CEJM 3(2) 2005 155–182
Miura opers and critical points of master functions Evgeny Mukhin1∗† , Alexander Varchenko2‡ 1
Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, 402 North Blackford St., Indianapolis, IN 46202-3216, USA 2 Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
Received 2 March 2004; accepted 13 January 2005 Abstract: Critical points of a master function associated to a simple Lie algebra g come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra t g . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY = 0 can be written explicitly in terms of critical points composing the population. c Central European Science Journals. All rights reserved.
Keywords: Bethe Ansatz, Miura opers, flag varieties MSC (2000): 82B23, 17B67, 14M15
1
Introduction
In [18] rational functions were considered which are products of powers of linear functions. It was discovered that under certain conditions all critical points of the rational functions are non-isolated and form non-trivial varieties. It is not clear yet how general that phenomenon is but the phenomenon certainly holds for products of powers of linear functions appearing in representation theory. Those products are called the master functions. Let h be the Cartan subalgebra of a simple Lie algebra g ; ( , ) the Killing form on h ∗ ; α1 , . . . , αr ∈ h ∗ simple roots; Λ1 , . . . , Λn ∈ h ∗ dominant integral weights; l1 , . . . , lr non∗ † ‡
E-mail:
[email protected] Supported in part by NSF grant DMS-0140460 Supported in part by NSF grant DMS-0244579
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negative integers; z1 , . . . , zn distinct complex numbers. The master function Φ associated with this data is given by formula (1). It is a rational function of l1 + · · · + lr variables (1) (1) (2) (r) t1 , . . . , tl1 , t1 , . . . , tlr , and n variables z1 , . . . , zn . We may think that l1 + · · · + lr + n (i) particles are given in C. The particle tj has weight −αi and the particle zs has weight Λs . The particles interact pairwise. The interaction of particles x and y with weights v and w, respectively, is given by (x − y)(v,w) . Then total interaction is the product of such terms over the set of all pairs. The master function describes the interaction of t-particles and z-particles. The master function appears in hypergeometric solutions to the KZ equations with values in the tensor product of irreducible highest weight representations LΛ1 , . . . , LΛn with highest weights Λ1 , . . . , Λn , respectively. The solutions have the form Z u(z) = Φ(t; z)1/κ A(t; z) dt ,
where κ is the parameter of the KZ equations and A(t; z) is some explicitly written rational function with values in the tensor product [19]. The master function also appears in the Bethe ansatz of the Gaudin model with values in the same tensor product [17]. In that case the value of the function A( · ; z) at a point t is an eigenvector of the commuting Gaudin Hamiltonians if t is a critical point of the master function. In this paper we study critical points of the master function on the set where all (i) (i) {tj , zs } are distinct. In other words we study those positions of distinct particles {tj } in the complement to {zs } which extremize the master function. Critical points of master functions associated to a simple Lie algebra g come in families called populations [18, 11]. In this paper we prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra t g . The proof is based on the correspondence between critical points and differential operators called the Miura opers. To every critical point t one assigns a certain linear differential operator Dt with coefficients in t g , called the Miura oper. The differential operators of that type were considered by V. Drinfeld and V. Sokolov in their study of the KdV type equations [5]. On opers and Miura opers see [1, 6, 7, 8, 11, 4, 14]. Different critical points correspond to different Miura opers. The Miura opers corresponding to critical points of a given population form an equivalence class with respect to suitable gauge equivalence. We show that the equivalence class of Miura opers is isomorphic to the flag variety of t g . In [11, 4] we considered Miura opers for Lie algebras of types Ar , Br , Cr , G2 and using the opers proved that a population of critical points of types Ar , Br , Cr , G2 is isomorphic to the corresponding flag variety. The proof, suggested in the present paper, is more direct and works for any simple Lie algebra. If Dt is the Miura oper corresponding to a critical point t, then the set of solutions of the differential equation Dt Y = 0 with values in a suitable space is an important characteristics of the critical point. We used that characteristics for slr+1 Miura opers in [11] to give a bound from above for the number of populations of critical points of the
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corresponding slr+1 master function. That statement in [11] was in some sense opposite to the Bethe ansatz conjectures, see [11]. It turns out that for any simple Lie algebra g and any critical point t of a g master function all solutions of the differential equation Dt Y = 0 can be written explicitly in terms of critical points composing the population originated at t. Thus the population of critical points “solves” the Miura differential equation Dt Y = 0. This is the second main result of the paper. When this paper was being written preprint [8] by E. Frenkel appeared. The preprint is devoted to the same fact that the variety of gauge equivalent t g Miura opers is isomorphic to the flag variety of t g . One of the main claims of [8] is Corollary 3.3. In our opinion the proofs leading to Corollary 3.3 in [8] are sometimes incomplete. Moreover in our paper [16] we construct a counterexample to the statement of Corollary 3.3. The idea of this paper was originated in discussions with E. Frenkel in the spring of 2002. As a result of those discussions two papers appeared: this one (see its preprint version in [15]) and [8]. We thank E. Frenkel for stimulating meetings which originated this paper. We thank P. Belkale and S. Kumar for numerous useful discussions. The paper is organized as follows. In Section 2 we introduce populations of critical points. In Section 3 we discuss elementary properties of Miura opers corresponding to critical points. In Section 4 we prove that the variety of gauge equivalent Miura opers is isomorphic to the flag variety, see Theorem 4.3. We discuss the relations between the Bruhat cell decomposition of the flag variety and populations of critical points in Section 5. The main result there is Corollary 5.4 describing the structure of connected components of the critical set of master functions. In Section 6 we give explicit formulas for solutions of the differential equation Dt Y = 0, see Theorems 6.5, 6.7, 6.8.
2
Master functions and critical points, [11]
2.1 Kac-Moody algebras Let A = (ai,j )ri,j=1 be a generalized Cartan matrix, ai,i = 2, ai,j = 0 if and only aj,i = 0, ai,j ∈ Z≤0 if i 6= j. We assume that A is symmetrizable, i.e. there exists a diagonal matrix D = diag{d1 , . . . , dr } with positive integers di such that B = DA is symmetric. Let g = g (A) be the corresponding complex Kac-Moody Lie algebra (see [10], §1.2), h ⊂ g the Cartan subalgebra. The associated scalar product is non-degenerate on h ∗ and dim h = r + 2d, where d is the dimension of the kernel of the Cartan matrix A. Let αi ∈ h ∗ , αi∨ ∈ h , i = 1, . . . , r, be the sets of simple roots, coroots, respectively. We have (αi , αj ) = di ai,j , hλ, αi∨i = 2(λ, αi )/(αi , αi ),
λ ∈ h ∗.
