on the algebraic group GL,.(;, where ll = det(a11 ).
Proof. We have already explained that, by the Stone-Weierstrass theorem. the representative functions on U,. are the polynomials in ali and all· But, being unitary, (aiJ) is the transposed inverse matrix to (a;1), so 7iij = Plil ll, where Pii is a polynomial in the ail. Finally, we have Propositioa 12.3 Every representation of U,. is the restriction of a unique
holomorphic representation of GL,.C Proof. The uniqueness is because a holomorphic map GL,.([;- Aut(Y) 108
12 The complexification of a compact group
109
is determined by its values on Un (e.g. a holomorphic function on CC-{0} is determined by its values on the unit circle T). The extendability is because CaJ1(Un) = CC[a1J>A- 1]: a more detailed argument will be given in the proof of (13.2) in the next section.
Remarb. (i) It is not true that any continuous function on Un extends to a holomorphic function on GLnCC. but the ones which do extend are dense in C(Un). (ii) For the standard representation V = G::n of Un the complexconjugate representation V is isomorphic to the dual representation y•. But V and y• are not equivalent representations of GLnG::: for y• is holomorphic and Vis not. (iii) The holomorphic representations of GLnG:: are a very different thing from its unitary representations, which are aU infinite dimensional A non-trivial holomorphic representation cannot be unitary, for if f : X - GLnG:: is holomorphic, where X is any complex manifold X, we cannot have /(X) c UN. (Otherwise, using the local chart on GLnCC given by the logarithm, we should have a holomorphic map into MNCC with values contained in the real vector subspace of skew hermitian matrices.) There is a complexification Gc of any compact group G, and aU four of the above characterizations of the relationship between G and Gc continue to hold. In the language of Macdonald's lectures, Gc is the linear algebraic group whose coordinate ring is Ca~a(G). Example. The complexification of On is
On(([)= {A E GLnCC: A'A= 1}.
13 The unitary groups and the symmetric groups
Weyl's correspondence Hermann Weyl showed that the irreducible representations of the unitary group Un are realized as spaces of "tensors" with various symmetry properties. This is a very beautiful and important theorem, with many ramifications. It is one of the starting-points of the modern theory of "quantum groups". The most obvious representation of Un is its natural action on C = V. This induces an action on
for each k. An element of vet is a "tensor", i.e. an array of numbers a = {at1iz.•il}, with 1 ~ i, ~ n. A matrix (Uij) E Un acts on yet by a 1-+ I, where
a,•..,. =
L
"hJ•"Ilh ... uuaJ•-lt .
}J..-J.
The representations vet of Un are reducible. An element of V ® V can be written as the sum of a symmetric and a skew tensor V ® V = S2 V E9 A2 V,
and both S 2 V and A2 V are irreducible representations of U,.. The case of V83 is a little more complicated. We have V®V®V=S 3 VE&A3 VE9W,
where W consists of the a11" such that atJk.
+ aJk.i + IJiciJ = 110
0.
13 The unitary and symmetric groups
111
Both S 3 V and A 3 V are irreducible under Un, but W breaks into two irreducible representations W = W+ E9 W_, where W± = {(a;JA:) E W : a;Jk = ±aJik} •
There is a better way of expressing this. The space
Q={(J.,p,v)eC :J.+p+v=O} is an irreducible representation of the symmetric group S3, and for each (A,p, v) e Q the subspace WA,p,, of W consisting of tensors (a11k) for which, for each triple i ~ j ~ k, the vector (aiJk,aJki.lllciJ) e C is proportional to (A,Jl, v) is an irreducible Un-invariant subspace. In this notation, W+ = Wt,t,-2. and w_ = Wt,-1.0· Each of the representations WA,p,, is equivalent to the representation of Un on the space VQ
= Homs,(Q; V~3 ) ,
of S3-equivariant linear maps Q - v~3 • (Here S3 acts on V~3 by permuting the factors. This action commutes with the action of Un on V~3 , so Un acts on VQ.) In fact the obvious map Q®VQ-+ W
of representations of S3 x Un is an isomorphism. The decomposition W = W+ E9 W_ corresponds to writing Q = C[ E9 C[ by using the basis {(1,1,-2),(1,-1,0)} for Q. In general the symmetric group Sk acts on that we can decompose under
v•
sk
v•, and we already know
v~ =EBQ®VQ.
(13.1)
Q
where Q runs through the irreducible representations of SJc and
vQ =
Homs.(Q; v•)
.
As before, Un acts on VQ, so (13.1) is an isomorphism of representations of 5Jc x Un. We call VQ the "tensors of degree k with symmetry of type
Q". Weyl's theorem is Theorem 13.2 VQ is an irreducible representation ofUn. and, up to multiplication by a power of the determinant, all irreducible representations of U,.
112
II Lie Groups
arise in this way for some k. Furthermore, all i"educible representations of Sk occur in y®k if dim( V) ;;a: k. The theorem establishes a 1-1 correspondence between the irreducible representations of U, contained in y®k and the irreducible representations of Sk contained in y®k. We shall see in the next section that irreducible representations of U,, in general, are indexed by sequences of integers ka ;;a: k2 ;;a: ••• ;;a: k,. The corresponding representation is contained in y®k if E k, = k and k, ;;a: 0. Multiplying by the determinant changes (kt. .•• , k,) to (ka + 1, ... , k, + 1).
Proof of (13.2). Essentially we have already seen why all representations of U, arise. For the matrix entries of all subrepresentations of all y®k form a subalgebra A of C8 ,8(U,) which would be orthogonal to the matrix entries of any hypothetical "missing" representation. But (if we put in arbitrary powers of the determinant A) the algebra A certainly contains G::[a,J,A- 1], which is dense in C(U,). It is also clear why all representations of Sk occur in y®k if k ;;a: n. For they all occur in the left-action of Sk on the group-ring G:;[Sk], and if {e,} is a basis of Y the orbit of e,1 ® · · · ® E y®k under Sk spans a copy of G::[Sk] if all the e, occur among the e,..
e,.
The hardest part is to prove that VQ is irreducible. The argument is short, but extremely ingenious. We know from Chapter 12 that it is enough to show that VQ is indecomposable under the action of G = GL,G::, and hence to show that EndG(VQ) = G::. Now G:: is the centre of the matrix algebra End(Va), and by Schur's lemma we have Ends.(V®k) = E9End(Ya) Q
and Endaxs.(V®k) = E9EndG(VQ) Q
It is therefore enough to show that the algebra Endaxs.(V®k) is contained in the centre of Ends.(V®k), and hence to show that the image of Gin Ends. ( y®k) spans it as a vector space. But Ends.(V®k) = (End(V)®k)St , where (End(V)®k)St denotes the part invariant under Sk. A linear form
13 The unitary and symmetric groups
113
on this is simplyt a homogeneous polynomial of degree k on the vector space End( Y). Such a polynomial certainly vanishes if it vanishes on the dense open subset G of End( Y). That completes the proof.
Quantum groups In recent years the relation between the representations of the unitary and symmetric groups has become the height of fashion, because of the emerging theory of "quantum groups". I cannot describe this development here. It must suffice to say that the action of the Lie algebra u, of U, on (C)®ic, and the commuting action of S1c on it, each possess a canbnical !-parameter family of deformations, indexed by q e The defotiilation is not, however, through representations of u, and S~c. The action of S~c is deformed to a representation of the braid group Br~c on k strings. A braid is an "enhanced" permutation, described by a diagram like
cr:x.
2
3
4
5
cr:x
For each q e there is an action of the group Br1c on (C£:")®k, and only when q = 1 does the action factorize through S~c. Meanwhile, the deformed action of u, is an action of a new kind of algebraic object (u,)9 which depends on q. This is the so-called quantum group. It remains true after the deformation that the actions of (u,)9 and of Br~c are commutants of each other, and so either can be used to construct the other. The representations of the braid groups are of great interest in three-
t We are using the fact that for any representation W of S. a linear form on wSt comes from an .S.-invariant linear form on W. This holds because a linear form on wSt can be extended arbitrarily to a linear form on W, and then averaged over S,. We are also using the fact that a homogeneous polynomial of depee k on a vector space B is the same as a S!fiiiDelric k-fold multilinear map B x · · · x B -+ i.e. an St.-invariant linear form on £8i.
«:.
114
II Lie Groups
dimensional topology, while those of the quantum group arise in twodimensional quantum field theory and statistical mechanics. The link so established between these areas has been one of the great surprises of recent mathematics.
14 The Borel-Weil theorem
I shall now give a systematic description of all the irreducible unitary representations V of Un. in a form which generalizes immediately to any compact group. We begin by observing that the Un-action on V extends to a holomorphic action of Gc = GLn(C· Then we decompose V under the action of the subgroup T = T' of diagonal matrices
•= ("' ·. ..) Because T is commutative and acts unitarily we can find a basis of V consisting of weight vectors, i.e. vectors v which are eigenvectors of the T-action. For such a vector we have uv = l(u)v, where
l(u) = ~·u~2 • • • """ for some multi-index k = (kt.· · ·,kn) e Z" called the weight ofv. The Lie algebra 94; of GLn(C acts on V. Let E;j E 9c: = Mn(C be the matrix with 1 in the (i,j) place and 0 elsewhere. If v is a weight vector with weight k then, because uEiJu- 1 = u1uj1Eli, the vector E1JV is either a weight vector or zero, and its weight is k + e11 , where e11 e Z" has 1 (resp. -1) in the ;th (resp. ft) place. By ordering the weights lexicographically we can find a highest weight e V, which has the property that E11v = 0 where i < j. It will tum out that there is only one such vector, up to a scalar multiple, but we do not know that yet.
vector v
115
II Lie Groups
116
Let B be the subgroup of upper triangular matrices in Gc:: its Lie algebra is spanned by the EiJ with i < j together with the diagonal matrices. So the highest weight vector v is an eigenvector of B, i.e. bv = .A.(b)v, where .A. : B -
ccx is the homomorphism such that .A.(b)= b~~ ... ~~ if bu
b= (
0 Now define a Gc;-map
v· -
:J
M-k = Map~(Gc: : CC)
by
oc .._.. {g.._.. oc(gv)} , where the right-hand side means the holomorphic maps f : Gc: - CC such that f(gb) = .A.(b)f(g). In other words, M-k is the representation of Gc: holomorphically induced from the 1-dimensional representation .A.- 1 of B. The BoreJ-Weil theorem is 1beorem 14.1 The map
v•- M-k is an isomorphism, and so V~M:".
Proof The map is clearly non-zero, and hence injective, as v• is irreducible. To prove it is surjective we show M-k is also irreducible. It is enough to see that M-k contains at most one vector fixed under the action of the subgroup N of strictly lower-triangular matrices, for if it split into two pieces each would contain a lowest weight vector, and such a vector is fixed under N. Iff e M-11. is fixed under N we have f(itb) = .A.(b)f(l). So fiNB is completely determined by its value /(1) e CC. But an open dense subset of elements g e Gc: can be factorized g = ilb with il e N and b e B (I shall discuss this further below), so f itself is completely determined by f(l), and dim(M~k) ~ 1, as we want. A corollary of the proof of the Borel-Weil theorm is Tbeorem 14.2 Any irreducible representation of U,. contains a unique high-
est weight vector.
14 The Borel-Weil theorem
117
The Borel-Weil theorem constructs and labels all the irreducible representations of Un. They are in 1- 1 correspondence with their highest weights k = (k,, ... ,kn) e zn. A possible highest weight must be dominant, i.e. for if k is a weight of a representation V then so is any k' got by reordering the k;. We shall see below that if k is not dominant then the holomorphically induced representation M -k is zero. We should also mention that the dual representation M:11 is easily seen to be isomorphic toM,, where I= (kn,kn-t, ... ,kt),
for M1 is an irreducible representation whose lowest weight is I= (l., ... ,ln),
and whose highest weight is therefore necessarily (In, ln-t. ... , lt). For any representation V of Gc: the orbit of the highest weight ray [v) in the projective space F( V) under the action of Ga: gives a holomorphic map i: Gc./B-+ P(V).
For most representations V the subgroup B is precisely the stabilizer of the ray [v], and so i is an embedding. But if A. : B -+ Cl:x extends to a larger subgroup P containing B then P stabilizes [v], and the map factorizes Ga:/ B -+ Ga:/ P -+ JP(V). In this case the representation v· is holomorphically induced from P, as well as from B. We saw in Chapter 11 that this occurs when V = (Skeen)" and P = GLt,n-tCC. Subgroups of Ga: containing B are called parabolic subgroups. Example. The representation (A,k C)" of GLnCl: is induced from the representation (
~
: )
~ det(a)-1
of P = GLk,n-/c, or alternatively from A. : B
-+
Cl:x, where
A.(b) = (bub22 · · · bkk)-1.
For iff : Gc:
-+
C£: satisfies f(gp) =/(g) when
p=(~
:)eP
II Lie Groups
118
then /(g) depends only on the first k columns g" · · · ,g1c of g. If also f(gp) == det(a)- 1f(g) when p==
(~ ~)
then f(g) depends only on g 1 /\g2 1\ · · · 1\g~c. So f is a holomorphic section of the line bundle on the Grassmannian Gr~c(CC") = Gt;/ P whose fibre at a subspace W of G::" is (/\lc W)*. On the other hand, an element of (/\lc V)* gives an element of (1\lc W)* for each W E Gr~c(CC") by restriction. The embedding
1-+ [w1 1\ · · · 1\ w~c], where {w1} is a basis for W, is called the Plucker embedding, and the (Z) components of w1 1\ · · · 1\ w1c are called the Pliicker coordinates of W.
given by W
It is worth analysing the general holomorphically induced representation M-k of Gt; == GL,.([. a little more explicitly, in order to relate it to Weyl's tensorial construction described in Chapter 13. By multiplying by a power of the determinant we can reduce to the case where k.. == 0. Then we have Theorem 14.3 (i) The elements f : Gt; -+ G:: of M-"· when regarded as functions/(,., ... ,,,.) ofn vectors, depend polynomially on each vector''' and are homogeneous of degree k; in ~ 1 • In particular, M_k is a subrepresentation of
(ii) M-k = 0 unless k is dominant, i.e. k1
~
· · · ~ k,..