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In particular, hαj , αi∨ i = ai,j . Let P = {λ ∈ h ∗ | hλ, αi∨i ∈ Z} and P + = {λ ∈ h ∗ | hλ, αi∨i ∈ Z≥0 } be the sets of integral and dominant integral weights. Fix ρ ∈ h ∗ such that hρ, αi∨i = 1, i = 1, . . . , r. We have (ρ, αi ) = (αi , αi )/2. The Weyl group W ∈ End(h * ) is generated by reflections si , i = 1, . . . , r, si (λ) = λ − hλ, αi∨ iαi ,
λ ∈ h ∗.
We use the notation w ∈ W, λ ∈ h ∗ ,
w · λ = w(λ + ρ) − ρ,
for the shifted action of the Weyl group. The Kac-Moody algebra g (A) is generated by h , e1 , . . . , er , f1 , . . . , fr with defining relations [ei , fj ] = δi,j αi∨ , ′
[h, h ] = 0,
i, j = 1, . . . r, ′
h, h ∈ h ,
[h, ei ] = hαi, hi ei , [h, fi ] = −hαi , hi fi,
h ∈ h , i = 1, . . . r, h ∈ h , i = 1, . . . r,
and the Serre’s relations (ad ei )1−ai,j ej = 0,
(ad fi )1−ai,j fj = 0,
for all i 6= j. The generators h , e1 , . . . , er , f1 , . . . , fr are called the Chevalley generators. Denote n+ (resp. n− ) the subalgebra generated by e1 , . . . , er (resp. f1 , . . . , fr ). Then g = n+ ⊕ h ⊕ n− . Set b± = h ⊕ n± . Let g = ⊕j g j be the canonical grading of g . Here we have ei ∈ g 1 , fi ∈ g −1 , n+ = ⊕j>0 g j , h = g 0 , n− = ⊕j<0 g j . For a vector space X we denote M(X) the space of X-valued rational functions on C. ¯ + ) the completion of the space M(n+ ) with respect to the canonical We denote M(n P j ¯ (n+ ) is a formal sum grading. An element of M j>0 uj , where uj : C → g are rational functions. The Kac-Moody algebra t g = g (t A) corresponding to the transposed Cartan matrix t A is called Langlands dual to g . Let t αi ∈ t h ∗ , t αi∨ ∈ t h , i = 1, . . . , r, be the sets of simple roots, coroots of t g , respectively. Then ht αi , t αj∨ i = hαj , αi∨ i = ai,j for all i, j.
2.2 The definition of master functions and critical points We fix a Kac-Moody algebra g = g (A), a non-negative integer n, a collection of dominant integral weights Λ = (Λ1 , . . . , Λn ), Λi ∈ P + , and points z = {z1 , . . . , zn } ⊂ C. We
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assume that zi 6= zj if i 6= j. We often do not stress the dependence of our objects on these parameters. In addition we choose a collection of non-negative integers l = (l1 , . . . , lr ) ∈ Zr≥0 . The choice of l is equivalent to the choice of the weight Λ∞ =
n X
Λi −
i=1
r X
lj αj ∈ P.
j=1
The weight Λ∞ will be called the weight at infinity. The master function Φ(t; Λ∞ ) is defined by Y Φ(t; Λ∞ ) = Φ(t; z, Λ, Λ∞ ) = (zs − zu )(Λs ,Λu ) ×
(1)
1≤s
li Y r Y n Y
(i) (tj
− zs )
−(Λs ,αi )
i=1 j=1 s=1
r Y
Y
(i) (tj
−
(αi ,αi ) t(i) s )
lj li Y Y Y
(j)
(αi ,αj ) (t(i) , s − tk )
1≤i<j≤r s=1 k=1
i=1 1≤j<s≤li
(i)
see [19]. The function Φ is a function of variables t = (tj ), where i = 1, . . . , r, and j = 1, . . . , li , of variables z = (z1 , . . . , zn ), weights Λ, and integers l. The main variables are t, the other variables will be considered as parameters. The function Φ is symmetric with respect to permutations of variables with the same upper index. A point t with complex coordinates is called a critical point if the following system of algebraic equations is satisfied −
n X (Λs , αi ) (i)
s=1
tj − zs
+
ls X X (αs , αi ) (i)
s, s6=i k=1
(s)
tj − tk
+
X (αi , αi ) (i)
s, s6=j
(i)
tj − ts
= 0,
(2)
where i = 1, . . . , r and j = 1, . . . , li . In other words, the point t is a critical point if ! ∂Φ Φ−1 (i) (t) = 0, i = 1, . . . , r, j = 1, . . . , li . ∂tj Note that the product of symmetric groups Sl = Sl1 × · · · × Slr acts on the critical set of the master function permuting the coordinates with the same upper index. All orbits have the same cardinality l1 ! · · · lr ! . We do not make distinction between critical points in the same orbit.
2.3
Polynomials representing critical points
Let t be a critical point of the master function Φ = Φ(t; Λ∞ ). Introduce an r-tuple of polynomials y = (y1 (x), . . . , yr (x)), li Y (i) yi(x) = (x − tj ). j=1
(3)
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Each polynomial is considered up to multiplication by a non-zero number. The tuple defines a point in the direct product P (C[x])r of r copies of the projective space associated with the vector space of polynomials in x. We say that the tuple y represents the critical point. It is convenient to think that the tuple (1, . . . , 1) of constant polynomials represents in P (C[x])r the critical point of the master function with no variables. This corresponds P to the case when l = (0, . . . , 0) and Λ∞ = ns=1 Λs . Introduce polynomials Ti (x) =
n Y
∨
(x − zs )hΛs ,αi i ,
i = 1, . . . , r.
(4)
s=1
We say that a given tuple y is generic with respect to weights Λ1 , . . . , Λn and points z1 , . . . , zn if • each polynomial yi (x) has no multiple roots; • all roots of yi (x) are different from roots of the polynomial Ti ; • any two polynomials yi(x), yj (x) have no common roots if i 6= j and ai,j 6= 0. A tuple is generic if it represents a critical point. Let W (f, g) = f ′ g − f g ′ be the Wronskian of functions f, g of x. A tuple y is called fertile, if for every i = 1, . . . , r there exists a polynomial y˜i satisfying the equation Y −a Y −hαj ,α∨ i i (5) = Ti yj i,j . W (yi , y˜i) = Ti yj j, j6=i
j, j6=i
The polynomial y˜i considered up to multiplication by a non-zero number has the form y˜i (x) = c1 yi(x)
Z
Ti (x)
r Y
yj (x)−ai,j dx + c2 yi (x) ,
(6)
j=1
where c1 , c2 are complex numbers, c1 6= 0. If y is fertile and i ∈ {1, . . . , r}, then the tuple y [i] = (y1 , . . . , y˜i , . . . , yr )
∈
P (C[x])r
(7)
is called the immediate descendant of y in the i-th direction. Theorem 2.1 ([11]). (i) A generic tuple y = (y1 , . . . , yr ), with deg yj = lj , represents a critical point of the P P master function Φ(t; Λ∞ ), with Λ∞ = ns=1 Λs − rj=1 lj αj , if and only if it is fertile. (ii) If y represents a critical point, i ∈ {1, . . . , r}, and y [i] is an immediate descendant of y, then y [i] is fertile. Let y represent a critical point of Φ(t; Λ∞ ). Let i ∈ {1, . . . , r} and let y [i] be an P [i] immediate descendant of y in the i-th direction. Denote ˜li = deg y˜i and Λ∞ = ns=1 Λs −
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˜li αi − Pr j=1,
161
Assume that y [i] is generic, then y [i] represents a critical point of [i] the master function Φ(t; Λ∞ ). If ˜li 6= li , then j6=i lj αj .
Λ[i] ∞ = si · Λ ∞ , where si · is the shifted action of the i-th reflection of the Weyl group.
2.4 Simple reproduction procedure Let y represent a critical point of Φ(t, Λ∞ ). The tuples y [i] = (y1 , . . . , y˜i, . . . , yr ) ∈ P (C[x])r , where y˜i is given by (6) and c1 , c2 are arbitrary numbers, not both equal to zero, form a one-parameter family. The parameter space of the family is identified with the projective line P 1 with projective coordinates (c1 : c2 ). We have a map Yy,i : P 1 → P (C[x])r , which sends a point c = (c1 : c2 ) to the corresponding tuple y [i] . Almost all tuples y [i] are generic. The exceptions form a finite set in P 1 . Thus, starting with a tuple y, representing a critical point of the master function Φ(t; Λ∞ ), and an index i ∈ {1, . . . , r}, we construct a family Yy,i : P 1 → P (C[x])r of fertile tuples. For almost all c ∈ P 1 (with finitely many exceptions only), the tuple Yy,i (c) represents a critical point of the master function associated with integral dominant weights Λ1 , . . . , Λn , points z1 , . . . , zn , and a suitable weight at infinity. We call this construction the simple reproduction procedure in the i-th direction.
2.5 General reproduction procedure Assume that a tuple y ∈ P (C[x])r represents a critical point of the master function Φ(t; Λ∞ ). Let i = [i1 , i2 , . . . , ik ], ij ∈ {1, . . . , r}, be a sequence of natural numbers. We define a k-parameter family of fertile tuples Yy,i : (P 1 )k → P (C[x])r by induction on k, starting at y and successively applying the simple reproduction procedure in directions i1 , . . . , ik . The image of this map is denoted Py,i . For a given i = [i1 , . . . , ik ], almost all tuples Yy,i(c) represent critical points of master functions associated to weights Λ1 , . . . , Λn , points z1 , . . . , zn , and suitable weights at infinity. Exceptional values of c ∈ (P 1 )k are contained in a proper algebraic subset. It is easy to see that if i′ = [i′1 , i′2 , . . . , i′k′ ], ij ∈ {1, . . . , r}, is a sequence of natural numbers, and the sequence i′ is contained in the sequence i as an ordered subset, then Py,i′ is a subset of Py,i. The union Py = ∪i Py,i ⊂ P (C[x])r ,
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where the summation is over all of sequences i, is called the population of critical points associated with the Kac-Moody algebra g , integral dominant weights Λ1 , . . . , Λn , points z1 , . . . , zn , and originated at y. If two populations intersect, then they coincide. If the Weyl group is finite, then all tuples of a population consist of polynomials of bounded degree. Thus, if the Weyl group of g is finite, then a population is a projective irreducible variety. Every population P has a tuple y = (y1 , . . . , yr ), deg yi = li , such that the weight P P Λ∞ = ns=1 Λs − ri=1 li αi is dominant integral, see [11].
Conjecture 2.2 ([11]). Every population, associated with a Kac-Moody algebra g , dominant integral weights Λ1 , . . . , Λn , points z1 , . . . , zn , is an algebraic variety isomorphic to the flag variety associated to the Kac-Moody algebra t g which is Langlands dual to g . Moreover, the parts of the family corresponding to tuples of polynomials with fixed degrees are isomorphic to Bruhat cells of the flag variety. The conjecture is proved for the Lie algebras with root systems of types Ar , Br , Cr , G2 in [11, 4]. In Theorems 4.3 and 5.3 we prove this conjecture for any simple Lie algebra.
2.6 Diagonal sequences of polynomials associated with a critical point and a sequence of indices We introduce notions which will be used in Chapter 6 to construct solutions of differential equations. Lemma 2.3. Assume that a tuple y of non-zero polynomials represents a critical point of the master function Φ(t; Λ∞ ). Let i = [i1 , i2 , . . . , ik ], ij ∈ {1, . . . , r}, be a sequence of na[i ] [i ] [i ,i ] [i ,i ] tural numbers. Then there exist tuples y [i1 ] = (y1 1 , . . . , yr 1 ), y [i1 ,i2 ] = (y1 1 2 , . . . , yr 1 2 ), [i ,i ,...,i ] [i ,i ,...,i ] . . . , y [i1 ,i2 ,...,ik ] = (y1 1 2 k , . . . , yr 1 2 k ) in P (C[x])r such that (i) Y −ai ,j [i ] yj 1 W (yi1 , yi11 ) = Ti1 j, j6=i1
[i ]
and yj 1 = yj for j 6= i1 ; (ii) for l = 2, . . . , k, we have [i ,...,il−1 ]
W (yil 1
[i ,...,il ]
, yil 1
) = Til
Y
[i ,...,il−1 ] −ai ,j l
(yj 1
)
j, j6=il [i ,...,il ]
and yj 1
[i ,...,il−1 ]
= yj 1
for j 6= il .
The tuples y [i1 ] , y [i1 ,i2 ] , . . . , y [i1 ,i2 ,...,ik ] belong to the population Py . The tuple y [i1 ] is obtained from y by the i1 -th simple generation procedure and for l = 2, . . . , k, the tuple y [i1 ,...,il ] is obtained from y [i1 ,...,il−1 ] by the il -th simple generation procedure.