Proof. (i) If k,. == 0 then an argument we have already used in Chapter 11 shows that/(,,, ... ,~~~) is independent of,,.. Now hold all fixed for j f i, and consider the dependence on ,,. We get a holomorphic function /(i) defined on CC"- W, where W is the (n- 2)-dimensional subspace of ([." spanned by the for j f i, n. By Hartogs's theorem ([Griffiths and Harris] page 7), which tells us that a holomorphic function cannot have singularities of codimension > 1, we conclude that /(i) extends to all of C. As it is homogeneous, it must be a polynomial of degree k;. (ii) Restrict f to the copy of GL2([. in GL,.G:: formed by the ;th and lh
'i
'i
14 The Borel- Weil theorem
119
rows and columns. We get a function of a 2 x 2 matrix
which is homogeneous of degrees k1 and k1 in the columns, and depends on the second column only through the determinant ad-be. It is therefore of the form
p(a, c)(ad- bc)kJ , where p is a homogeneous polynomial of degree k1 - k1. In particular,
14 ~ kj. Everything said in this section generalizes from Un to any compact group G. The diagonal matrices T are replaced by a maximal torus T, and the upper-triangular matrices B by the group B generated by Tc: and the !-parameter subgroups corresponding to the positive rootvectors in the Lie algebra of Gc;. The homogeneous space Gc:/ B is always a compact complex algebraic variety, and, as I mentioned in Chapter 4C, it possesses a beautiful decomposition into cells, called the Bruhat decomposition. The interplay between the geometry of these cells and the structure of the representations of Gc: is fundamental in representation theory. I shall give one more example of the Borel-Weil construction. Example. The spin representation of 02n. We saw on page 61 that the homogeneous space .3n = 02n/Un of complex structures on R 2" has a complex description as 02n(f:,)jP, where P is a parabolic subgroup. The group 02n has a double covering 02n, whose restriction to the subgroup Un is the group Un of pairs (u, A) with u e Un and A2 = det(u) e G::x. Using the complex structure of 02n/Un = 02n/Un we can form the representation of 02n holomorphically induced from the onedimensional representation C of Un given by {u,A) ...... A.- 1• This is the spin representation. It has dimension 2", and when restricted to Un it becomes
15 Representations of non-compact groups
This is a huge subject, and I can only make some orientational remarks. For excellent comprehensive introductions to the subject, from contrasting standpoints, I recommend the books by KiriUov and Knapp. The first problem one meets is that infinite dimensional representations of a group come in families of roughly equivalent representations which one wants to lump together. For example, PSU1,1 acts on the circle S 1, and hence on the functions on S 1• But one might want to consider its action on the continuous functions C(S 1), the smooth functions C00 (S 1), the L2-functions L2(S 1), or perhaps some other class. For most purposes these representations of PSU1,1 are not interestingly different. All of them are irreducible in the sense that when they are given their natural topologies there is no closed invariant subspace. If a representation can be made unitary it is natural to consider the Hilbert space version (which is unique if the representation is irreducible), but that has the disadvantage that the Lie algebra of the group does not act on it. For example, the !-parameter group of rigid rotations of S 1 is generated by the Lie algebra element which acts on C00 (S 1) by the differentiation operator/, . But -!9 is not everywhere defined on C(S 1) or L2(S 1). There is a standard way to deal with this problem in the case of semisimple groups G. When given a representation of G on a topological vector space Y we first consider the action on Y of the maximal compact subgroup K of G. As we saw in Chapter 9 this picks out the dense subspace yfin of K-finite vectors. In PSU1,1 the maximal compact subgroup is the group T of rigid rotations of S 1, and we get the same space yfin whether we start withY= C00(S 1 ) or Y = C(S 1) or Y = L2(S 1): in each case yfin consists of the trigonometric polynomials, i.e. the Fourier series 120
121
15 Representations of non-compact groups
'£a~cei" 8 with only finitely many a" non-zero. Usually the group G does
not act on V60 • In our example, a typical clement of PSU1,1 takes the function 118 to aei0 +b
bei0 +a' which is not a trigonometric polynomial. On the other hand, if M is a finite dimensional K-invariant subspace of V then so is 9M, where 9 is the Lie algebra of G. So we have
Proposition 15.1 The Lie algebra 9 of G acts on V60 • In our example, the basis elements of the Lie algebra act by
e±io
.Ju
and
d
ill
A less obvious result, which I shall not prove, is
Proposition 15.2 If an irreducible representation V of a semisimple group G is decomposed into isotypical parts ylin
=E9Vp p
under the action of the maximal compact subgroup K of G, then dim(Vp) is finite for each P, i.e. each irreducible representation P of K occurs with finite multiplicity.
For semisimple groups the spaces yfin with their simultaneous action of K and 9 seem to be the right objects to study and classify. Passing from V to V60 eliminates most of the analysis from the picture, and reduces the representation theory to algebra. For groups which are not semisimple the position is quite different, and (15.2) is far from true, as we shall see when discussing the Heisenberg group in Chapter 17. The idea of focussing on the (K,9)-action on yfin is a fundamental step for another reason too: we should get very little information by considering the 9-action alone. The representation theory of a group G and of its Lie algebra g are not at all well related when infinite dimensional representations are considered. For example, the Lie algebras of SL 2R and SU2 have the same complexification, and hence the same representation theory, but the representations of the two groups are quite different from each other. Most infinite dimensional representations of the Lie algebra g do not come from representations of the group G. The basic example arises when G acts on a smooth manifold X. The group
122
II Lie Groups
G acts on C 00 (X), but if Y is an open subset of X not stable under G it certainly does not act on C 00 (Y). Nevertheless the Lie algebra g acts on C 00(Y), because functions on Y can be differentiated along the vector fields which generate the G-action.
Examples (i) The additive group R. acts on C 00 (lR) by translation, and the Lie algebra generator acts by fx, which acts on C 00(a,b) for any open interval (a, b) c JR. But the group JR. does not act on C 00 (a,b). (ii) If Gc: is a complex algebraic group the Borel-Weil theorem (see Chap. 14) constructs the irreducible representations as
Map~01 ( Gc:; C[J ). In the group Gc: there is an important dense open subset U = spaces
NB. The
Map~01 ( U; C[J)
are called dual Verma modules. They are representations of 9c: but not of Gc;. A central role in the representation theory of any group G is played by the Plancherel theorem, which describes the decomposition of the Hilbert space L2(G) under the left- and right-action of G. For compact groups this was accomplished by the Peter-Weyl theorem, which enabled us to write any f e L 2G as a sum
t=EtP
(15.3)
p
of functions transforming according to the unitary irreducible representations P of G. The main difference in the non-compact case is that the sum (15.3) must be replaced by an integral
f
=
J
fpdJ.t(P)
(15.4)
with respect to a measure dJ.t(P) on the space of irreducible representations, just as Fourier series are replaced by Fourier integrals when one passes from the compact group T to the non-compact group R.. As for Fourier integrals the Plancherel theorem tells us that II
f
11 2=
j
II
fp
11 2
dJ.t(P).
15 Representations of non-compact groups
This formula, which generalizes the result 2""
1
II! II= L..,dim(P) II!P II
2
for compact groups, accounts for the name of the theorem.
123
16 Representations of S L2R
The representation theory of SL2R is, of course, simpler than that of an arbitrary semisimple group, but nevertheless it exhibits the main features of the general case. In this section I shall describe the most important irreducible unitary representations of G = SL2R, namely the ones which are needed for the Plancherel theorem, or, equivalently, which "occur" in L2(SL2.R). (It would be more accurate to say that the representations not described form a set of measure zero for the Plancherel measure dJl(P) of (15.4).) A natural family of representations to consider are those induced from the subgroup B of upper-triangular matrices. Then G/B is the real projective line P~ ~ S 1 • The !-dimensional representations of B are of the form (
~ a~ 1 )
....._
(sign(aW I a 1',
where p e cr:: and e = 0 or 1. The induced representation of G will be denoted by Ep.E· If e = 0 its elements are "~-densities" f(O) 1dO 1''2 on the circle S 1 = G/B, forB acts on the tangent space to G/B at its base-point by
(~ a~l) ~a2. If e = 1 the elements of Ep.e are "twisted ~- densities" f(O)I dO 1''2 on S 1, where f is not a function but a cross-section of the Mobius band. As representations of the maximal compact subgroup T = S0 2 the spaces E,.E are independent of p :for G/B = Y/{±1}, and E,.E is simply the space of functions qJ : T -+ cr:: such that cp(-z) = (-l)ecp(z).
124
16 Representations of SL2R
125
Each isotypical piece for the T-action is !-dimensional. If p is not an integer then Ep.e is an irreducible representation of G, as is easily seen by considering the action of the Lie algebra 9 on (Ep,e)fin. But it is not usually a unitary representation. There is an obvious multiplication map Ep,e
X
Ep,e -
Ep+p,D·
The space of densities on S 1 is E2.o. so when p invariant inner product on Ep.e given by
(h/2}
+p =
2 there is an
1
= ld2.
Taking p = 1 + is, and either value of e, we get two families of irreducible unitary representations parametrized by the real number s. They are called the principal series representations. (Actually Eu should be excluded, as it is reducible : we shall see in a moment that ~ E90hol·) E1,1 = Ohol
-
-+
The other important unitary representations are the discrete series, which are holomorphically induced from the compact subgroup T = S02• The space G/T is the upper half-plane H, and the action of G preserves the Poincare metric and area element on H. Corresponding to the !dimensional representations z ~ zP ofT (with p E Z) we have the spaces '1&~ of square-summable holomorphic ~-forms f(z)(dz)PI 2 on H. The fractional powers (dz)PI 2 make sense, as under the Mobius transformation induced by
in SL2R we have dz ~ (cz + d)- 2dz and
To analyse these representations further - and in particular to prove they are irreducible - we consider the action of the compact subgroup T. It is best to replace SL2R by SU1,1, and hence the upper half-plane H by the unit disc D, on which T acts by rotation. Then an element of '1&~ has a Taylor expansion f(z)(dz)P/ 2 = L:a,.z11 (dz)PI 2• n>O
(16.1)
126
II Lie Groups
Because u e T acts on z"(dz)PI 2 by multiplication by un+p/ 2, we can identify (~~f) 60 , as a representation ofT, with the trigonometric polynomials (16.2) where m runs through the sequence ~. ~ + I, ~ + 2,.... These form an irreducible representation of the Lie algebra, generated by the lowestweight vector eip812. The invariant norm off= f(z)(dz)PI 2 is II
f
=
11 2
L
lf(z)l 2 (1-1 z
I2 )P-2
1dzdz
I
(16.3)
if p > 1 , as the invariant Poincare area element on D is
In terms of the expansion (16.1) this means that II
f
11 2
=
:L;K!P11 an
12,
n~
where for each p, {K!Pl} is a sequence of positive numbers which is 0(1/nP-1) as n- oo. If p ~ 1 there are no non-zero square-summable holomorphic (p/2)forms. But the case p = 1 is borderline: one can define a unitary representation 0~01 by completing the holomorphic !-forms on the closed disc with respect to the invariant norm
which is a renormalization of the divergent expression (16.3). The complex-conjugate representations fi"~~ for p ~ I form another discrete series. From (16.2) we see that the discrete series representations ~~~ are roughly "half the size" of the principal series representations Ep.e· In fact if e(p) is the parity of p then ~~~ is a closed invariant subspace of the non-unitary representation Ep,e(pl• which is the space QP/ 2 of all ~-forms on S 1• We have .
E9 ;r.p/2 - nP/2 c nD/2 a..rhol - ...0)
01.&
'
16 Representations of S L2JR and (16.2) shows that the quotient representation V = (p - 1)-dimensional, spanned by f~mO with
r
127
QPI2Jr1(612 is
r
-2 <m< 2· But QP/ 2 does not decompose as a sum nP/2 '-11/2 ·~ol $ uhol $
v·
The simplest case to understand is when p = 2. Then V is the trivial !dimensional representation of G, and 01 is the kernel of the equivariant map 0 1 - CC given by integration over S 1• But there is no G-invariant 1-form on S 1, and so the trivial representation does not occur in 0 1•
0/
To treat the general case we observe that G has a natural non-unitary irreducible representation on the (q +I)-dimensional space Pq ~ Sq(G::2 ) of homogeneous polynomials of degree q in two variables (u, v). The representation Pq is a subrepresentation of n-q/ 2 = E-q,e(qJ by the map
.9(u, v) .--- .9(z, l)(dz)-q/2• We saw that if q = p - 2 the representation n-q/2 is dual to QPI 2, and the annihilator of Pp-2 is easily seen to be r1(612. Thus nD/2 / r.D/2 :u· ll.'jo)
-o.:
p*p-2·
P;_
Actually 2 ~ Pp- 2, because the G-invariant skew form on «:2 induces an invariant bilinear form on Sq(CC2) for each q. But there is no (p- 2)dimensional invariant subspace in QP/ 2• The two kinds of irreducible representations of SL2R we have described correspond, roughly speaking, to the two kinds of so-called Cartan subgroups (see page 12) of SL2R. For the present purpose a Cartan subgroup can be defined as a subgroup A whose comple:xification is an algebraic maximal torus ccx (see page 179) of the comple:xified group SL2 C There are two kinds, for we have either A ~ R x or A ~ T. That concludes our account of the representations of SL2R. There is another series of irreducible unitary represenatations, called the complementary series, but it is of measure zero for the Plancherel measure. A very careful and elementary account of aU this material can be found in [Gelfand et al. Vol. 5].
17 The Heisenberg group, the metaplectic representation, and the spin representation
A basic problem in quantum mechanics is to find self-adjoint operators P and Q acting on a Hilbert space :¥f such that (17.1)
PQ-QP =-i
The problem, however, is not very weD-posed, as P and Q are necessarily unbounded operators, and cannot be everywhere defined in :¥f. Weyl pointed out that it is better to think of the !-parameter groups {eiaP} and {eibQ} of unitary operators they generate. These operators are defined everywhere in :¥f and satisfy (17.2) and so generate a 3-dimensional group H, which is called the Heisenberg group. Looking for unitary representations of H is a better way of formulating the basic problem (17.1). We have met the group H in Chapter 1 as N/Z, and also as the group of unitary operators in the Hilbert space :¥f = L2(R) generated by translations Ta and multiplications Mb by the functions eibx on R. These operators Ta and Mb satisfy the relations (17.2). A more symmetrical description of H is as a central extension
'I--+ H--+ R 2 , with the circle-group T as a normal subgroup, and HfT ~ R 2 • We can write the elements of H as uexp(e) with u e T and E R 2 • The group-law is
e
u exp(e).v exp(rr) = uvts<~,l exp(e + '7),
128
17 The Heisenberg group
129
where S is the skew bilinear form on R 2 given by
We are interested only in unitary representations of H in which the centre T acts by scalar multiplication. Stone and von Neuman proved that the only such representation is the one we already know on J'f = L2(R). Nevertheless, there are other interesting ways of describing it. If we define a=
~(P + iQ)
and a*
= ~(P -
iQ)
the relation ( 17.1) becomes [a*,a] = 1.
(17.3)
We observe that a* acts on J'f as the operator
i
d
- .Ji(dx +x),
and annihilates the vector n = e-!r. In fact the elements {a"O},.~o form an orthogonal basis for J'f. Thus J'f contains a dense subspace Jt'60 which is isomorphic to the polynomial ring «::[a], or, better, to the symmetric algebraS(«::). (Abstractly, what we have done in writing a= ~(P+iQ) is choose a complex structure on the space R 2, compatible with its skew form, and thereby we have identified R 2 with «::.) When described as a completion of «::[a] the Hilbert space J'f is known as the oscillator representation, for in quantum mechanics it describes the states of a simple harmonic oscillator. (The reason for the notation Jl"60 will appear.) Notice that a acts on CC[a] by multiplication, and a* acts as -Js. The choice of n such that a*n = 0 is reminiscent of choosing a lowestweight vector in a representation of SL2R. and this is no accident. For SL2R acts on R 2 preserving the skew formS which defines H, and so it acts on H as a group of automorphisms. As H has only one irreducible representation Jlt', Schur's lemma tells us we can find T1 : Jt' -+ J'f for each g e SL2R such that T1 o h = (gh) o T1
for all g e SL2.R and he H. The operators T1 are defined up to scalar multiplication, and define a projective representation of SL2R on J'f. We can find them without using the Stone-von Neumann theorem. The·
II Lie Groups
130 operators
(17.4)
are well-defined on Jf"60 , and have the standard commutation relations of sl2R. They act on L2R by
i ( d )2 d 1 dx ' x dx + 2'
-2
i
2
2x ·
(These formulas should be compared with those of(10.7).) They generate an action of a double covering of SL2R which is called the metaplectic group Mph. Its action on Jf" is the metaplectic representation. We see that it is a double covering of S L2R which acts on Jf" as follows. The operator corresponding to the Lie algebra element ~ which generates the circle subgroup S02 in SL2R is ~ = iA, where
A
= 21(P 2 + Q2) = aa* + 21,
which acts on L 2(R) as
!{-(.!)2 + x2} 2 dx !·
and on ([;[a] as afa + (In quantum mechanics this is the Hamiltonian, or energy operator, of the harmonic oscillator.) The eigenvectors of A are the basis vectors {a"O}, and the eigenvalues are {n +!}for n ~ 0. So exp (2n~) = -1, whereas exp (2n~) = 1 in SL2R. The group Mph is not a ·matrix group. For we know the finite dimensional representations of the Lie algebra sl2R quite explicitly, (see page 30) and in any of them the operator ~ representing has eigenvalues of the form ni, where n is an integer. This means that exp(2n~) = 1, and so a finite dimensional representation of Mpl2 cannot be faithful.
e
The circle T = {exp O~}oE;9E;4n in Mpl2 which is the double covering of S02 c: SL2R is a maximal compact subgroup. Its action on L 2R by the operators e10A is very beautiful. For most values of 0 the operator is an integral operator with a kernel of the form K9eiQo(x,y)
17 The Heisenberg group
131
where Qs is a quadratic formt on R 2, and when 0 = n/2 we have
where~ : L2(R)- L2(R) is the Fourier transform, which is of order four. But when 0 is a multiple of n the kernel becomes a cS-function;
thus (ei~rA f)(x) = if( -x),
and (;xiA f)(x) = -f(x).