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The sequence of tuples y [i1 ] , y [i1 ,i2 ] , . . . , y [i1 ,i2 ,...,ik ] satisfying Lemma 2.3 will be called associated with the critical point y and the sequence of indices i. The sequence of po[i ] [i ,i ] [i ,i ,...,i ] lynomials yi11 , yi21 2 , . . . , yik1 2 k will be called the diagonal sequence of polynomials associated with the critical point y and the sequence of indices i. For a given y the diagonal sequence of polynomials determine the sequence of tuples y [i1 ] , y [i1 ,i2 ] , . . . , y [i1 ,i2 ,...,ik ] uniquely. There are many diagonal sequences of polynomials associated with a given critical point and a given sequence of indices.
3
Opers
Let g = g (A) be a Kac-Moody algebra with simple roots α1 , . . . , αr and simple coroots α1∨ , . . . , αr∨. Let t g = g (t A) be the Langlands dual algebra with Chevalley generators t h , E1 , . . . , Er , F1 , . . . , Fr , simple roots t α1 , . . . , t αr and simple coroots t α1∨ , . . . , t αr∨ . Set H1 = t α1∨ , . . . , Hr = t αr∨ and I = F1 + · · · + Fr ,
∂ = d/dx .
A t g -oper is a differential operator of the form D = ∂ + I + V + W ¯ (t n+ ). A Miura t g -oper is a differential operator of the form with V ∈ M(t h ) and W ∈ M D = ∂ + I + V with V ∈ M(t h ). The differential operators of that type were considered by V. Drinfeld and V. Sokolov in their study of the KdV type equations [5]. On opers and Miura opers see [1, 6, 7, 8, 11, 4, 14]. ¯ (t n+ ) and a t g -oper D, the differential operator For u ∈ M 1 ead u · D = D + [u, D] + [u, [u, D]] + . . . 2 is a t g -oper. The opers D and ead u · D are called gauge equivalent. Let X be a t g -module with locally finite action of t n+ . Let D be a t g -oper and ¯ (t n+ ). Then D, ead u · D, e± u determine linear operators on M(X). Moreover, we u∈M have ead u · D = eu D e−u . Lemma 3.1. Let D = ∂ + I + V be a Miura t g -oper. Let g ∈ M(C) and i ∈ {1, . . . , r}. Then ead (gEi ) · D = ∂ + I + (V + g Hi ) − (g ′ + ht αi , V ig + g 2 ) Ei .
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The proof is straightforward. Corollary 3.2. Let D = ∂ + I + V be a Miura t g -oper. Then the t g -oper ead (gEi ) · D is a Miura oper if and only if the scalar rational function g satisfies the Ricatti equation g ′ + ht αi , V ig + g 2 = 0 .
(8)
We say that the Miura t g -oper D is deformable in the i-th direction if equation (8) has a non-zero solution which is a rational function. Fix a collection of dominant integral weights Λ = (Λ1 , . . . , Λn ) of the Kac-Moody algebra g , and numbers z = {z1 , . . . , zn } ⊂ C, zi 6= zj if i 6= j. Introduce polynomials T1 (x), . . . , Tr (x) by formulas (4). Let y = (y1 , . . . , yr ) be a tuple of non-zero polynomials. We say that a Miura t g -oper D = ∂ + I + V is associated with weights Λ, numbers z, and the tuple y = (y1 , . . . , yr ), if for every i ∈ {1, . . . , r} we have ! ! r r Y Y ∨ −hα ,α i −a (9) yj i,j , = − log′ Ti yj j i h t αi , V i = − log′ Ti j=1
j=1
cf. (5). If a Miura oper D is associated with weights Λ, numbers z, and a generic tuple y = (y1 , . . . , yr ) ∈ P (C[x])r , then the tuple y is determined uniquely. Indeed, the residues of the rational function h t αi , V i are positive exactly at the roots of the polynomial yi and the residues are equal to the multiplicities of roots of yi multiplied by two. If g is a simple Lie algebra, then D determines y uniquely even if y is not generic. That fact follows from the invertibility of the Cartan matrix of g . Theorem 3.3. Let the Miura t g -oper D = ∂ + I + V be associated with weights Λ, numbers z, and the tuple y = (y1 , . . . , yr ). Let i ∈ {1, . . . , r}. Then D is deformable in the i-th direction if and only if there exists a polynomial y˜i satisfying (5). Moreover, in that case any non-zero rational solution g of the Ricatti equation (8) has the form g = log′ (˜ yi/yi ) where y˜i is a solution of (5). If g = log′ (˜ yi/yi ), then the Miura t g -oper ead (gEi ) · D = ∂ + I + (V + g Hi )
(10)
is associated with weights Λ, numbers z, and the tuple y [i] = (y1 , . . . , y˜i, . . . , yr ), where the tuple y [i] is called in Section 2.3 an immediate descendant of y in the i-th direction, see (7). Proof 3.4. Write (8) as g ′ /g + g = log′
Ti
r Y j=1
−a yj i,j
!
.
(11)
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If g is a rational function, then g → 0 as x → ∞ and all poles of g are simple. Moreover, the residue of g at any point is an integer. Hence g = c′ /c for a suitable rational function c. Then Z r Y c = Ti (x) yj (x)−ai,j dx (12) j=1
and equation (5) has a polynomial solution y˜i = −cyi . Conversely if equation (5) has a polynomial solution y˜i, then the function c in (12) is rational. Then g = c′ /c is a rational solution of equation (8). Let g = log′ (c) = log′ (˜ yi/yi ), where y˜i is a solution of (5). Then ead (gEi ) · D = ∂ + I + (V + log′ (˜ yi/yi ) Hi ) and h t αk , V i + log′ (˜ yi /yi) h t αk , t αi∨ i = − log′
−hαi ,α∨ ki
Tk y˜i
r Y
j=1, j6=i
−hαj ,α∨ ki
yj
!
.
Note that if equation (8) has one non-zero rational solution g = c′ /c with rational c, then other non-zero (rational) solutions have the form g = c′ /(c + const). Corollary 3.5. Let the Miura t g -oper D = ∂ + I + V be associated with weights Λ, numbers z, and the tuple y = (y1 , . . . , yr ). Then D is deformable in all directions from 1 to r if and only if the tuple y is fertile. Corollary 3.6. Let the Miura t g -oper D = ∂ + I + V be associated with weights Λ, numbers z, and the tuple y = (y1 , . . . , yr ). Let the tuple y = (y1 , . . . , yr ) be generic in the sense of Section 2.3. Then D is deformable in all directions from 1 to r if and only if the tuple y represents a critical point of the master function (1) associated with parameters z, Λ, Λ∞ . Let the Miura t g -oper D = ∂ + I + V be associated with weights Λ, numbers z, and the tuple y = (y1 , . . . , yr ). Let the tuple y = (y1 , . . . , yr ) represent a critical point of the master function (1) associated with parameters z, Λ, Λ∞ . Let OmD 0 be the variety of all Miura opers which can be obtained from D by a sequence of deformations in directions i1 , . . . , iN where N is any positive integer and all ij lie in {1, . . . , r}. Corollary 3.7. For a simple Lie algebra g the variety OmD 0 is isomorphic to the population of critical points originated at y.
4
Miura opers and flag varieties
In this section we assume that g and t g are simple Lie algebras although most of considerations can be extended to Kac-Moody algebras.
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Let t G be the complex simply connected Lie group with Lie algebra t g . Let t B± , t N± , t H be the subgroups with Lie algebras t b± , t n± , t h , respectively.
4.1 Triviality of the monodromy Let D = ∂ + I + V be a Miura t g -oper. Let P1 be the complex projective line. Consider D as a t G-connection ∇D on the trivial principal t G-bundle p : t G × P1 → P1 . The connection has singularities at the set Sing ⊂ P1 where the function V has poles. Choose a regular point x0 ∈ P1 − ∞ of the connection. Parallel translations with respect to the connection define the monodromy representation π(P1 − Sing) → t G. Its image is called the monodromy group. Theorem 4.1. Assume that the Miura t g -oper D is associated with weights Λ, numbers z, and a tuple y = (y1 , . . . , yr ) as in Section 3. Assume that the tuple y = (y1 , . . . , yr ) is generic in the sense of Section 2.3. Assume that the Miura oper D is deformable in all directions from 1 to r. Then the monodromy group of ∇D belongs to the center of t G. Proof 4.2. It is known that the intersection of all of the Borel subgroups in t G is the center of t G, see [3, 9]. We show that the monodromy of ∇D lies in the intersection of all of the Borel subgroups. Let id ∈ t G be the identity element. Let Y¯ (x) ∈ t G be the (possibly multi-valued) solution of the equation DY = (∂ + I + V )Y = 0 such that Y (x0 ) = id. Since D is a Miura oper we have for any regular x the equality of sets Y¯ (x) t B− = t B− . Hence if m ∈ t G is an element of the monodromy group of ∇D , then m t B− = t B− and hence m ∈ t B− . Let i ∈ {1, . . . , r} and let gi ∈ M(C) be a solution of the Ricatti equation (8). Assume that gi is regular at x0 . Then the t G-valued function egi (x)Ei Y¯ (x) e−gi (x0 )Ei is the solution of the equation (ead (gi (x)Ei ) · D) Y = 0 such that Y (x0 ) = id. Since [i] ead (gi (x)Ei ) · D is a Miura oper we have an equality of sets Ygi (x) t B− = t B− for any x at which ead (gi (x)Ei ) · D is regular. Hence for any element m of the monodromy group of ∇D we have egi (x0 )Ei m e−gi (x0 )Ei t B− ⊂ t B− or m ∈ e−gi (x0 )Ei t B− egi (x0 )Ei . Now consider the Miura oper ead (gi (x)Ei ) · D = ∂ + I + Vi;gi where Vi;gi is the t h -part of ead (gi (x)Ei ) · D. Let j ∈ {1, . . . , r} and let gi,j;gi ∈ M(C) be a solution of the j-th Ricatti equation g ′ + ht αi , Vi;gi ig + g 2 = 0
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associated with the Miura oper ead (gi (x)Ei ) · D. Assume that gi,j;gi is regular at x0 . Then the t G-valued function egi,j;gi (x)Ej egi (x)Ei Y¯ (x) e−gi (x0 )Ei e−gi,j;gi (x0 )Ej is the solution of the equation (ead (gi,j;gi (x)Ej ) ead (gi (x)Ei ) · D) Y = 0 such that Y (x0 ) = id. Repeating the previous argument we conclude that any element m of the monodromy group of ∇D lies in the Borel subgroup e−gi (x0 )Ei e−gi,j;gi (x0 )Ej t B− egi,j;gi (x0 )Ej egi (x0 )Ei . Every u ∈ t N+ is a product of elements of the form eci Ei for i ∈ {1, . . . , r} and ci ∈ C. Every ci can be taken as the initial condition for a solution of the suitable i-th Ricatti equation. Therefore the iteration of the previous reason shows that every element of the monodromy group of ∇D lies in every Borel subgroup of the form u−1 ( t B− )u, u ∈ t N+ . The Borel subgroups in t G of the form u−1 ( t B− )u, u ∈ t N+ , form an open dense subset in the flag variety of all of the Borel subgroups. Hence the monodromy lies in the intersection of all of the Borel subgroups.
4.2 Gauge equivalent Miura opers As in Section 4.1, let D be the Miura t G-oper associated with weights Λ, numbers z, and a tuple y = (y1 , . . . , yr ). We assume that the tuple y = (y1, . . . , yr ) is generic in the sense of Section 2.3 and the Miura oper D is deformable in all directions from 1 to r. Consider the variety OmD of all Miura opers gauge equivalent to D. If D ′ ∈ OmD , then there exists a rational N+ -valued function v on P1 such that D ′ = v D v −1 . In that case we denote D ′ by D v . Let OmD 0 ⊆ OmD be the subvariety of all Miura opers which can be obtained from D by a sequence of deformations in directions i1 , . . . , iN where N is a non-negative integer and all ij lie in {1, . . . , r}. By Corollary 3.7 the subvariety OmD 0 is isomorphic to the population of critical points originated at y. The connection ∇D is regular at x0 ∈ P1 if x0 does not lie in {z1 , . . . , zn , ∞} and x0 is not a root of some of polynomials y1 , . . . , yr . Consider the trivial bundle p′ : (t G/t B− ) × P1 → P1 associated with the bundle p. The fiber of p′ is the flag variety t G/t B− . The connection ∇D induces a connection ∇′D on p′ . The monodromy of ∇′D is trivial by Theorem 4.1. Thus the variety Γ of global horizontal sections of ∇′D is identified with the fiber (p′ )−1 (x0 ) over any x0 which is a regular point of the connection. Thus Γ is isomorphic to t G/t B− . Any t G-valued rational function v defines a section Sv : x 7→ v(x)−1 t B− × x
(13)
of p′ over the set of regular points of v. The section Sv is also well defined over the poles of v since t G/t B− is a projective variety.
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If D v ∈ OmD , then the section Sv is horizontal with respect to ∇′D , cf. the proof of Theorem 4.1. Thus we have a map S : OmD → Γ,
D v 7→ Sv .