In the notation of Chapter 16 the metaplectic representation Yf is 0~01 , the space of holomorphic !-forms f(z)(dz)i on the disc or upper half-plane. The map Yf-
takes
tp E
n~ol(upper half-plane)
L 2R to f(z)(dz)t, where f(z)
= j tp(x)e~i:xl dx.
The metaplectic representation is of great importance not only in quantum mechanics but also in many other branches of mathematics, such as number theory (where the double covering is related to the law of quadratic reciprocity). It can be generalized in various ways. The most immediate is that a non-degenerate skew bilinear formS on R 211 defines a Heisenberg group H 2,. which is an extension T-H2,.-R211 • Studying the representations of H 2,. is essentially the same as looking for operators P~. ... ,P,.,Q~. ... ,Q,. such that [Pk, PI] = [Qk, Qd = 0, } [Pko Q,] = -icSkt·
(17.5)
As when n = 1, the group H211 has a unique faithful irreducible representation, on a Hilbert space .1'f which is a completion of .1'f6n = S(C[11 ). The symplectic group Sp2,.R acts on H2n as a group of automorphisms, and a double covering Mpl2,. of Sp211 R acts on .1'f.
t
In fact Q9 is the generating function of the "contact transformation" exp (0~) of R 2, in the sense of classical mechanics.
132
II Lie Groups
The spin representation The metaplectic representation is an exact analogue of the spin representation of a double covering of S02,. If we replace the skew form S on R 2" by a symmetric one we can pose the problem, analogous to (17.1) and (17.5), of finding self-adjoint operators {P;}tE::iE::2n on a Hilbert space Jlt' such that P;Pi
+ PjP; = ~ii·
In this case the operators P; are automatically bounded (because = and they generate a finite dimensional algebra C2n. which is called the Clifford algebra. It has dimension 22", and turns out to be isomorphic to the algebra of all 2" x 2" matrices. Looking for representations of Cz, corresponds to looking for representations of the Heisenberg group Hz,. There is a unique irreducible representation, and it can be realized on A(C').
Pl
!),
The elements t
Linear Algebraic Groups I. G. Macdonald
Contents Linear Algebraic Groups
Preface Introduction Affine algebraic varieties 1 Definition and elementary properties 2 Interlude Projective algebraic varieties 3 Tangent spaces. Separability 4 Lie algebra of a linear algebraic group 5 Homogeneous spaces and quotients 6 Borel subgroups and maximal tori 7 Root structure 8
134
135 137 139 146 154 157 162 166 172 177 182
Preface
These notes follow the course of the lectures I gave at Lancaster, but contain rather more detail than it was possible to include in seven hours' lecturing time. Even so, since any adequate account of the theory of linear algebraic groups requires a book of two to three hundred pages, it is obvious that I have had to leave a lot out I have attempted only to convey the flavour of the subject, though I am painfully aware that the experts (who ought not to be reading this anyway) may well disagree with my choice of what to put in and what to leave out, what to prove and what not to prove. In preparing these notes I have relied heavily on the books of Borel [B], Humphreys [H] and Springer [S], to which these notes may perhaps serve as an introduction. At the end, under Notes and References, I give references to these books for the proofs of theorems not proved in the text. [B] A. Borel, Linear Algebraic Groups (Math. Lecture Note Series, W.A. Benjamin, Inc. New York, 1969). [H] James E. Humphreys, Linear Algebraic Groups (Graduate Texts in Mathematics, Springer-Verlag, New York, 1975). [S] T. A. Springer, Linear Algebraic Groups (BirkhaUser, Boston, 1981)
135
Introduction
Let K be a field (in fact K will always be algebraically closed, for example the field of complex numbers, or an algebraic closure of a finite field), and let M,(K) denote the space of all n x n matrices x = (x;1) with entries XiJ inK. The determinant of x E M,(K) is denoted by detx, and the transpose of x by x'. Let GL,(K) = {x E M,(K) : detx =/= 0},
the genera/linear group; SL,(K) = {x E M,(K): detx = 1}, the special linear group;
the orthogonal group; SO,(K) = O,(K) n SL,(K), the special orthogonal group; Sp2n(K) = {x E M2,(K) : x' jx = j} (where j = (
-~, ~
) ), the symplectic group.
All of these are examples of linear algebraic groups. Each of them is (a) a group, the group operation in each case being matrix multiplication; and (b) an algebraic variety, that is to say defined by polynomial equations in the matrix coordinates x;1• For example, O,(K) is defined 137
111 Linear Algebraic Groups
138
by the equations II
L
X;jXkj
= ~ik
(l~i,k~n)
jal
where
~ik =
1 or 0 according as i = k or i i= k. (The general linear group
GL,.(K) appears to be an exception: however, if we introduce another
coordinate z, we may regard GL,.(K) as the set of points (xij,z) E M,.(K) X K = K" 2 +1 that satisfy z det(Xij) = 1.) Moreover, if x andy are elements of any one of these groups, the coordinates of xy (resp. of x- 1 ) are polynomial functions of the coordinates of x andy (resp. of x). Some knowledge of algebraic geometry is an essential prerequisite to any study of linear algebraic groups, and in §1 we shall present the reader with just enough to get the subject started. Later sections will provide additional doses as and when required.
1 Affine algebraic varieties
Let K be an algebraically closed field, of any characteristic (for example, the field CC of complex numbers, or an algebraic closure of a finite field). Let A =An =K[tt, ... ,tn]
where t 1, ••• ,tn are independent indeterminates over K. The elements of A are polynomials in the t; with coefficients in K, and we may regard them as K-valued functions on the affine space K" = K x ... x K (n factors): iff E A and x = (Xt. .•. , Xn) E K", then /(x) = f(xt, ... , Xn) E K is the result of substituting x; for t; (1 ~ i ~ n) in f. In particular, t;(x) = x;, so that t; is the ith coordinate function on K". An algebraic set inK" is traditionally defined by a finite set of polynomial equations, say /;(X)= 0
(1
~
i ~ r)
where the /; are in A. More precisely, it is the set (perhaps empty) of all x e K" satisfying the equations ( *). However, in making the definitions there is no need to restrict the number of equations to be finite; so we start with any subset S c A and define V(S) = {x E K" : f(x) = 0 for all f E S} so that V(S) is the set of all points inK" at which all the polynomials in S vanish. This operation V has the following properties, in which S, S,, S2, S; are arbitrary subsets of A. The proofs are straightforward.
139
III Linear Algebraic Groups
140
(1.1) (i) V(St) U V(S2) = V(StS2), where S1S2 is the set of products !1/2 with /1 ESt and /2 E S2. (ii) n V(S1) = V( u S1) for any index set J (finite or infinite). leJ
ieJ
(iii) V(A) = 0. V(0) = K". (iv) St c: S2 => V(S!) ::l V(S2). (v) Let a be the ideal in A generated by S; then V(S) = V(a).
(Remark. The ring A is Noetherian (Hilbert's basis theorem), hence the ideal a in (v) above has a finite basis, say So, and V(S) = V(a) = V(So). Thus there would be no loss of generality in assuming that the set of equations defining V is finite.) Consider in particular the statements (iHiii) above. They show that the sets V(S), S c: A, satisfy the axioms for closed sets in a topological space. The resulting topology on K" is called the Zariski topology, and the induced topology on an algebraic set X c: K" is the Zariski topology of X. Thus the open sets in K" are the complements of the algebraic sets V(S). Intuitively, the non-empty open sets are very large: for example in the affine line (n = 1) they are the complements of finite subsets of K, and so in particular any two non-empty open sets in the affine line always intersect. By (l.l)(v) we may restrict our attention to subsets of A that are ideals, and the operation V takes ideals in A to (certain) subsets of K". In the opposite direction, let E be any subset of K" and define l(E) = {/ E A : f(x) = 0, all x E E}.
Clearly J(E) is an ideal in A, and so the operation I takes subsets of K" to ideals in A. It is not difficult to check that if E c: K" then V(J(E)) = E, the closure of E in the Zariski topology. Next, if a is an ideal of A we have I(V(a))
= r(a)
where r(a) is the radical of a, namely the set of all f e A some power of which lies in a. This is not an obvious result: it is a famous theorem of Hilbert (the Nullstellensatz). Now r(o) is a radical ideal (i.e., equal to its radical). It follows from what we have said that the operations V and I are mutually inverse
1 Affine algebraic varieties
141
order-reversing bijections between the set of algebraic subsets of K" and the set of radical ideals in A, : (1.2)
I
(alg. subsets X c K") +t(radical ideals a c A,). I'
As remarked earlier, the ring A,= K[t 1, ••• ,t11] is Noetherian, so that the ideals in A, satisfy the ascending chain condition. From (1.2) it follows that the closed sets in K" (i.e., the algebraic subsets) satisfy the descending chain condition. In general, a topological space X is said to be Noetherian if it satisfies the descending chain condition on closed subsets, that is to say if every strictly decreasing sequence of closed sets in X:
x, :::::~ x2 :::::~ ... :::::~ x, :::::~ ... (with strict inclusions at each stage) is finite. (1.3) Let X be a Noetherian topological space. Then (i) X is quasi-compact (i.e., every open covering of X has a finite subcovering). (ii) Every closed subset of X (with the induced topology) is Noetherian. Next, a topological space X is i"educible if it is not the union of two proper closed subsets, or equivalently if any two non-empty open subsets of X intersect. Irreducible implies connected, but not conversely (the subvariety x 1x 2 = 0 of K 2 is connected but not irreducible). (1.4) Let E be a subset of a topological space X. (i) E (with the induced topology) is i"educible if and only if E is i"educible. (ii) Iff : X - Y is a continuous mapping and E is irreducible then f(E) is i"educible.
( 1.5) Let X be a Noetherian topological space. Then X is the union of finitely many i"educible closed subsets, say
X=Xt U ... UX,.
If Xi¢ Xj for all pairs i =I= j, then this decomposition of X
is unique, and the Xi are the maximal i"educible subsets of X (relative to inclusion).
The closed subsets Xi in (1.5) are the i"educible components of X. In particular, if X c K" is an algebraic set, then X (with the induced
III Linear Algebraic Groups
142
topology) is Noetherian by (1.3)(ii) (since K" is Noetherian), and (1.5) applies to X. Instead of concentrating attention on the ideal a =I (X), where X c: K" is an algebraic set, it is better to factor it out and consider the K -algebra A,./a. This is called the affine algebra (or coordinate ring) of X, and it may be described as follows. A function cp on X with values in K is said to be regular if it is the restriction to X of some f e A,.. The regular functions on X form a K -algebra K (X) under pointwise addition and multiplication, and restriction to X defines a surjective homomorphism of A,. onto K[X] whose kernel is precisely I(X) =a, so that K [X] ~ A,./ a. As a K -algebra, K [X] is generated by the restrictions to X of the coordinate functions t, (1 :E;; i :E;; n). Each point x e X determines a K -algebra homomorphism Ex
namely f
..... f(x)
:K[X) -K,
(evaluation at x). Thus we have a mapping X- HomK-alg.(K[X],K)
which in fact is a bijection. Thus X can be reconstructed from its affine algebra K[X], and the closed subsets of X are the sets Vx(S) = {x eX: f(x) = 0 for allf e S} for all subsets S of K[X]. Abstractly, then, X is a topological space carrying a ring K[X) of K-valued functions on X. The pair (X,K[X]) is an affine algebraic variety, and every finitely generated commutative K-algebra with no nilpotent elements =/= 0 occurs as K[X]. In practice we shall habitually drop K [X] from the notation, and speak of X as an affine algebraic variety. ( 1.6) Let X be an affine algebraic variety. Then X is i"educible only if K [X] is an integral domain.
if and
In particular, affine space K" is irreducible. If X is irreducible, the integral domain K [X] has a field of fractions, denoted by K(X) and called the field of rational functions or function field of X. It is a finitely generated field extension of K.
1 Affine algebraic varieties
143
Morphisms First of all, a mapping f : K" --+ K"' is a morphism (or regular mapping) if there exist polynomials ft.· .. ,fm in A,.= K[t,, ... ,t,.] such that f(x) = (ft(X), ... ,fm(x)) for all x E K". More generally, if X c K" and Y c K"' are affine algebraic varieties, a mapping qJ : X --+ Y is a morphism if qJ = fiX for some morphism f : K" --+ K"' as above. Equivalently, qJ is a morphism if and only if g o qJ is a regular function on X whenever g is a regular function on Y. It follows that qJ determines a K-algebra homomorphism qJ• : K[Y] --+
K[X],
namely qJ*(g) = g o qJ. It is easily verified that if S is any subset of K[Y], then qJ- 1(Vr(S)) = Vx(qJ*S), which shows that qJ is a continuous mapping. Conversely, every K-algebra homomorphism of K[Y] into K[X] is equal to qJ• for some morphism qJ : X --+ Y. If qJ : X --+ Y and 1p : Y --+ Z are morphisms, then 1p o qJ : X --+ Z is a morphism, and (VJ o qJ)* = qJ• o 1p•.
( 1. 7) Let qJ : X --+ Y be a morphism of affine algebraic varieties. (i) · (/)• is injective if and only if qJ(X) = Y (one says that qJ is dominant). (ii) (/)• is surjective if and only if qJ is an isomorphism of X onto a closed subvariety of Y. An isomorphism of affine algebraic varieties is a bijective morphism :X--+ Y such that qJ- 1 : Y --+X is also a morphism: equivalently, qJ is an isomorphism if and only if qJ• : K[Y] --+ K[X) is an isomorphism of K-algebras. It should be remarked that it can happen that qJ :X--+ Y is bijective, and indeed a homeomorphism, but not an isomorphism. qJ
(1.8) Example. Let X = Y = K (the affine line), and qJ(x) = xP, where p > 0 is the characteristic of K. Then qJ is a homeomorphism, but qJ• : K[t] --+ K[t] is the homomorphism defined by t t-+ tP, and therefore qJ is not an isomorphism. Products Let X c K", Y c K"' be affine algebraic varieties. Then X x Y c K" x K"' = K"+m is a closed subset (i.e., subvariety) of K"+"'. Let f E K[X], g e K[Y], and define a function/ ·g on X x Y by (f·g)(x,y) = f(x)g(y). Then f · g is a regular function on X x Y, and the mapping (f, g) t-+ f · g
Ill Linear Algebraic Groups
144
from K[X] x K[Y] to K[X x Y] is K-bilinear. Hence by the universal property of tensor products it gives rise to a mapping IX: K[X] ®K K[Y]- K[X
X
Y]
such that IX(/® g)= f ·g. This mapping IX is in fact an isomorphism of K -algebras, so that
(1.9) K[X x Y]
~
K[X] ®K K[Y].
Next we have
(1.10) If X. Y are irreducible affine varieties. then X x Y is irreducible. It should be remarked that if X, Y are affine varieties, the Zariski topology on X x Y is in general finer (i.e., has more open sets) than the product of the Zariski topologies on X and Y. (Consider the case X= Y =K 1.)
( 1.11) Let X be an affine algebraic variety. Then the diagonal
l!..x = {(x,x): X eX} is a closed subset of X x X.
This is a sort of substitute for the Hausdorff axiom: if X is a topological space and X x X is given the product topology, then X is Hausdorff if and only if llx is closed in X x X. Finally, if cp : X -
Y and cp' : X' -
Y' are morphisms, then
cp x cp' : X x X' - Y x Y' is a morphism, and (cp x cp')" = cp• ® cp'* (when K[X x X1 and K[Y x Y'] are identified with K[X] ®K[X'] and K[Y] ®K[Y'] respectively, via the isomorphism (1.9)).
1be image of a morphism If cp : X - Y is a morphism of affine algebraic varieties, the image cp(X) of cp need not be either open or closed in Y. For example, if X = Y = K 2 and cp(xt,x2) = (x 1x 2,x2), then the image of cp consists of the complement of the line x 2 = 0, together with the point (0, 0), hence is the union of an open set and a closed set. A subset E of a topological space X is locally closed if each x e E has an open neighbourhood Ux in X such that E (') Ux is closed in Ux. Equivalently, E is locally closed if and only if E is the intersection of an
I Affine algebraic varieties
145
open set and a closed set, or again if and only if E is open in its closure
E. Next, E c X is constructible if it is a finite union of locally closed subsets. If E is constructible and not empty, then E contains a non-empty open subset of E. If E and F are constructible, so also are E u F, En F and the complement X- E. The constructible subsets of X are precisely the elements of the Boolean algebra generated by the open subsets of X. (1.12) Let (/) : X -
Y be a morphism of affine algebraic varieties, E a constructible subset of X. Then ip(E) is a constructible subset of Y. The proof of (1.12) can be reduced to showing that
(1.12') If X is an irreducible affine variety and (/) : X - Y a dominant morphism (1.7), then (f)( X) contains a non-empty open subset of Y. This in turn rests on the following proposition from commutative algebra: Let A c B be integral domains such that B is finitely generated as an Aalgebra, and let K be an algebraically closed field. Let b e B, b =I= 0. Then there exists a =I= 0 in A with the following property: every homomorphism e :A- K such that e(a) =I= 0 extends to a homomorphism e' : B- K such that e'(b) =I= 0. Dimension Let X be an irreducible affine algebraic variety, K(X) its function field. The dimension dim X of X is defined to be the transcendence degree of K(X) over K, that is to say the maximum number of elements of K(X) that are algebraically independent over K. For example, when X= K" we have K(X) = K(t,, ... ,tn) and hence dimK" = n. If now X is reducible, with irreducible components x,, ... ,X, (1.5), we define dim X to be the maximum of the dimensions of the components X 1• In particular, dim X= 0 if and only if X is a finite set. (1.13) Let X be an i"educible affine algebraic variety, Y an i"educible closed subvariety of X. If Y =I= X then dim Y
(1.14) Let X, Y be i"educible affine varieties. dimX +dim Y.
Then dim(X x Y) =
2 Linear algebraic groups: definition and elementary properties
A linear algebraic group G is (1) an affine algebraic variety and (2) a group, such that multiplication (p(x,y) = xy) and inversion are morphisms of affine algebraic varieties. (The reason for the adjective "linear" (rather than "affine") will become plain later, in (2.5).) If G, H are linear algebraic groups, so is their direct product G x H. A mapping cp : G - H is a homomorphism of algebraic groups if cp is (1) a morphism of affine varieties and (2) a homomorphism of groups. Let G be a linear algebraic group, A = K(G) its affine algebra The K-algebra A carries a "comultiplication"
p" :A -A®K A which makes it into a (coassociative) Hopf algebra with antipode ,. A - A. From this point of view, the study of linear algebraic groups is equivalent to the study of a certain class of Hopf algebras; but we shall not pursue this line of thought.
Examples 1. G = K with addition as group operation: p(x,y) = x + y, r(x) = -x. We have K[G] = K[t] with
p" : K[t]- K[t] ®K[t] = K[t1,t2] given by p"(t) = t1
+ t2, and r" : K[t] - K[t] 146
2 Linear algebraic groups: definition and elementary properties 147 given by r•(t) = -t. G is the additive group, usually denoted by CG,. Clearly dim CG, = 1. 2. G = K• = K- {0} with multiplication as group operation: p(x,y) = xy, r(x) = x- 1. We may identify G with the closed subvariety of K 2 given by the equation x1x2 = 1, so that K(G] = K[t,t- 11 with p• : K[t,t- 11-+ K[tt.t2,tj" 1,t2 11
given by p•(t) = t1 t2, and r•(t) = t- 1. G is the multiplicative group, usually denoted by CGm (or GL1). 3. G = GLn(K) is the group of nonsingular n x n matrices over K (i.e., matrices x such that det x =I= 0). We identify G with the closed subset of points (x,y) e Kn2 x K = Kn2+1 such that ydet(x) = 1. Thus K[G1 = K[ttj(1 ~ i,j ~ n),d-11 where d = det(t;J). and n
p·(tij> =
L: t1k ® tkj· k~l
GLn(K) is irreducible, of dimension n2. 4. Any Zariski-closed subgroup of GLn(K) is a linear algebraic group. Thus, apart from SLn, On, SOn, Sp2n mentioned in the introduction, the following are linear algebraic groups: (a) any finite subgroup of GLn(K); (b) Dn, the group of nonsingular diagonal matrices
isomorphic to CG!; (c) Bn, the group of upper triangular matrices x = (x;J) E GLn(K) such that XtJ = 0 if i > j; (d) Un. the group of upper unipotent matrices (x;J = 0 if i > j; x;; = 1 (1 ~ i ~ n)).
(2.1) Let G be a linear algebraic group. Then G has a unique irreducible
component Go containing the identity element e. and Go is a closed normal subgroup of finite index in G. The irreducible components of G are also the connected components of G, and are the cosets of Go in G.
148
Ill Linear Algebraic Groups
As a general remark before coming to the proof of (2.1), if x mappings
e G the
Ax : g 1-+ xg, Px : g 1-+ gx are automorphisms of the algebraic variety G, and in particular are homeomorphisms of the underlying topological space. For example, Ax is the composition g 1-+ (x,g) ~-+ Jl(x,g) = xg, hence is a morphism of affine varieties, with inverse Ax-•·
Proof of (2.1). Let X, Y be irreducible components of G containing
e. Then XY = Jl(X x Y) is irreducible, by (1.4) and (1.9). But XY contains X and Y, hence (as X, Y are maximal irreducible subsets of G) X=XY=Y.
It follows that XX = X, whence X is closed under multiplication; also x- 1 = 1(X) is irreducible and contains e, so that x- 1 c X. Hence X = Go is a subgroup of G, and is closed (because irreducible components are closed). Again, if x E G then xGox- 1 = AxPx-•(Go) is an irreducible component of G containing e, hence is equal to Go. So Go is a closed normal subgroup of G. By translation, the unique irreducible component of G containing a given x e G is AxGo = xGo. It follows that the irreducible components of G are the cosets of GoinG, and so by (1.5) Go has finite index in G. Each coset xG0 is closed and therefore G0 , being the complement of the union of the cosets xGo Go. is open in G. Finally, Go is connected (because irreducible), and since it is both open and closed in G it is the connected component of e in G.
+
From (2.1) it follows that for a linear algebraic group, irreducibility is equivalent to connectedness. It is customary to speak of a connected (rather than irreducible) algebraic group. The groups G", Gm. GL,., SLn. Dn, Bn. Un are all connected; the group On is not (if char. K -=/= 2). The groups SOn and Sp2n are in fact connected (but this is not obvious at this stage). (2.2) Let G be a linear algebraic group, H a subgroup of G. (i) H is a subgroup of G. (ii) If H is constructible then it is closed in G.
Proof. (i) Let x E H, so that H = xH = AxH. Take closures: H = AxH = A:xH = xH, whence HH c H and therefore Hy c H for all y e H. Take closures again: Hy = Hy c H, whence H.H c H. Also -1 -- H = H-1 = H, and hence His a group.
2 Linear algebraic groups: definition and elementary properties 149
(ii) Since H is constructible it contains a non-empty open subset U of H. But then H is a union of translates of U, hence is open in H. Hence the cosets of H in Hare open in H, and therefore H, being the complement in H of the union of the cosets xH f H, is closed in H; so H = H is closed in G.
(2.3) Let cp : G - H be a homomorphism of linear algebraic groups. Then (i) Ker
The next fact to be established is that every linear algebraic group G is isomorphic (as an algebraic group) to a closed subgroup of some GL,.(K). (Thus the general linear groups GL,.(K) play the same role in the theory of linear algebraic groups as the symmetric groups do in the theory of finite groups.) Let G be a linear algebraic group, A = K [G) its affine algebra. G acts on A by right translations: (p(x)f)(y) = f(yx),
and by left translations: (l(x)f)(y) = /(x-• y),
for f e A and x, y e G. In what follows we shall use right translations rather than left translations. This is purely a matter of choice. In (2.4) below, therefore, "G-stable" means "stable under p(x) for all x E G". (2.4) Let V be a finite-dimensional K-vector subspace of A= K[G). Then (i) V is contained in a finite-dimensional G-stable vector subspace of A. (ii) V is G-stable if and only if Jl·(v) c V ®K A.
150
III Linear Algebraic Groups
Proof (i) It is enough to consider the case where dim V /EA. Let
= 1, say V = Kf,
n
p·(f> =
'LJ' ® g, i=l
say, with f 1, g1 E A. Then for x, y E G we have
=
(p(x)f)(y)
=
f(yx) = f(p(y,x)) (p.f)(y,x) 2:/;(y)g;(x)
and therefore n
p(x)f = 2:g;(x)f, E 2:Kf,. I
i
Thus the G-orbit of f is contained in the subspace of A spanned by f 1, ••• , f n. and therefore spans a finite-dimensional K -vector subspace of A. (ii) Let v., ... ,Vn be a K -basis of V, and adjoin (infinitely many) elements w11 E A to obtain a K-basis of A. Let f E V, then p•f can be written in
the form n
p•f = 2:v; ®v;
+ 2:w~~ ® w~
i=l
II
for suitable U,, w~ E A (and almost all w~ = 0). If now x E G it follows as above that p(x)f = 2: v;(x)v; + 2: w~(x)w~~ i
II
so that p(x)f E V if and only if w~(x) = 0 for p(x)f E V for all x E G <=> <=>
all~-
Hence
w~
= 0 for all a p• f = 2: v; ®
v;
i
<=>
JJ•f E V ®K A.
(2.5) Let G be a linear algebraic group. Then G is isomorphic (as an algebraic group) to a closed subgroup of GLn(K) for some n ~ 1.
Proof The affine algebra A = K[G] is a finitely-generated K-algebra, say A = K[v 1, ••• ,vn1· By (2.4)(i) we may assume that the subspace V = EKv1 of A is G-stab1e (G acting by right translations), and that the
2 Linear algebraic groups: definition and elementary properties 151 v; are linearly independent over K. By (2.4)(ii) we have p"(V) c V ®K A, and hence equations (1)
p"(vJ)
" ® lfJIJ = Ev; i•l
for suitable (/JIJ
e A.
From ( 1) it follows that
(2)
p(x)vJ =
L" (/J;J(x)v; I• I
for all x E G, and hence (since p(xy) = p(x)p(y)) that cp : x 1-+ (cp11(x)) is a homomorphism of algebraic groups mapping G into GLn(K). To show that cp is an isomorphism of G onto a closed subgroup of GLn(K), it is enough by (1.7) to check that cp" is surjective. We have K[GLn] = K[tiJ(1 :s;;; i,j :s;;; n),d-1], where t;1 are the coordinate functions on GLn(K), and d = det(tiJ); and (cp"tiJ)(x) = t;J(cp(x)) = cp11(x), so that cp"(tiJ) = lfJiJ· From (2) above we have v1(x)
= (p(x)v1)(e) = L"
cp11(x)v;(e),
i=l
so that
v1 =
~v;(e)cp;1 = cp" ( ~v1(e)t11).
Hence each generator v1 of A lies in the image of cp", and so cp" is surjective, as required.
Jordan decomposition A matrix x E Mn(K) is said to be semisimple if it is diagonalisable, i.e. if there exists g e GLn(K) such that gxg- 1 is a diagonal matrix; nilpotent if x'" = 0 for some positive integer m, i.e. if the only eigenvalue of xis 0; unipotent if x -ln is nilpotent, i.e. if the only eigenvalue of xis 1. (2.6) Let x, y E Mn(K) commute (xy = yx). (i) If x, y are semisimple then x + y and xy are semisimple. (ii) If x, y are nilpotent then x + y and xy are nilpotent. (iii) If x, y are unipotent then xy is unipotent.
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III Linear Algebraic Groups
Proof (i) follows from the fact that commuting semisimple matrices can
be simultaneously diagonalized. (ii) is true in any ring. (iii) We have x = 1 + a and y = 1 + b where a, b are nilpotent and commute. Hence xy = 1 + c where c = a + b + ab is nilpotent by (ii). (2.7) Let x E GLn(K). Then there exist x 5 , Xu E GLn(K) such that Xs is semisimple, Xu unipotent, and x = X.sXu = XuX 5 • Moreover x 5 and Xu are uniquely determined by these conditions.
It follows from the definitions that a matrix x E GLn(K) is semisimple or unipotent if and only if gxg- 1 has the same property, for any g E G. Hence to define x 5 and Xu we may replace x by any conjugate of x in GLn(K). Thus we may replace x by its Jordan canonical form: there exists g E GLn(K) such that the matrix gxg- 1 is a diagonal sum of Jordan blocks
J,(l)- ( l
~ .~. ~
)
(r being the size of the block, and l e K an eigenvalue of x). Thus it is enough to define x 5 and Xu when xis a Jordan block J,(l) as above. In this case Xs = ll, and Xu= A,- 1x clearly satisfy the conditions of (2.7), and hence Xs and Xu are defined for all x E GLn(K): they are called respectively the semisimple part and the unipotent part of x . . Now let G be any linear algebraic group, and let x e G. By (2.5) there exists an injective homomorphism of algebraic groups cp : G- GLn(K) for some n. In this situation it can be shown that the semisimple and unipotent parts of the matrix cp(x) lie in cp(G), and more precisely that the elements X 5 , Xu E G defined by cp(x5 ) = cp(x)5 and cp(xu) = cp(x)u depend only on x and not on the embedding cp of G in a general linear group. As in the previous case we have X
= X.sXu =
XuXs
and x 5 , xu are called respectively the semisimple part and the unipotent part of x E G. Moreover, (2.8) Let cp : G -
H be a homomorphism of linear algebraic groups, and
2 Linear algebraic groups: definition and elementary properties 153 let x e G. Then q>(X)s = q>(Xs), q>(X)u = q>(Xu). An element x e G is semisimple if x = x, (i.e., if xu = e, the identity element of G), and x e G is unipotent if x = Xu (ie., if Xs =e). Let Gs (resp. Gu) denote the subset of G consisting of semisimple (resp. unipotent) elements. (2.9) (i) Gu is closed in G. (ii) G, is a constructible subset of G. (iii) Gs n Gu = {e}.