Theorem 4.3. The map S : OmD → Γ is an isomorphism and OmD 0 = OmD . Proof 4.4. Let D v1 , D v2 ∈ OmD . Assume that the images of D v1 and D v2 under the map S coincide. Assume that v1 , v2 , D are regular at x0 ∈ P1 . The equality Sv1 (x0 ) = Sv2 (x0 ) means that v1 (x0 )−1 t B− = v2 (x0 )−1 t B− . Then v1 (x0 ) = v2 (x0 ). Hence v1 = v2 and D v1 = D v2 . That proves the injectivity of S. Let x0 be a regular point of D in P1 − ∞. For any u ∈ t N+ there exists a rational t N+ -valued function v such that v(x0 ) = u and D v ∈ OmD and D v is obtained from D by a sequence of deformations in some directions i1 , . . . , iN , see the proof of Theorem 4.1. Thus the set Im(x0 ) = {Sv (x0 ) ∈ (t G/t B− ) × x0 | D v ∈ OmD 0 } contains the set ((t N+ t B− )/t B− ) × x0 ⊂ (t G/t B− ) × x0 . The set Im(x0 ) is the image with respect to S of a population of critical points. Hence it is closed as the image of a closed variety. On the other hand the set ((t N+ t B− )/t B− ) × x0 is dense in (t G/t B− ) × x0 . Hence Im(x0 ) = (t G/t B− ) × x0 and S(OmD 0 ) = Γ. Therefore OmD 0 = OmD since the map S is injective.
4.3 Remarks on the isomorphism Let g be a simple Lie algebra. Let Py0 be the population of critical points originated at a tuple y 0 . We assume that the tuple y 0 is generic in the sense of Section 2.3. Theorem 4.3 says that the population Py0 is isomorphic to the flag variety t G/t B− . The isomorphism is constructed in three steps. If y ′ ∈ Py is a point of the population, then one assigns to it the associated Miura oper Dy′ as in Section 3, see also Lemma 6.1. By Theorem 3.3 we have Dy′ = v Dy0 v −1 for a suitable rational function v : P1 → t N+ . To the Miura oper D v one assigns the section Sv ∈ Γ by formula (13). Then one chooses a point x0 ∈ C, regular with respect to the connection ∇Dy 0 , and assigns to a section S v its value S v (x0 ) ∈ (t G/t B− ) × x0 . The resulting composition φy0 ,x0 : Py → t G/t B− is an isomorphism according to Theorem 4.3. Lemma 4.5. If x0 , x1 ∈ C are points regular with respect to ∇Dy 0 , then there exists an element g ∈ t B− such that φy0 ,x1 = g φy0 ,x0 . Proof 4.6. Let Y be the t G-valued solution of the equation Dy0 Y = 0 such that Y (x0 ) = id. Then Y (x) ∈ t B− for all x. If D v ∈ OmDy0 , then S v is a horizontal section of
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∇′Dy 0 . Thus it has the form x 7→ (Y (x) u t B− ) × x for a suitable element u ∈ t G. Hence φy0 ,x0 (y ′ ) = Y (x0 ) u t B− and φy0 ,x1 (y ′ ) = Y (x1 ) u t B− . We conclude that φy0 ,x1 = Y (x1 )Y (x0 )−1 φy0 ,x0 . Let y 1 be a point of Py0 . Let Py1 be the population originated at y 1 . We have Py0 = Py1 . Lemma 4.7. Let x0 ∈ C be regular with respect to both connections ∇Dy 0 and ∇Dy 1 . Then there exists an element g ∈ t B+ such that φy1 ,x0 = g φy0 ,x0 . Proof 4.8. We have Dy1 = w Dy0 w −1 for a suitable rational function w : P1 → t N+ . If Y0 (x) is the t G-valued solution of the equation Dy0 Y = 0 such that Y0 (x0 ) = id, then Y1 (x) = w(x) Y (x) w(x0 )−1 is the t G-valued solution of the equation Dy1 Y = 0 such that Y1 (x0 ) = id. Let y ′ ∈ Py0 and Dy′ = v Dy0 v −1 for a suitable rational function v : P1 → t N+ . Then Dy′ = vw −1 Dy1 wv −1 . Hence φy0 ,x0 (y ′ ) = v(x0 )−1 t B− and φy1 ,x0 (y ′ ) = w(x0 )v(x0 )−1 t B− . Therefore, φy1 ,x0 = w(x0 ) φy0 ,x0 .
5
Bruhat cells
5.1 Properties of Bruhat cells Let g be a simple Lie algebra. For an element w of the Weyl group W , the set Bw = t B− w t B− ⊂ t G/t B− is called the Bruhat cell associated to w. The Bruhat cells form a cell decomposition of the flag variety t G/t B− . For w ∈ W denote l(w) the length of w. We have dim Bw = l(w). Let s1 , . . . , sr ∈ W be the generating reflections of the Weyl group. For v ∈ t G/t B− and i ∈ {1, . . . , r} consider the rational curve C → t G/t B− ,
c 7→ ecEi v .
The limit of ecEi v is well defined as c → ∞, since t G/t B− is a projective variety. We need the following standard property of Bruhat cells. Lemma 5.1. Let si , w ∈ W be such that l(si w) = l(w) + 1. Then Bsi w = { ecEi v | v ∈ Bw , c ∈ {P1 − 0} } . Corollary 5.2. Let w = si1 · · · sik be a reduced decomposition of w ∈ W . Then Bw = { lim0 . . . lim0 ec1 Ei1 · · · eck Eik t B− ∈ t G/t B− | c01 , . . . , c0k ∈ {P1 − 0}}. c1 →c1
ck →ck
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Introduce the map fi1 ,...,ik : (C − 0)k → Bsi1 ··· sik ,
(c1 , . . . , ck ) 7→ ec1 Ei1 · · · eck Eik t B− .