Proof. We may assume that G is a closed subgroup of GLn(K). Hence it is enough to prove (2.9) when G = GLn(K). Now x e GLn(K) is unipotent if and only if (x-ln)" = 0, which shows that the set of unipotent matrices is closed in GLn(K), proving (i). Next, when G = GLn(K) we have x e Gs if and only if x e gDng- 1 for some g e G, where Dn is the diagonal subgroup of G; hence G, is the image of the morphism q>: G x Dn- G defined by q>(g,t) = gtg- 1, hence is constructible by (1.12). Finally, if X e Gs n Gu then Xu = x, = e and therefore X = e. In general, G, and Gu are not subgroups of G. However, (2.10) Let G be a commutative linear algebraic group. Then G, and Gu are closed subgroups of G, and p : Gs x Gu - G is an isomorphism of algebraic groups.
Proof. Since G is commutative it follows that Gs and Gu are subgroups of G; Gu is closed by (2.9)(i), and G, is closed by (2.9)(ii) and (2.2)(ii). So certainly p : Gs x Gu - G is a bijective homomorphism of linear algebraic groups. On the other hand, the mapping x ~--+ Xs is a morphism, . JJ-1 .• x ~--+ (Xs, xx,-1). hence so IS
Interlude
A linear algebraic group G is said to be unipotent if G = Gu, i.e, if each x eGis unipotent. For example, the group Un (§2, Ex. 4(d)) is unipotent, and so are all its closed subgroups. Conversely, in fact, every unipotent group is isomorphic to a closed subgroup of some Un. Next, G is said to be solvable if it is solvable as an abstract group, that is to say if the "derived series" (D"G)n;;.o reaches {e} in a finite number of steps, where D0 G = G and (for n ~ 0) D"+ 1G = (D"G,D"G) is the group generated by all commutators (x,y) = xyx-ly- 1 with x, y e D"G. In particular, unipotent groups are solvable; on the other hand, the group Bn (§2, Ex. 4(c)) of upper triangular n x n matrices is solvable but not unipotent. Now let G be any linear algebraic group. The radical R(G) (resp. unipotent radical Ru(G)) of G is the unique maximal, closed, connected, solvable (resp. unipotent), normal subgroup of G. We have .R,(G) c R(G), and indeed Ru( G) = R( G)u. . If Ru(G) = {e}, the group G is said to be reductive. If R(G) = {e}, it is said to be semisimple. We now have the following chain of subgroups in an arbitrary linear algebraic group G: G (finite) I (connected) Go (semisimple) I R(G) (solvable) (torus) I Ru(G) (unipotent) (unipotent) I
{e} 154
2 Interlude
155
where (as in §2) Go is the identity component of G. The entries on the left of the chain describe the successive quotients: thus (as we have already seen in §2) G/Go is a finite group, Go/R(G) is semisimple, and R(G)/R.,(G) is an (algebraic) torus, isomorphic to a product of copies of the multiplicative group CGm. One of the aims of these lectures (although I shall have to skip a lot of the details) will be indicate how a connected reductive linear algebraic group (i.e. the quotient Go/Ra(G) in the chain above) is classified up to isomorphism by a combinatorial object called its root datum, which is a slightly more elaborate version of the root systems of Roger Carter's lectures. This classification is independent of the underlying (algebraically closed) field K. At the same time this classifies the compact connected Lie groups. For if U is a compact connected Lie group then the CC-algebra Ca~1( U) spanned by the matrix coefficients of the finite-dimensional representations of U is the affine algebra G;[G] of a connected reductive linear algebraic group G over G; (the complexification of U, cf. [Segal, 3.9]). In the other direction, U is (isomorphic to) a maximal compact subgroup of G. So we have a one-one correspondence between (isomorphism classes of) compact connected Lie groups U and (isomorphism classes of) connected reductive linear algebraic groups G over the field of complex numbers. In terms of Lie algebras, if u (resp. g) is the Lie algebra of U (resp. G), then g is the complexification of u, and u is the compact real form of g. The group G is semisimple if and only if U has finite centre, and G is an (algebraic) torus if and only if U is a (geometric) torus, i.e. a product of copies of the circle group {z E; G;: lzl = 1}. In talking of the successive quotients in the chain of subgroups of G above, I have run ahead of myself because I have not yet shown how to factor out a normal closed subgroup H of a linear algebraic group G. Let A== K[G) be the affine algebra of G; then H acts on A as follows: hqJ(x)
== qJ(xh)
for h E H, x E G and (/) E A. Hence
A8 == {lf' E A :hlp= lf', all hE H} consists of the functions lf' e A constant on each coset of H in G. In fact A 8 is the affine algebra of a linear algebraic group G/H, and the embedding A 8 '-+A is dual to a suljective homomorphism of G onto G/H with kernel H. However, we shall also need to consider GI H when H is a closed (but
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Ill Linear Algebraic Groups
not normal) subgroup of G. (The analogous situation in Lie theory is that of a Lie group G and a closed subgroup H, and one shows that the set G/H = {xH : x e G} has a natural structure of a smooth manifold.) In the algebraic context, we shall show that GI H can be given the structure of an algebraic variety, which in general is not affine. Thus the little algebraic geometry covered in §1 is no longer adequate; we need to introduce projective and quasi-projective varieties.
3 Projective algebraic varieties
If G is a linear algebraic group and H is a closed subgroup of G, we shall see in §6 that the set X = GI H of cosets xH (x e G) can be endowed with the structure of an algebraic variety. However, X is not always an affine variety: in general it is a quasi-projective variety (to be defined below). Thus we need to develop a more general notion of algebraic variety. Let X be an affine algebraic variety, let U be a non-empty open subset of X, and let x e U. A function f on U with values inK is said to be regular at x if there exists an open neighbourhood U' of x contained in U, and functions g,h e K[X] such that h vanishes nowhere on U' and f(y) = g(y)h(y)-1 for ally e U'. Then it is a basic fact (which requires proof) that (3.1) A function xeX.
f :X
-+
K is regular
if and only iff is regular at each
For each non-empty open set U c: X let l9(U) = l9x(U) denote the K-algebra of functions f : U-+ K that are regular at each x e U. Then: (a) If V c: U are non-empty open sets in X and
f e
l9(U) then
II V
E
l9(V).
(b)
If a non-empty open set U c: X is covered by non-empty open subsets is such that IIU1 e l9( U,) for each i, then I e l9( U).
u,, and iff : U -+ K
These two conditions (a) and (b) say that the assignment U ....... l9x(U) (for U open in X) is a sheaf l9x of functions on X, called the structure sheaf of the affine variety X. From this point of view, the pair (X,l9x) is a ringed space, i.e. a topological space carrying a sheaf of functions (satisfying the conditions (a) and (b) above). From (3.1) it follows that
157
158
III Linear Algebraic Groups
the affine algebra K[X] of X is just (!Jx(X), and it may appear to the reader that all we have achieved by this discussion is to replace a simple object K[X] by a more complicated object, namely the sheaf (!)x. But this elaboration has a purpose, namely to enable us to define a more general notion of algebraic varieties.
Prevarieties and varieties Let (X, £9) be a ringed space and let Y be a subset of X. Give Y the induced topology, and for each non-empty open subset V of Y let
f : V- K such that for each x E V there is an open neighbourhood Ux of x in X and a function fx E (!J(Ux) which agrees with f on V () Ux. The assignment V ~--+ ((!JIY)(V) is a sheaf (!)IY on Y (i.e., it satisfies conditions (a) and (b) above), called the sheaf induced by (!) on Y. In particular, if Y is open in X we have ((!)I Y)(V) = (!)(V) for all V open in Y. Next let (X,(!Jx) and (Y ,(!)y) be ringed spaces, and let fP: X- Y be a continuous map. Then fP is a morphism of ringed spaces if, for each open V c: Y and each f E (!)y(V), the function f o fP : (('- 1V - K belongs to (!)x(fl'-t V). When X and Y are affine algebraic varieties, this notion of morphism agrees with that defined in §1. After these preliminaries, a prevariety (over K) is a ringed space (X,(!Jx) such that X is covered by a finite number of open sets U; With the property that each induced ringed space (U,£9IU1) is isomorphic (as ringed space) to an affine algebraic variety. Intuitively, X is obtained by patching together a finite number of affine varieties in such a way that the regular functions agree on the overlaps. It follows from this definition that X is a Noetherian topological space, so that (1.5) applies to X. We shall habitually drop (!)x from the notation, and speak of X (rather than (X,(!Jx)) as a prevariety; and the reader may be relieved to be told that (except in the present section) he will never see the structure sheaf (!)x explicitly referred to; but he should bear in mind that it is always implicitly present, as an essential part of the structure of X. One shows next that if X and Y are prevarieties, the product X x Y (satisfying the usual universal property) exists and is unique up to isom n morphism. Briefly, if X = .u U1 and Y = u V1, where the U1 and v1 are ((!JIY)(V) be the set of functions
r=l
J-1
affine open sets, then X x Y is covered by the products U; x V1, which are themselves affine varieties.
3 Projective algebraic varieties
159
A variety is a prevariety X satisfying the separation property (1.10); the diagonal Ax= {(x,x) : x eX} is closed in X x X. If X is an irreducible variety covered by affine open sets U;, each U; is irreducible by (1.4) (since the closure of U; is X) and each intersection U; n u1 is non-empty. It follows that U; and u1 have the same function field, which is called the function field K(X) of X. The dimension of X is defined, as in the affine case, to be the transcendence degree of K(X) over K. Finally, propositions (1.12)- (1.14) remain true for arbitrary varieties.
Projective Varieties The most important examples of non-affine varieties (and the only ones we shall encounter) are the projective and quasi-projective varieties, to which we now tum. If V is a finite-dimensional K-vector space, the projective space P(V) of V is the set of all lines (i.e. 1-dimensional subspaces) in V. If V = K"+ 1, P(V) is denoted by Pn(K). A line in Kn+l is determined by any point (.xo, .•• ,xn) =/= 0 on it, and hence a point x e Pn(K) has n + 1 homogeneous coordinates (xo, ... , Xn). not all zero, and such that (xo, ... ,xn) and (lxo, ... ,A.xn), where A. is any non-zero element of K, represent the same point of Pn(K). In projective geometry, equations of varieties are homogeneous; so if S c: K[t0 , ••• ,tn] is any set of homogeneous polynomials (not necessarily of the same degree), let V(S)
= {x E Pn(K)
: /(x)
= 0,
all f E S}.
Just as in the affine case, the V(S) are the closed sets in a topology (the Zariski topology) on Pn(K). Consider in particular H; = V(t;), a hyperplane in Pn whose complement U; is the open set consisting of all x = (xo, ... , Xn) E Pn such that x; =/= 0. By homogeneity we may assume that x1 = 1, so that U; consists of all x E Pn with coordinates (.xo, ... ,X;-t. 1,x;+t.····xn) and hence is in bijective correspondence with affine space K". Thus we have Pn(K) = Uo U Ut U ... U Un the union of n+ 1 open sets each identified with K". We have a structure sheaf ~u, on each U;, and since the restrictions of ~u, and lJu1 to U; n U1 coincide, the sheaves lJu, are the restrictions to U1 of a well-defined sheaf lJ = lJp. on Pn. the structure sheaf of Pn. Thus projective space Pn(K) is
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Ill Linear Algebraic Groups
a prevariety as defined above, and it may be verified that it is in fact a variety (the diagonal of Pn x Pn is a closed set). A projective (resp. quasi-projective) variety X is now defined to be a closed (resp. locally closed) subset of a projective space Pn(K), together with its induced structure sheaf (;x = ()p.IX. Both affine and projective varieties are quasi-projective. On an affine variety X, as we saw in §1, there are plenty of regular functions defined on all of X; indeed enough to determine the structure of X. On a projective variety, on the other hand, this is not the case: if X is an irreducible projective variety, we have (;x(X) = K, ie. the only regular functions defined on all of X are the constant functions. (Compare Liouville's theorem: the only holomorphic functions on the Riemann sphere(= P1(CC)) are the constants.)
Complete varieties An algebraic variety X is said to be complete if for any variety Y the projection morphism X x Y - Y is a closed mapping, i.e. maps closed sets to closed sets. (This notion is an analogue for algebraic varieties of the notion of compactness in the category of locally compact (Hausdorff) topological spaces: if X is locally compact, then X is compact if and only if, for all locally compact spaces Y, the projection X x Y - Y is a closed mapping.) (3.2) Let X, Y be varieties. (i) If X is complete and Y is closed in X, then Y is complete. (ii) If X and Y are complete, then X x Y is complete. (iii) If cp : X - Y is a morphism and X is complete, then cp(X) is closed
in Y and is complete. (iv) If Y is a complete subvariety of X, then Y is closed in X. (v) If X is complete and irreducible, the only regular functions on X are the constant functions. (vi) If X is affine and complete then X is finite (i.e. dim X= 0). Proof. (i) and (ii) are immediate from the definitions. As to (iii), let r = {(x, cp(x)) : X E X} c: X X y be the graph of cp. Then r is the inverse image of the diagonaiAy under the morphism (x,y) ~-+ (cp(x),y) of X x Y into Y x Y, hence is closed in X x Y (because Y is a variety). Since cp(X) is the image of r under the projection X X y - y. it follows that cp(X) is closed in Y. To show that cp(X) is complete, we
3 Projective algebraic varieties
161
may assume that q~(X) = Y. If Z is any variety, let p : X x Z - Z, q : Y x Z - Z be the projections; then if W is closed in Y x Z, we have q(W) = p((q~ x l)-1(W)) closed in Z, because X is complete. (iv) now follows from (iii), applied to the inclusion morphism Y -X. Next, a regular function f on X may be regarded as a morphism of X into the affine line K 1, which is also a morphism of X into the projective line P1• If X is irreducible, then /(X) is an irreducible proper closed subset of P1, by (1.4) and (iii) above, hence consists of a single point (because the only proper closed subsets of P1 are the finite subsets). Hence f is constant, which proves (v). Finally, (vi) is a direct consequence of (v). A basic fact (which we shall not prove here) is that
(3.3) Projective varieties are complete. In view of (3.2)(i}, it is enough to show that projective space P = Pn(K) is complete, and for this it is enough to show that the projection P x K"' - K"' is a closed mapping, for each m ~ 0. In view of (3.3), the assertions of (3.2) apply to projective varieties. In particular, the image of a projective variety X under a morphism cp : X - Y is a closed subset of Y (contrast with the analogous statement (1.12) for affine varieties).
4 Tangent spaces. Separability
Let X be an affine algebraic variety, embedded as a closed subset inK", and let f1, ... ,f, e K[t., ... ,t,.] be a set of generators of the ideal I(X), so that for x e K" we have x eX if and only if /i(x) = 0 (1 ~ i ~ r). Suppose for the moment that K = CC. To say that a vector v e CC" is a tangent vector to X at x eX means that /;(x + EV) is O(e2) for small e e CC and i = 1, 2, ... , r, or equivalently that (1
(*)
~
i ~ r)
as polynomials in e. This condition makes sense for any field K, and we may reformulate it as follows. The algebra D = D(K) of dual numbers over K is defined to be D
= K[t]/(t2 ) = K
eKe
where e is the image oft in D, so that e2 = 0. Thus the elements of D are of the form a+ be with a, be K, and add and multiply as follows: (a+ be)+ (a'+ b'e) =(a+ a')+ (b + b')e, (a+ be)(a' + b'e) = aa' + (ab' + a'b)e.
Hence a + be 1-+ a is a K -algebra homomorphism of D onto K. The condition (•) for a tangent vector v e K" is now replaced by (4.1)
/;(x +EV) = 0
(1
~
i ~ r)
and we are therefore led to the following definition: a vector v e K" is a tangent vector to X at x if and only if the mapping f ~--+ f(x + EV) of K [X] into D is a K -algebra homomorphism. Accordingly we define the tangent bundle of X to be T(X) = HomK-alg(K[X],D) 162
4 Tangent spaces. Separability
163
and the projection a+ be.....,. a of D onto K projects T(X) onto X = HomK-alg(K(X],K). For each x EX, the fibre Tx(X) of T(X) over x is the K-vector space of tangent vectors at x. If' E Tx(X) and f E K [X], then e(f) is of the form
e
(4.2)
where d~ : K [X] - K satisfies
ddfg) = ddf)g(x)
(4.3)
+ f(x)d~(g)
for all/, g E K[X], since e(fg) = '(f),(g). Conversely, each K-linear mapping d~ : K[X] - K satisfying (4.3) determines a tangent vector E Tx(X) by the formula (4.2). One should think of dd as the derivative of f at X in the direction Now let cp : X - Y be a morphism of affine algebraic varieties, so that we have a K-algebra homomorphism cp• : K[Y]- K[X], and hence
e
e.
T(cp) : T(X)- T(Y)
e.....,. eo
e
namely cp• for E T(X). Over each x E X the restriction of T(cp) to Tx(X) is a K-linear mapping Tx(cp) : Tx(X)- Tfll(xl(Y).
If
eE Tx(X) and '1 = Tx(cp)e, then
(4.4) If tp : Y - Z is another morphism of affine algebraic varieties we have T(tp o cp) = T(tp) o T(cp), so that (4.5)
Tx(V' o cp) =
Tq~(x)(tp)
o Tx(cp)
which is just the "chain rule" of differential calculus. Finally, if X is any algebraic variety as in §3, covered by affine open sets U;, one checks that the tangent bundles T(U;) patch together to form the tangent bundle T(X) of X, and that T(X) depends only on X and not on the affine open covering chosen. For each x E X, the fibre Tx(X) of T(X) over xis a finite-dimensional K-vector space. (4.6) Example. Let V be a finite-dimensional K-vector space, P = P(V) the projective space of V (§3), and let 1t : V- {0}- P be the projection. Let x e V - {0}, so that x(x) is the line L = Kx generated by x. The tangent space Tx(Y) may be identified with V; with this identification Tx(1t) : V - TL(P) is surjective with kernel L. (We may take V = Kn+l
Ill Linear Algebraic Groups
164
and x = (Xo.XJ, ... ,x,.) with Xo ::/= 0, so that n(x) e Uo in the notation of §3, and n(x) = (X0 1Xt.····X0 1x,.). Let e = (~o, ... ,~,.) E Tx(Y). Since X;
+ ee;
Xo + eeo
= =
1 E'~O Xo (x; + ee;)( 1 -
x;)
X;
Xo
+ -~' (Xo~i - X;~o)
xo
it follows that Tx(n)~ = 'I = ('ft, ... , ,,.), where 'f; = x02 (.xo~;- x;~o), and hence that Tx(n)e = 0 if and only if .xoe; = x;eo (1 ~ i ~ n), that is to say if and only if e E Kx.) If X is an arbitrary variety, in general the tangent spaces Tx(X) will not all have the same dimension, even if X is irreducible. For example, if X is the plane curve with equation XtX2 + xf + x~ = 0, which has a double point at the origin, then Tx(X) is 1-dimensional for all x e X except (0, 0), and T(o,o)(X) = K 2 is 2-dimensional. In fact we have
(4.7) Let X be an i"educible variety of dimension d. Then (i) dim Tx(X);;;::: d for all x EX. (ii) S = {x eX :dim Tx(X) > d} is a proper closed subvariety of X. Sis called the singular locus of X. If dim Tx(X) = d, we say that xis a simple point of X, or that X is smooth at x. If every x e X is simple (i.e. if S is empty) we say that X is smooth. Now let G be a connected linear algebraic group. For each x e G, let A.x : G- G be left multiplication by x, as in §2. Then Te(A.x) : Te(G)Tx(G) is an isomorphism, so that dim Tx(G) is the same at all x e G, hence by (4.7) is equal to dim G, and G is smooth.
Separability At this point it is instructive to return to Example (1.8). cp : Ga - Ga is the mapping defined by cp(x) = x', where p > 0 is the characteristic of K. This mapping cp is a homomorphism because (x + y)P = xP + yP, hence is a bijective homomorphism of algebraic groups, but is not an automorphism of the algebraic group Ga. Let e e Tx(Ga) = K: since (x+ee)P = x' = xP, it follows that T(cp) maps each tangent space Tx(Ga) to 0; thus cp has an 'infinitesimal' kernel To(G0 ), invisible to the naked eye. The notion of separability, which we shall now introduce briefly, is designed to avoid this sort of situation. If E c F are fields such that F is finitely generated over E, then F is said to be separably generated over E
+ePe'
4 Tangent spaces. Separability
165
if there is an intermediate field E' such that E' IE is a pure transcendental extension and FIE' is a finite separable (algebraic) extension (so that F = E'(tX) where the minimal polynomial for tX over E' has no repeated roots). Now suppose that X and Y are irreducible varieties, with respective function fields, K(X), K(Y). A dominant morphism(/) :X- Y (ie., such that qJ{X) is dense in Y) induces an embedding (/)• of K(Y) in K(X), and (/)is said to be separable if K(X) is separably generated over qJ•K(Y). A more geometrical criterion for separability is contained in the following proposition:
(4.8) Let (/) :X- Y be a morphism between irreducible varieties. (i) Suppose that x e X and y = qJ(x) e Y are simple points of X, Y
respectively and that Tx((f)) : Tx(X) - Ty(Y) is surjective. Then (/) is dominant and separable. (ii) Conversely, suppose that (/) is dominant and separable. Then there is a non-empty open subset U of X such that for each x e U the point y = qJ(x) is simple on Y and Tx((/)): Tx(X)- T1 (Y) is surjective. Remark. In characteristic zero all finitely generated field extensions are separably generated, and hence all dominant morphisms are separable: inseparability is a "characteristic p" phenomenon.
5 The Lie algebra of a linear algebraic group
Let G be a linear algebraic group, A = K [G) its affine algebra, and let g = Te( G) be the tangent space to G at the identity element e. From §4 we know that g is a K -vector space of dimension equal to that of G. We shall show that g can be given the structure of a Lie algebra over K, by interpreting the elements of g as derivations. More precisely, let L(G) denote the space of all K-linear maps ~ : A - A which (a) are derivations, i.e. satisfy ~(fg)
(5.1)
=
~(f)g
+ f ~(g)
for all f, g E A; and (b) are left-invariant, i.e. satisfy A(x)~
(5.2)
= cU(x)
for all x E G, where as in §2 A(x)f is the function y ...... f(x- 1y). If ~" ~2 e L(G) one checks immediately that~= [~1.~21 = ~1~2 -~2~1 satisfies (5.1) and (5.2), and hence that L(G) is a Lie algebra over K. Recall from §4 that each X E g determines a K-linear mapping dx :A-K
+ dx(f)e, so that dx(fg) = dx(f)g(e) + f(e)dx(g)
by the rule X(f) = f(e) (5.3)
for f,g EA. Now define, for each X E g and f E A, a function by the rule (5.4)
~xf
on G
(~xf)(x) = dx(A(x- 1)!).
Let us first check that product map and Jet
~xf
e A. As in §2, let Jl : G x G - G be the (gi,hi E A)
166
5 The Lie algebra of a linear algebraic group so that f(xy) =
161
E g;(x)h;(y) for all x, y e G, and therefore ..t(x- 1)/ = Lg;(x)h;,
from which it follows that
~x/ =
L g;dx(h;) E A.
Thus an equivalent definition of ~x is ~x
= (1 ® dx) o J.l•.
(5.5) The mapping X 1-+ ~x is a K -linear isomorphism of g onto L( G).
Proof. From (5.3) and (5.4) it follows that ~x is a K-derivation of A which is left-invariant, because for all f e A and x,y e G we have (..t(y)~xf)(x)
= =
(~xf)(y- 1 x) = dx(..\((y- 1x)- 1)f) dx(..\(x- 1)..\(y)f) = ~x(..\(y)f)(x)
that ..\(y) 0 ~X= ~X 0 A(y). Conversely, if~ E L(G) the mappingd :A- K defined by df = satisfies SO
(~f)(e)
d(fg) = d(f)g(e) + f(e)d(g) and hence (§4) d = dx for a unique X e g; and by left-invariance we have (~f)(x)
= =
so that
~
=
~x.
(..t(x- 1 )~/)(e) = ~(..t(x- 1 )/)(e) dx(..\(x- 1)/) = (~xf)(x)
completing the proof.
In view of (5.5) we may transport the Lie algebra structure of L(G) to g by defining
(5.6)
~(X,Y] = (~x.~y] = ~X~Y- ~Y~X
for all X, Y e g. L(G) (or g) is the Lie algebra of G. (5.7) Example. Let G = GLn(K), so that A= K[G] = K[tij(l :E;; i,j :E;; n), d- 1], where the t 1i are the coordinate functions on G, and d = det(t;i). Since G is an open subset of matrix space Mn(K) = K"2 , the tangent vectors at the identity ln E G are the homomorphisms f ~--+/On+ eX),
Ill Linear Algebraic Groups
168
where X= (XtJ) E Mn(K), and we may identity 9 = T1.(G) with Mn(K). We have dx(tiJ) = XiJ and hence, for x = (x11 ) e G, (c5xtiJ)(x) =
dx(A.(x- 1)tiJ)
=
dx(L X;ktkJ)
n
n
=
k•l
LXikXicj k-1
so that c5xT = TX, where Tis the matrix (tiJ)· From (5.6) it follows that the Lie algebra structure on 9 = Mn(K) is the usual one: [X, Y] = XV- YX. This Lie algebra is denoted by 91n(K). More generally, if Vis any finite-dimensional K-vector space, the Lie algebra gi{V) of the algebraic group GL(V) is the Lie algebra of the associative algebra EndK(V) of all K-linear maps V-+ V. Let now qJ : G -+ H be a homomorphism of linear algebraic groups, and let g, ~be the Lie algebras of G, H respectively. The differential of qJ is dqJ = T
From the definitions it follows that if X e 9 andY= (dqJ)X e 1), then (5.8) (i) dv = dx o (/)•, (ii) qJ" o c5v = c5x o qJ", from which it is easy to check that dqJ : g -+ I) is a Lie algebra homomorphism. Let V be a finite-dimensional subspace of A = K[G], stable under right translation p(x) for each x e G, so that we have a homomorphism of algebraic groups p: G-+ GL(V). Let X e 9 and let Y = (dp)X e 9I(V). Then we have (5.9) c5x/
= Y/
for all
f e
V.
Proof. Choose a K-basis /J, ... ,fn of V, thereby identifying GL(V) with GLn(K). As in §2 we have n
(1)
p· iJ = 'LJ~ ® (/)1} I• I
5 The Lie algebra of a linear algebraic group
for suitable Cf'ii
169
e A, so that p(x)/j
= L Cf'ij(x)f; i
and p(x) is identified with the matrix (Cf'ij(X)) Also from (1) we have
e GLn(K).
).(x- 1)/i = Lf;(x)cp;i i
so that (2)
bx/j = L:dx(Cf';j)/;.
But dx'Pii
= dxcp•(tiJ) = dvtiJ = YiJ by (5.7) and (5.8), hence (2) becomes lJx/j = LYiJ/i = Yfi. i
Let G be a linear algebraic group, H a closed subgroup of G, and let a = {f e A : /IH = 0} be the ideal in A = K [G) defined by H. Restriction of functions to H defines a K-algebra isomorphism A/a~ K[H], and hence the tangent bundle of H may be identified with the set of e T(G) such that e(a) = 0. In particular, the Lie algebra ~ = Te(H) of H is identified with the vector subspace of g consisting of the X e g such that dxa =0.
e
(5.10) (i) H = {x e G : p(x)a c a} = {x e G : ).(x)a c a}. (ii) ~={X e g : lJx(a) c a}. Proof. (i) Let x e H, f e a. For ally e H we have (p(x)f)(y) = f(yx) = 0, whence p(x)f e a and therefore p(x)a c a. Conversely, if p(x)a c a and f e a, then f(x) = (p(x)f)(e) = 0, so that x e H. Likewise with p replaced by )., (ii) Let X E g. Then
Xe ~
'* '* '* '*
dxa =0 dx().(x- 1)a) = 0 for all x E H (by (i)) (lJxf)(x) = 0 for all f E a, x e H lJxa ca.
(5.11) Let cp : G - H be a bijective homomorphism of linear algebraic groups, and let g, ~ be the Lie algebras of G, H respectively. Then cp is an isomorphism if and only if dcp : g - ~ is an isomorphism of Lie algebras.
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III Linear Algebraic Groups
Clearly, if cp is an isomorphism so is dcp. In the other direction we shall merely indicate the proof. We may assume that G, and therefore also H, is connected. If E = K(G) and F = K(H) are the function fields, it can be shown that the field extension E I cp •F is purely inseparable. If now dcp is an isomorphism, it follows from (4.3)(i) that E/cp•F is separable; hence E = cp• F, so that cp• and hence also cp is an isomorphism. In particular, if K has characteristic 0, every bijective homomorphism is an isomorphism (because there are no inseparable field extensions). The adjoint representation Let G be a linear algebraic group, g its Lie algebra. For each x E G we have an inner automorphism Intx : G - G, namely g 1-+ xgx- 1• Its differential is denoted by Adx:
Adx = d(Intx): g- g. This is a linear transformation of the vector space g, with inverse (Adx)- 1 = Ad(x- 1), so that Adx e GL(g). Moreover, since Int(xy) = (lntx)o(Inty) for all x, y e G, it follows that Ad(xy) = (Ad x) o (Ad y) and hence that Ad =AdG: G- GL(g) is a representation of G, called the adjoint representation. (5.12) Let X E g, Y = (Adx)X. Then {Jy = p(x) o {Jx o p(x- 1).
Proof. Let cp = lntx, so that (cp.f)(y) = f(xyx- 1) for f e A= K[G] and y e G, and therefore cp• = l(x- 1)p(x- 1). The formula (5.12) now follows from (5.8)(ii), being in mind that {Jx commutes with l(x- 1). (5.13) Example. Let G = GL11 (K), x E G, X e g. Then (Ad x)X = xXx- 1•
Proof. Let T = (tij) be the matrix of coordinate functions on G, as in (5.7). If Y = (Adx)X we have from (5.7) and (5.12) TV= {JyT
= =
p(x){Jxp(x- 1)T = p(x){JxTx- 1 p(x)TXx- 1 = TxXx- 1•
5 The Lie algebra of a linear algebraic group
171
From (5.13) it follows that AdG : G- GL(g) is a homomorphism of algebraic groups in the particular case that G = GL,.(K). If now G is any linear algebraic group, we may by (2.5) embed G is a closed subgroup in some GL,.(K), and correspondingly g as a subspace of gi,.(K). One checks that AdG is the restriction toG of AdGL.,.(Kl• and hence (5.14) For any linear algebraic group G, Ad : G- GL,.(g) is a homomorphism of algebraic groups.
6 Homogeneous spaces and quotients
Let G be a linear algebraic group, X an algebraic variety. An action of x X - X, written (g, x) ~---+ gx, such that
G on X is a morphism G
(i) (ii)
g(hx) = (gh)x for all g,h e G and x eX, ex= x for all x eX.