5.2 Populations and Bruhat cells Let P be a population of critical points associated with weights Λ, numbers z. Let T1 , . . . , Tr be the polynomials defined by (4). Let y 0 = (y10, . . . , yr0) ∈ P with li = deg yi0 for i ∈ {1, . . . , r}. Assume that the weight at infinity of y 0 , Λ∞ =
n X
Λi −
i=1
r X
li αi ,
i=1
is integral dominant, see Section 2. Such y 0 exists according to [11]. For w ∈ W consider the weight w · Λ∞ , where w· is the shifted action of w on h ∗ . Write w · Λ∞ =
n X i=1
Λi −
r X
liw αi .
i=1
Set Pw = { y = (y1 , . . . , yr ) ∈ P | deg yi = liw , i = 1, . . . , r } . Consider the trivial bundle p′ : (t G/t B− ) × P1 → P1 with connection ∇′Dy 0 . Consider the Bruhat cell decomposition of fibers of p′ . Let x0 ∈ C be such that Ti (x0 ) 6= 0 and yi0 (x0 ) 6= 0 for i = 1, . . . , r. The point x0 ∈ C is a regular point of the connection ∇′Dy 0 . Let φy0 ,x0 : P → t G/t B− be the isomorphism defined in Section 4.3. Theorem 5.3. For w ∈ W we have φy0 , x0 (Pw ) = Bw−1 . Corollary 5.4. Let Λ1 , . . . , Λn , Λ∞ be integral dominant g -weights. Let z1 , . . . , zn be distinct complex numbers. Let w ∈ W . Consider the master function Φ(t; z, Λ, w · Λ∞ ). Let K be a connected component of the critical set of the master function. For each t ∈ K consider the tuple y t ∈ (C[x])r of monic polynomials representing the critical point t. Then the closure of the set { y t | t ∈ K } is an l(w)-dimensional cell.
5.3 Proof of Theorem 5.3 Lemma 5.5. For w ∈ W , the subset Bw × P1 ⊂ (t G/t B− ) × P1 is invariant with respect to the connection ∇′Dy 0 .
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Proof 5.6. Let Y be the t G-valued solution of the equation Dy0 Y = 0 such that Y (x0 ) = id. Then Y (x) ∈ t B− for all x. The horizontal sections of ∇′Dy 0 have the form x 7→ (Y (x) u t B− ) × x for a suitable element u ∈ t G. Hence if u t B− ∈ Bw , then Y (x)u t B− ∈ Bw for all x. Let w = sik · · · si1 be a reduced decomposition of w ∈ W . For b = 1, . . . , k set (sib · · · si1 ) · Λ∞ =
n X
Λi −
i=1
r X
lib αi .
i=1
From [2] it follows that li11 > li1 and libb > lib−1 for b = 2, . . . , k. b For b = 1, . . . , k define by induction on b a family of tuples of polynomials depending on complex parameters c1 , . . . , cb . Namely, let y˜i1 be a polynomial satisfying equation Y W ( yi01 , y˜i1 ) = Ti1 ( yj0 )−ai1 ,j . j, j6=i1
We fix y˜i1 assuming that the coefficient of xli1 in y˜i1 is equal to zero. Set y 1; c1 = (y11; c1 , . . . , yr1; c1 ), where yi1;1 c1 (x) = y˜i1 (x) + c1 yi01 (x)
and yj1; c1 (x) = yj0(x) for j 6= i1 . b−1; c ,...,c
1 b−1 Assume that the family y b−1; c1 ,...,cb−1 is already defined. Let y˜ib be a polynomial satisfying equation Y b−1; c ,...,c b−1; c1 ,...,cb−1 b−1; c1 ,...,cb−1 −ai ,j W ( yib 1 b−1 , y˜ib ) = Tib ( yj ) b .
j, j6=ib
ld−1
d−1; c ,...,c
1 d−1 We fix y˜id assuming that the coefficient of x id zero. Set y b; c1 ,...,cb = (y1b; c1 ,...,cb , . . . , yrb; c1 ,...,cb ), where
d−1; c1 ,...,cd−1
in y˜id
b−1; c1 ,...,cb−1
yib;b c1 ,...,cb (x) = y˜ibb−1; c1 ,...,cb (x) + cb yib
is equal to
(x)
and b−1; c1 ,...,cb−1
yjb; c1 ,...,cb (x) = yj
(x) for j 6= ib .
The b-th family is obtained from the (b − 1)-st family by the generation procedure in the ib -th direction, see Section 2.5. For any c1 , . . . , ck the tuple y k; c1 ,...,ck lies in P . For any c1 , . . . , ck and any i ∈ {1, . . . , r}, we have deg yik; c1 ,...,ck (x) = liw . Set P [i1 ,...,ik ] = { y k; c1 ,...,ck | c1 , . . . , ck ∈ C } .
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Proposition 5.7. We have φy0 ,x0 (P [i1,...,ik ] ) = Bw−1 . Proof of Proposition 5.7. Let Dyk; c1 ,...,ck be the Miura oper associated with the tuple y k; c1 ,...,ck , then (14) yik;k c1 ,...,ck k−1; c ,...,c yik 1 k−1
Dyk; c1 ,...,ck = exp ad log′
yi2;2 c1 ,c2
exp ad log′
exp −ad
exp −ad
yi1;2 c1 Tik
Ti2
Q
!
Ei2
exp ad log′
Eik
!
yi1;1 c1 yi01
!
k−1; c1 ,...,ck−1 −ai ,j k
j6=ik (yj
j, k; c1 ,...,ck yik
Q
!
!
)
k−1; c1 ,...,ck−1
yik
1; c1 −ai ,j ) 2 j6=i2 (yj
j, yi2;2 c1 ,c2 yi1;2 c1
!
!
!
Eik
...
Ei1
!
Ei2 exp −ad
!
· Dy0 =
...
Ti1
Q
j,
0 −ai1 ,j j6=i1 (yj )
yi1;1 c1 yi01
!
!
Ei1 ·Dy0 ,
see Theorem 3.3. Introduce the rational map g : Ck+1 → Ck ,
(x; c1 , . . . , ck ) 7→ (g1 (x; c1 ), . . . , gk (x; c1 , . . . , ck ))
where g1 (x; c1 ) =
gb (x; c1 , . . . , cb ) =
Ti1 (x)
0 −ai1 ,j j, j6=i1 ( yj (x) ) yi1;1 c1 (x) yi01 (x)
Q
,
b−1; c1 ,...,cb−1 (x) )−aib ,j j, j6=ib ( yj b−1; c ,...,c yib;b c1 ,...,cb (x) yib 1 b−1 (x)
Tib (x)
Q
for b = 2, . . . , k. From (14) it follows that the tuple y k; c1 ,...,ck corresponds to the rational section S(c1 ,...,ck ) : x 7→ fi1 ,...,ik (g(x, c1 , . . . , ck )) × x of the bundle p′ . This section is horizontal with respect to the connection ∇′Dy 0 , and we have φy0 , x (y k; c1 ,...,ck ) = fi1 ,...,ik (g(x, c1 , . . . , ck )) .