The variety X, equipped with an action of G as above, is called a Gvariety or G-space. If X and Y are G-spaces, a morphism qJ : X - Y is a G-morphism if qJ(gx) = gqJ(x) for all g E G and x e X. Let X beaG-space, x EX. The isotropy group of x in G is the group Gx = {g E G : gx = x}, the subgroup of G that fixes x. It is a closed subgroup of G, because it is the inverse image of {x} under the morphism g ~---+ gx of G into X. The orbit of xis Gx = {gx : g e G}, which is the image of Gunder the same morphism g ~---+ gx, hence by (1.12) contains a non-empty open subset U of Gx. Since Gx is G-stable, it follows that the translates gU of U cover Gx, and hence that Gx is open in its closure Gx. This proves the first part of
(6.1) (i) Each orbit of G in X is a locally closed subvariety of X. (ii) If G is connected, all orbits of minimal dimension are closed. In particular, there is at least one closed orbit. As to (ii), let Y be an orbit of minimum dimension d. (Since G is connected, Y is irreducible.) Now G acts on Y, hence Z = Y - Y is a union of orbits of G, all of dimension < d, since dim Z < d. We conclude that Z is empty, i.e. that Y is closed. A homogeneous space for G is a G-space X on which G acts transitively (i.e., there is only one orbit). In that case the isotropy groups Gx, x eX, are all conjugate in G. If we fix a point Xo e X and write H = G"" (a 172
6 Homogeneous spaces and quotients
173
closed subgroup of G) and GI H for the set of cosets gH (g e G), the mapping gXQ ~--+ gH : X - GI H is bijective. Let H be a closed subgroup of G. A quotient of G by H is a pair (X,Xo) consisting of a homogeneous space X for G together with a point Xo e X with isotropy group H, such that the following universal property holds: for any pair (Y ,yo) consisting of a homogeneous space Y for G and a point Yo e Y whose isotropy group contains H, there is a unique G-morphism cp :X - Y such that cp(Xo) = Yo· Clearly, if a quotient exists, it is unique up to G-isomorphism. When is a homogeneous space for G a quotient? One answer to this question is provided by (6.2) below, but we shall have to omit the proof. Suppose X is a homogeneous space for G, and that Xo e X has isotropy group GXo = H. Let 1p : G - X be the mapping g 1-+ gxo, and let Xo = VJ(Go), where Go is the identity component of G. Let VJo :Go- Xo be the restriction of 1p to Go. (6.2) (X,Xo) is a quotient ofG by H
if and only if the morphism VJo :Go-
Xo is separable. Consider the morphism 1p. Its fibres are the cosets gH of H in G, hence are all of dimension dim H, and X is a variety of dimension dim G-dim H. Moreover, by homogeneity all the tangent spaces Tx(X) (x e X) have the same dimension, so that X is smooth and dim Tx(X) = dim G -dim H by (4.2). Let 9, I) be the Lie algebras of G, H respectively. Then we can reformulate (6.2) as follows: (6.3) Let X be a homogeneous space for G. and let Xo eX have isotropy group H. Then (X,xo) is a quotient of G by H if and only if the kernel of Te(VJ): 9 - TXo(X) is equal to l).
Proof. We may assume that G is connected. Since
maps H to the point X(), the kernel of Te(VJ) contains(). Since the dimension of TXo(X) is dim G - dim H = dim 9 - dim 1), it follows that ker Te(1p) = I) if and only if Te(1p) is surjective, that is to say (4.3) if and only if 1p is separable. Hence (6.3) follows from (6.2). 1p
Now let G be any linear algebraic group and H a closed subgroup of G. Let 9, I) be the Lie algebras of G, H respectively, so that I) is a subalgebra of g. We shall show how to construct a quotient of G by H.
174
III Linear Algebraic Groups
For this purpose let A = K[G] and let a = {/ e A : /IH = 0} be the ideal defined by H. Since A is a Noetherian ring, the ideal a is finitely generated, say by f 1, ... , f,. By (2.4) there is a finite-dimensional vector subspace V of A, stable under G (acting by right translations) and containing ft, ... ,f,. Let p : G- GL(V) be the representation of G so defined, and let U = V n a, which is a subspace of V containing the generators / 1 of the ideal a. Then we have (6.4) (i) H (ii) I)= {X
= {x e G : p(x)U =
U}.
e g : dp(X)U c: U}.
Proof. (i) Let x e H, f e U. Then (p(x)f)(y) = f(yx) vanishes for all y e H (since f e a), hence p(x)f e V n a = U and therefore p(x)U c: U. But p(x) is bijective, hence p(x)U = U. Conversely, if p(x)U = U then p(x)/1 e a for each generator /;, and therefore / 1(x) = (p(x)/;)(e) = 0, whencex e H. (ii) By (5.9) we have dp(X)U = ~xU for all X e g. If X e I) then ~xU c: V n a= U by (5.10). Conversely, if ~xU c: U then ~x/1 e a for each i. Since the /; generate a and ~x is a derivation, it follows that ~xa c: a and hence that X e I) by (5.10) again. The next step is to squash U down to a line (i.e. a one-dimensional space). To do this we take exterior powers: if U has dimension d then L = AdU is a line contained in E = AdV. The group G acts on E via Adp = lf', say: (,O(X)(vl A ... A Vd)
=
p(x)vt A .•. A p(x)vd,
dqJ(X)(vl A .•. A Vd)
=
Evt A ... A~xv;A ... Avd
d
i=l
for x e G, X E g and v~o ... ,vd not difficult to deduce that
e
V. From these formulas and (6.4) it is
(6.5) (i) H = {x e G : p(x)L = L}, (ii) I)= {X e g : dqJ(X)L c: L}. We now pass to the projective space P = P(E), in which Lis a point, say Xo e P. The group G acts on P via lf', and by (6.5)(i) the isotropy group G:lq) is precisely H. Let X= GXo be the orbit of Xo(= L) in P. By (6.1), X is locally closed in P, that is to say it is a quasi-projective variety on which G acts transitively. In fact (X,xo) is a quotient ofG by H. To show this, it is enough by (6.3) to verify that the kernel of Te(tp) is equal to 1). Choose a non-zero element uo e L and let co : G - E be the
6 Homogeneous spaces and quotients
175
mapping g 1-+ q>(g)Uo; also let 1t : E- {0} - P be the projection, so that 1t(u) = Ku for u E E, u =I= 0. Then tp = 1t ow and hence Te(tp)X = T.,.(1t)(dq>(X)Uo) for X E g. Now by (4.6) the kernel ofT110 (1t) is L = Kuo, hence by (6.5)(ii) the kernel of Te( tp) is ~. as required. We have therefore proved (6.6) If G is any linear algebraic group, H a closed subgroup of G, then a quotient of G by H exists and is a quasi-projective variety. We denote the quotient of G by H by G/H. (6.7) Example. Let G = GL,.(K), acting on the space V = K 11 of column vectors of length n, with basis consisting of the unit vectors e1, ••• , e,.. A (complete) flag in V is a sequence of subs paces
f : 0 = Uo < Ut < ... < U,.
= V
such that dim U; = i for each i. In particular,
fo : 0 =
Vo < Vt < ... < V,. = V
is a flag, where V; is spanned by e1, ... , e1• If g E G, then gf is the flag
0 = Uo j. The set F may be given the structure of a projective algebraic variety, as follows. With u1, ... ,u,. as above, the exterior product Uti\ ... /\ u1 depends (up to a nonzero scalar multiple) only on U1, and hence the tensor product (•) Ut
®(uti\ U2) ® (ut 1\ U21\ U3) ® ... E V ® /\ 2 ¥ ® I\ 3 V ® ... = E,
say, depends (up to a non-zero scalar multiple) only on the flag f. Hence if P(E) is the projective space of E, we have a mapping q> : F - P(E), namely f~-+ image of(•) in P(E). It can be verified that q> is injective and that X= q>(F) is a closed subvariety of P(E), hence a projective variety.
176
III Linear Algebraic Groups
Moreover if Xo eX is the image of the flag f0 , then (X,.xo) is a quotient of G by B. Thus G/B is a projective variety, called the .flag variety. Finally, when His normal in G, we can say more: (6.8) Let G be a linear algebraic group, H a closed normal subgroup of
G. Then G/H is an affine variety and, when provided with the usual group structure, a linear algebraic group.
7 Borel subgroups and maximal tori
The key to unravelling the structure of a compact connected Lie group G is the theorem that the maximal tori of G are all conjugate in G. In the case of a linear algebraic group, the key to unravelling its structure is the study of its connected closed solvable subgroups, and for this purpose the following fixed-point theorem is crucial: (7.1) Let G be a connected solvable linear algebraic group acting on a
nonempty projective variety X. Then G has a fixed point in X. Proof (sketch). We proceed by induction on d = dim G. If d = 0 then G = {e}, and the result is obvious. So assumed> 0 and let G' = DG be the derived group of G, generated by all commutators xyx- 1y- 1 (x,y e G). In fact G' is closed and connected and (of course) solvable, and is a proper subgroup of G, so that dim G' < d. Hence, by the inductive hypothesis, G' has a fixed point in X. For each g e G' the set XI = {X e X : gx = X} is closed in X' because it is the inverse image of the diagonal A= {(x,x) : x eX} under the morphism x ~-+ (x,gx) of X into X x X, and A is closed in X since X is projective. Hence
X'= n X 1 , geG'
the set of fixed points of G', is closed (and non-empty), hence projective. Moreover, G' is normal in G and hence X' is G-stable. By (6.1 ), there exists x e X' such that the orbit Gx is closed, hence projective. On the other hand, the isotropy group Gx of x contains G', hence is normal in G (because G/G' is an abelian group). To complete the proof we need the following result, whose proof we omit: (7.2) Let Y, Z be i"educible transitive G-spaces, and let bijective G-morphism. If Z is complete, so is Y. 177
qJ :
Y - Z be a
178
Ill Linear Algebraic Groups
We take Y = GIGx and Z = Gx, (/) being the morphism gGx ~--+ gx. Z is projective, hence complete, and therefore Y is complete. But also Y is affine, since it is the quotient of G by a normal subgroup; and irreducible, since G is connected. Hence by (3.2)(vi) Y is a single point, so that Gx = G and x is a fixed point. An immediate corollary of (7.1) is (7.3) Let G be a connected closed solvable subgroup of GLn(K). Then there exists g E G such that G c gBg- 1, where B = Bn is the group of upper triangular matrices. Proof. G acts on the flag variety F of K" (6.7), which is a projective variety. By (7.1) there is a flag f fixed by G. Iff= gf0 in the notation of (6.7), we have G c gBg- 1•
Borel subgroups Let G be a connected linear algebraic group. Consider the set of all connected closed solvable subgroups of G, ordered by inclusion. This set has maximal elements, for reasons of dimension. These maximal elements are the Borel subgroups of G. For example, if G = GLn(K) it follows from (7.3) that the Borel subgroups are precisely the conjugates of Bn, and therefore are all conjugate in G. We shall now show that this is so in general. (7.4) Let G be a connected linear algebraic group and let B be a Borel
subgroup of G. Then (i) GI B is a projective variety. (ii) All Borel subgroups of G are conjugate to B. Proof. Choose a Borel subgroup Bo of largest possible dimension. By (6.5) there exists a K-vector space of dimension (say) n, a line Vt in V and a linear action of G on V such that Bo is the stabiliser of V1• Thus Bo acts on V IV1, and by (7.3) there is a flag in V IVt. say 0 < V21V1 < · · · < V IV., stabilised by B0• It follows that the stabiliser of the flag
f : 0 < Vt <
V2 < · · · < Vn = V
is precisely B0 , and hence we have a bijective morphism of GI Bo onto the orbit Gf. Iff' is any other flag in V, its isotropy group is solvable, hence has
7 Borel subgroups and maximal tori
179
dimension ~ dim Bo, and its orbit Gf' has dimension ;;?; dim G -dim Bo = dim Gf. Hence Gf is an orbit of minimum dimension in the flag variety F(V), hence by (6.1) is closed in F(V) and therefore projective. By (7.2) it follows that GIBo is complete; since it is also quasi-projective (6.11) it follows that GI Bo is a projective variety. Now let B be any Borel subgroup of G. It acts on GIBo by left multiplication. By the fixed point theorem (7.1) there exists x e G such that BxBo = xBo, whence B c: xBoX- 1• But xBoX- 1 is a Borel subgroup (since Bo is), hence B = xB0 x- 1• (7.5) Let P be a closed subgroup of G. Then GIP is a projective variety
if and only if P contains a Borel subgroup. Proof. Suppose G I P is projective, and let B be a Borel subgroup of G. Then B acts on GIP by left multiplication, and by (7.1) this action has a fixed point, i.e. there exists x e G such that BxP = xP and hence Bx c: xP, i.e. P :::::> x- 1Bx. Conversely, if P :::::> B there is a surjective morphism GI B - GI P. By (3.2)(iii), GI P is complete and therefore projective.
Subgroups P of G satisfying the equivalent conditions of (7.5) are called parabolic subgroups of G. (7.6) Example. Let G = GLn(K), acting on V = Kn as usual, and let v = (nt, ... ,nr) be any sequence of positive integers such that n1 +... +nr = n. A flag of type v in V is a sequence
f : 0 = Uo < U1 < ... < Ur = V of subspaces of V such that dim Ui-dim Ui-1 = ni (1 ~ i ~ r). The group G acts transitively on the set Fv(V) of flags of type v, and the stabiliser of any flag of type v is a parabolic subgroup Pv of G. If v = (1, ... , 1), Pv is a Borel subgroup, at the other extreme, if v = (n), then Pv = G. There are 2n- 1 choices for v, and correspondingly 2•-• conjugacy classes of parabolic subgroups in GLn(K). Maximal tori From here on, for lack of space and time, our account will become increasingly sketchy. An (algebraic) torus is a linear algebraic group isomorphic to CG~ = CGm x ... x CGm (n factors) for some n ;;?; 1; or, equivalently, to the group Dn(K) of diagonal matrices. (Warning: in the theory of Lie groups, a torus is a compact abelian group isomorphic to a product of copies of the circle S 1 : not the same thing at all.)
180
Ill Linear Algebraic Groups
A torus is connected and abelian, hence solvable, and all its elements are semisimple. Conversely (but not obviously) if G is a connected abelian linear algebraic group all of whose elements are semisimple, then G is a torus. Let now G be any linear algebraic group. Just as in the case of solvable subgroups, the set of closed tori in G, ordered by inclusion, has maximal elements, for reasons of dimension. These are the maximal tori of G. We want to show that they are all conjugate in G. Since they all lie in the identity component Go of G, we may as well assume that G is connected. Each maximal torus is connected and solvable, and hence is contained in some Borel subgroup B of G. Hence it is enough to show that all maximal tori in B are conjugate in B, i.e. we reduce to the situation where G is solvable (and connected). We need now to consider the structure of connected solvable groups. (7.7) Example. Let G = Bn, the group of n x n upper triangular matrices. This is the semidirect product Dnl><. Un. where Dn is the group of diagonal matrices, and Un consists of the upper triangular matrices with l's down the diagonal. Thus Un = Gu, the set of unipotent elements of G, and therefore Gu is a connected closed normal subgroup of G. Moreover, Dn is a maximal torus in G. For if Tis a closed torus in G, then TnUn = {ln}. because each element of T is semisimple and each element of Un is unipotent, hence (since Un is normal) TUn is a constructible subgroup of G, hence closed, and T = TUn/Un has dimension :s;; dim G- dim Un = dimDn. This example is typical. The following theorem (which I shall not prove here) elucidates the structure of connected solvable group: (7.8) Let G be a connected solvable linear algebraic group. Then (i) Gu is a connected closed normal subgroup of G. (ii) The maximal tori in G are all conjugate, and if T is one then G is the semidirect product of T and Gu.
(By (7.3) we may assume that G is a closed subgroup of Bn, whence Gu = G nUn, which is a closed normal subgroup of G since by (7.7) Un is a closed normal subgroup of Bn. But it is less obvious that Gu is connected, and the proof of (ii) is quite complicated: there is apparently no nice fixed-point theorem as in the case of Borel subgroups ((7.4) above).)
7 Borel subgroups and maximal tori
181
From (7.8) and the conjugacy of Borel subgroups (7.4) it follows immediately that (7.9) Let G be any linear algebraic group. Then (i) The maximal tori in G are all conjugate.
(ii) The maximal connected unipotent subgroups of G are all conjugate.
(In (ii), a maximal connected unipotent subgroup U of G is contained in a Borel subgroup B, hence is equal to Bu.)
8 The root structure of a linear algebraic group
Characters and one-parameter subgroups of tori A character of a torus T is a homomorphism of algebraic groups X: T-+ Cl1m.
If t e T we shall write tX rather than x(t) for the image oft under X· If x. 1p are characters of T then so are x + 1p and -x, defined by tX+¥> = tXt¥' and t-x = (tX)- 1. Hence the set X(T) of characters of T is in a natural way an abelian group. Dually, a one-parameter subgroup of T (by abuse of language) is a homomorphism of algebraic groups '1: CIJm-+ T. As above, we shall write Y!' in place of 'I(X) (x e CIJm). The set Y(T) of one-parameter subgroups ofT is again an abelian group: x"+C = x".xC, x-" = (x")- 1• The structure of these groups X(T), Y(T) is easily described. Consider first the case T = Cl1m. It is easily verified that the characters of Cl1m are just the mappings x 1--+ x', r E Z. If now T is a torus of dimension d, say T =~.then X(T) ~ X(CIJm)d ~ zd and Y(T) ~ Y(CIJm)d ~ zd. More precisely, for each v = (nt, ... ,nd) E zd define Xv E X(T) and 'lv E Y(T) by Xv(x., ... ,Xd) = x~• ... xd" and 'lv(x) = (x"•, ... ,x""). Then v 1--+ Xv and v . . . . . 'lv are isomorphisms of zd onto X(T) and Y(T) respectively. Finally we have a pairing X(T) x Y(T)-+ Z,
say
(x,,)~--+
(x,'f), defined as follows:
(x")x = x<M>. 182
8 The root structure of a linear algebraic group
183
This pairing is linear in each variable, and puts the two lattices (=free abelian groups) X(T), Y(T) in duality. In the notation introduced above d
if p = (m., ... ,lnd) and v = (n., ... ,nd) we have
(x.,.,,.) = Em;n;. I
The root system R(G, T). Let G be a linear algebraic group, T a maximal torus in G. Recall from §5 the adjoint representation Ado : G- GL(g), where 9 is the Lie algebra of G. Since AdG is a homomorphism of algebraic groups, Ado(T) consists of commuting semisimple elements, hence is diagonalizable: this means that, relative to the action of T, g decomposes as a direct sum
g=
EB n ueX(T) :>«
where for each character !X e X(T), 911 = {X E g : Ado(t)X = !X(t)X,
all t E T}.
The nonzero !X e X(T) such that 9« =I= 0 are called the roots of G relative to T, and the set of roots is denoted by R( G, T): it is a finite subset of X(T). Let ZG(T) and NG(T) denote respectively the centralizer and normalizer of T in G. Then in fact ZG(T) is the identity component of NG(T) (this follows from the so-called "rigidity" of tori: a torus has no connected set of automorphisms consisting of more than one element). Hence W = W(G, T) = Na(T)/Za(T) is a finite group, which acts on T, hence also on X(T), and permutes the roots !X e R(G, T). Specifically, if w e W is the image of n e NG(T) then (w!X)(t) = !X(n- 1 tn) forte T. W is the Weyl group of R(G, T). The root datum Bl(G, T) The notion of a root datum is a fancier version of that of a root system, used for example to classify semisimple Lie algebras. Roughly speaking, it consists of a root system R embedded in a lattice X (but R need not span X) together with the dual root system Rv (obtained by reversing the arrows in the Dynkin diagram of R) embedded in the dual lattice Y = Hom(X,Z). The formal definition runs as follows. A root datum 91 = (X, Y, R, Rv) consists of
184
III Linear Algebraic Groups
(1) Lattices (i.e., free abelian groups) X and Y and a bilinear mapping X x Y - Z, written (e,,) .... (e,,), inducing isomorphisms X ....
Hom(Y,Z)
andY~--+
Hom(X,Z).
(2) finite subsets R c X and Rv c Y, and a bijection oc .... ocv of R onto
Rv. For each oc e R define s« : X - X by
s«e = e-(e. ocv)oc and
S11v
:
Y - Y by
Then the axioms are (RDl) (RD2)
{oc, ocv) = 2, s«R c R, SlllvRV c
RV
for all oc e R. Let Q be the subgroup of X generated by R and let V = Q ® R. If R
:/= 0 then R is a root system in V in the usual sense, and Rv is the dual
root system (in the dual vector space V"). The root datum 91 is reduced if (RD3) Starting with a linear algebraic group G and a maximal torus T in G (which, as we have seen, is unique up to conjugacy) we have already constructed three of the four ingredients X, Y, R, Rv: X is X(T), Y is Y(T) and R is R(G, T) as defined above. It remains to construct Rv, and one way of doing this is as follows. Let oc e R and let T11 = (Keroc)o, the connected component of the identity in the kernel of oc; then T11 is a subtorus ofT of codimension 1. Let G« be the centralizer ofT« in G; then it turns out that G11 is connected, and clearly T is a maximal torus of G11• The Weyl group W(G11 , T) in fact has order 2, and embeds in W(G, T). Let s11 E W(G, T) be the non-identity element of W(G«, T); then s11 acts on X(T) as follows: s«(X) = X- {x, ocv)oc for a unique ocv E Y(T). Since s~ = 1 we have (oc,ocv) = 2. Then Rv = Rv(G, T) is by definition the set of these ocv, for all oc e R. In this way we can associate with any linear algebraic group G and a maximal torus T in G a root datum 91(G, T), which depends up to
8 The root structure of a linear algebraic group
185
isomorphism only on G, since different choices ofT are conjugate in G, and hence may be denoted simply by ai(G). It can be shown that ai(G) is reduced, i.e. satisfies (RD3).
Exmnple. Let G = GLn(K), T
= Dn(K). Then we may identify X(T) and Y ( T) with zn, the pairing being the usual scalar product on zn. The roots oc E R(G, T) are the characters t 1-+ t;tj 1 (i-=/= j), where t = diag(tt, ... , tn) (because (Adt)x = txt- 1 for x e G). Hence if e,, ... ,En is the standard basis of we have R(G, T) = {e; -ei : i-=/= j}. If we follow through the construction of Rv c: Y (T) given above, we find that with oc as above we have ocv = Ej.
zn,
e,-
We recall that a linear algebraic group G is said to be reductive if Ru(G) = {e}, that is to say if G contains no connected closed unipotent normal subgroup other than {e}. We shall conclude with the statement of the following existence and uniqueness theorem for connected reductive groups over an algebraically closed field K : (8.1) (Uniqueness) Let G1, G2 be connected reductive linear algebraic groups over K. Then G., G2 are isomorphic as algebraic groups if and only if ai(G1) ~ ai(G2). (Existence) Let al be a reduced root datum. Then there exists a connected reductive linear algebraic group G over K such that al( G) ~ al.
Notes and references
§1. The background algebraic geometry is developed ab initio in [S], Chapter 1 and (modulo some commutative algebra) in [H], Chapter I and [B], Chapter AG. These references contain proofs of all the propositions of §1. §2. See [B], Chapter I; [H], Chapter II and Chapter VI (for the Jordan decomposition); [S] Chapter 2. §3. See [B], Chapter AG; [H], Chapter I; [S] Chapter 1.
§4. See [B], Chapter AG §§16, 17; [H), Chapter I, §S; [S], Chapter 3. §S. See [B), Chapter I, §3; [H], Chapter III; [S], Chapter 3. §6. See [B), Chapter II; [H], Chapter IV; [S], Chapter 5. §7. See [B), Chapter IV: §11; [H], Chapter VIII, §21; [S], Chapters 6 and 7. §8. The proof of the existence and uniqueness theorem is given in full in [S], Chapter 11 and 12.
186
Bibliography
J.F. Adams. Lectures on Lie groups. Benjamin, 1969. Also avaliable as Midway reprint, University of Chicago, 1982. A.
Bore~ Linear Algebraic Groups. Math. Lecture Note Series, W.A. Benjamin, Inc. New York, 1969.
N. Bourbaki, Groups et algebras de Lie IV. V. VI, Masson, Paris, 1981. E. Cartan, Oeuvres Compleres, Gauthier-Villars, Paris, 1952. R.W. Carter, Simple groups and simple Lie algebras, J. London Math. Soc. 40, 193-240, 1965. R.W. Carter, Finite groups of Lie type, conjugacy classes and complex characters, Wiley Classics Library Edition, J. Wiley, New York, 1989. R.W. Carter, Simple Groups of Lie Type. Wiley Classics Library Edition, J. Wiley, New York, 1989. C. Chevalley, Theory of Lie groups. Princeton University Press, 1946. C. Chevalley, Sur certains groups simples, Tohuku Math. J. 7, 14-66, 1955. I.M. Gelfand, M.I. Graev, N.Ya. Vilenkin, Generalized functions. Academic Press, 1966. P. Griffiths and J. Harris, Principles of algebraic geometry. Wiley, 1978. S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Academic Press, 1978. G. Hochschild, The structure of Lie groups. Holden-Day, 1965. J.E. Humphreys, Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.
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Bibliography
J.E. Humphreys, Unear Algebraic Groups. Graduate Texts in Mathematics, Springer-Verlag, New York, 1975. N. Jacobson, Ue Algebras. Interscience Publishers, J. Wiley, New York, 1962. W. Killing, Die Zusammensetzung der stetigen endlichen Transformationsgruppen I-IV, Math. Ann. 31, 252-290, 1988; 33, 1-48, 1889; 34, 57-122, 1889; 36. 161-189, 1890. A. Kirillov, Elements of the theory of representations. Springer-Verlag, Berlin, 1976. A.W. Knapp, Representation theory of semisimple groups. Princeton University Press, 1986. J. Milnor [l], Morse theory. Ann. of Math. Studies 51, Princeton University Press, 1963. J. Milnor [2], Introduction to algebraic K-theory. Ann. of Math. Studies 72, Princeton University Press, 1971. J. Milnor [3], Remarks on i'!finite-dimensional Ue groups. In Proc. of Summer School on Quantum Gravity, ed. B. DeWitt, Les Houches, 1983. J. Milnor and J. Stasheff, Characteristic classes. Ann. of Math. Studies 76, Princeton University Press, 1974. D. Montgomery and L. Zippin, Topological transformation groups. Interscience, 1955. A. Pressley and G.
Sega~
Loop Groups. Oxford University Press, 1986.
R. Ree. A family of simple groups associated with the simple Lie algebra of type (F4 ), Amer. J. Math. 83,401-420, 1961. R. Ree. A family of simple groups associated with the simple Lie algebra of type (G 2), Amer. J. Math. 83,432-462, 1961. J.-P. Serre, Ue algebras and Lie groups. Benjamin, 1965. T.A. Springer, Linear Algebraic Groups. Birkhiiuser, Boston, 1981 R. Steinberg, Variations on a theme of Chevalley, Pacific J. Math. 9, 875-891, 1959. M. Suzuki, On a class of doubly transitive groups, Ann. of Math. 75, 105-145, 1962. J. Tits, Algebra alternatives, algebras de Jordan et alegbras de Lie exceptionnelles, Indag. Math. lB. 223-237, 1966.
Index
additive group. 147 adjoint representation, 170 affille algebra, 142 affiDe algebraic variety, 142 algebraic set, 139 algebraic torus, ISS anti-self-dual, SS adas, 69
Dirac's spanner, 76 discrete series, 12S dominant, 143 dominant weight, 117 dual numbers, 162 Dynkin diagram, 18 Euclidean geometry, 49 exponential map, 73
Borel subgroups, 178 Borei-Weil theorem, 116 braid group. 113 Bruhat decomposition, 66, 119 Campbeii-Baker-Hausdorft' series, 75 Cartaa decomposition, 14 Cartan matrix, 17 Cartan subalgebra, 12 Cartaa subgroups, 127 Cayley parametrization, 70 celestial sphere, 56 characters, 95, 182 charts,69 Cbevalley basis, 37 Chevalley group. 38 classification of finite dimensional irreducible g-modules, 28 classification or finite dimensional simple Lie algebras, 22 classification of the finite simple groups, 44 Clifford algebra, 132 complementary series, 127 complete, 160 complex structures, 61, 119 connected, 148 constructible, 145 coordinate ring, 142 differential, 168 dimension, 145
ftag, 175 ftag of type v, 179 flag variety, 176 flag-manifold, 66 function field, 142, 1S9 fundamental representations, 32 G-morpbism, 172 G-space, 172 general linear group, 137 Gram-Schmidt process, 63 Grassmanniaa, 59, 71, 118 Grassmanniaa, isotropic, 61 Heisenberg group, SO, 83, 128, 131 highest weight vector, liS Hilbert's fifth problem, 72 holomorphically induced representation, 106,116 homogeneous coordinates, 70 homogeneous space for G, 172 homogeneous space, functions oa, 97 homogeneous spaces, 59 homomorphism or algebraic groups, 146 induced representations, lOS inner automorphism, 170 irreducible, 141 irreducible components, 141 irreducible representation, 84 isomorphism, 143
189
190
Index
isotropy group, 172 isotypical part, 91 Jacobi identity, 5, 75 Jordan decomposition, 151 K-finite vectors, 120 Killing form, lS, 58 Laplacian, 101 lattices, 59 left-invariant integral, 85 linear algebraic group, 146 locally closed, 144 Lorentz group, 54 MiSbius transformation, 56 manifold, 69 matrix group, 92 maximal compact subgroup, 63, 120 maximal torus, 67, 180 metaplectic group, 130 metaplectic representation, 130 module for a Lie algebra, 7 MONSTER,44 morphism, 143 multiplicative group, 147 nilpotent, lSI nilpotent Lie algebra, 8 Noetherian topological space, 141 one-parameter subgroup, 73, 182 orbit, 172 orders of finite Chevalley group, 40 orders of finite twisted groups, 42 orders of the finite Suzuki and Ree groups, 44 orthogonal group, 72, 137 orthogonality of characters, 96 oscillator representation, 129 parabolic subgroups, 117, 179 Feter-Weyl theorem, 91 PlUcker embedding, 118 Plancherel theorem, 122 polar decomposition, 63 prevariety, 158 principal series, 125 projective, 160 projective space, 70, lOS, 117, 159
reduced echelon fonn, 64 reductive, 154 Ree groups, 43 regular, 142 representation of a Lie algebra, 7 representative function, 92 Riemann sphere, SS ringed space, I 57 root datum, 183 root system, 14, 183 Schur's lemma, 84 self-dual, SS semiumple, 151, 153, 154 semiumple group, 58 semisimple part, I 52 separability, 164 Siegel generalized upper half-plane, 62 simply connected covering group, 58, 76 singular locus, 164 smooth, 164 soluble Lie algebra, 9 solvable, 154 special linear group, 137 special orthogonal group, 137 spherical harmonics, 100, lOS spin representation, 119, 132 structure sheaf, 157 Suzuki groups, 43 symmetric space, 61 symplectic group, 137 tangent bundle, 162, 163 tangent space, 72 tangent vector, 162 tensors, 110 torus, 179 twisted groups, 41 unipotent, 151, 153, 154 unipotent part, 152 unipotent radical, 154 unitary group, 73 unitary representation, 84 universal enveloping algebra, 25 upper half-plane, 59, 125 variety, 159, 160 vector bundles, lOS Verma module, 27 Verma modules, dual, 122
quantum groups, 113 quasi-projective, 160 quaternions, 53 quotient of G by H, 173
weight vectors, liS Weyl denominator formula, 88 Weyl group, 16, 66 Weyl's character formula, 30 Weyl's dimension formula, 31
radical, 154 Radon transform, 103
Zariski topology, 140