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This means that φy0 ,x0 (P [i1 ,...,ik ] ) ⊂ Bw−1 . It remains to show that every point in Bw−1 is the limit of points of φy0 ,x0 (P [i1 ,...,ik ] ), where the limit is taken in the sense of the limit in Corollary 5.2, but that statement follows from Lemma 5.8. Let x0 ∈ C be such that Ti (x0 ) 6= 0 and yi0(x0 ) 6= 0 for i = 1, . . . , r. Then for any (c11 , . . . , c1k ) ∈ (C − 0)k there exists a unique (c21 , . . . , c2k ) ∈ Ck such that (c11 , . . . , c1k ) = g(x0 ; c21 , . . . , c2k ) . The proposition is proved. Theorem 5.3 is a direct corollary of Proposition 5.7.
6
Solutions of differential equations
As we observed earlier, the Miura opers, associated with a population of critical points, help to study the structure of the population. In addition to that it turns out that for a Miura oper D associated with a critical point of a population, all solutions of the differential equation DY = 0 with values in the corresponding group can be written explicitly in terms of critical points composing the population. First we give formulas for solutions of the equation DY = 0 for opers associated with Lie algebras of types Ar , Br , and then consider more general formulas for solutions which do not use the structure of the Lie algebra. In this section g = g (A) is a simple Lie algebra with Cartan matrix A = (ai,j ).
6.1 Elimination of polynomials Ti Let B = (bi,j ) be the matrix inverse to A. Let D be the Miura t G-oper associated with weights Λ, numbers z, and a tuple y = (y1 , . . . , yr ). Introduce polynomials T1 (x), . . . , Tr (x) by formulas (4). Introduce a tuple of functions y¯ = (¯ y1 , . . . , y¯r ) by y¯i = yi
r Y
−bi,l
Tl
.
(15)
l=1
Lemma 6.1. We have D = ∂ + I + V where V =
r X j=1
log′ (¯ y j ) Hj .
(16)
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Proof 6.2. If V is given by (16), then t
h αi , V i =
r X
′
t
log (¯ yj ) h αi ,
t
αj∨ i
=
j=1
r X
′
log ( y¯j
ai,j
′
) = log (
Ti−1
j=1
r Y
yj
ai,j
).
j=1
Assume that y represents a critical point of the master function (1) associated with parameters z, Λ, Λ∞ . Let i = [i1 , . . . , ik ], ij ∈ {1, . . . , r}, be a sequence of natural numbers. Let y [i1 ] = [i ] [i ] [i ,i ] [i ,i ] [i ,...,i ] [i ,...,i ] (y1 1 , . . . , yr 1 ), y [i1 ,i2 ] = (y1 1 2 , . . . , yr 1 2 ), . . . , y [i1 ,...,ik ] = (y1 1 k , . . . , yr 1 k ) be a sequence of tuples associated with the critical point y and the sequence of indices i, see [i ,...,i ] Section 2.5. Introduce functions y¯i 1 l by [i ,...,il ]
[i ,...,il ]
y¯i 1
= yi 1
r Y
−bi,l
Tl
.
(17)
l=1
Lemma 6.3. We have [i ]
W (¯ yi1 , y¯i11 ) =
Y
−ai1 ,j
y¯j
(18)
j, j6=i1 [i ]
and y¯j 1 = y¯j for j 6= i1 ;
for l = 2, . . . , k, we have Y [i ,...,i ] [i ,...,i ] [i ,...,i ] W (¯ yil 1 l−1 , y¯il 1 l ) = (¯ yj 1 l−1 )−ail ,j
(19)
j, j6=il
[i ,...,il ]
and y¯j 1
[i ,...,il−1 ]
= y¯j 1
for j 6= il .
[i ]
[i ]
[i ,i ]
[i ,i ]
The sequence of tuples y¯[i1 ] = (¯ y1 1 , . . . , y¯r 1 ), y¯[i1 ,i2 ] = (¯ y1 1 2 , . . . , y¯r 1 2 ), . . . , [i ,i ,...,i ] [i ,i ,...,i ] y¯[i1 ,i2 ,...,ik ] = (¯ y1 1 2 k , . . . , y¯r 1 2 k ) will be called the sequence of reduced tuples associated with the critical point y and the sequence of indices i. [i ]
[i ,i ]
[i ,i ,...,i ]
The sequence of functions y¯i11 , y¯i21 2 , . . . , y¯ik1 2 k , each defined up to multiplication by a non-zero number, will be called the reduced diagonal sequence of functions associated with the critical point y and the sequence of indices i. The reduced diagonal sequence of functions determine the sequence of tuples y¯[i1 ] , y¯[i1 ,i2 ] , . . . , y¯[i1 ,i2 ,...,ik ] uniquely. In the next sections we use the following lemma. Lemma 6.4. Q −H • Consider the product rj=1 y¯j j as a function of x with values in the group t G. Then r Y j=1
y¯j
−Hj
=
r Y
yj
−Hj
Tj
wj
,
j=1
where w1 , . . . , wr ∈ t h are the fundamental coweights, i.e. h t αi , wj i = δi,j .
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• Let D = ∂ + I + V be the Miura oper with V given by formula (16). Define ¯ = ∂ + D
r X j=1
(
r Y
y¯l
−aj,l
) Fj .
l=1
Then D(
r Y
y¯j
−Hj
r Y
) = (
j=1
y¯j
−Hj
¯ . )D
j=1
6.2 The Ar critical points and Ar opers The Lie algebra slr+1 is of Ar -type. The Langlands dual to slr+1 is slr+1 . Let F1 , . . . , Fr , H1 , . . . , Hr , E1 , . . . , Er be the Chevalley generators of slr+1 . Let w1 , . . . , wr be the fundamental coweights of slr+1 . We start with two examples. Let g = sl2 . Let y = (y1 ) represent a critical point of the sl2 master function (1) −1/2 associated with parameters z, Λ, Λ∞ . Introduce the function y¯1 = y1 T1 , see (15). Then the Miura oper associated with y has the form D = ∂ + F1 + log′ (¯ y1 )H1 . [1]
Let y¯1 be the reduced diagonal sequence of functions associated with y and the sequence [1] of indices [1], in other words, W (¯ y1 , y¯1 ) = 1. Then [1] y ¯ 1 F
Y = y¯1−H1 e y¯1
1
is a solution of the differential equation DY = 0 with values in SL (2, C). Indeed, DY = y¯1−H1 (∂ +
1 F1 ) e (¯ y 1 )2
[1] y ¯ 1 F 1 y ¯1
= Y (∂ + (
[1] y¯1
y¯1
!′
+
1 )F1 ) id = Y ∂ id = 0. (¯ y 1 )2
Let g = sl3 . Let y = (y1 , y2 ) represent a critical point of the sl3 master function −2/3 −1/3 (1) associated with parameters z, Λ, Λ∞ . Introduce the functions y¯1 = y1 T1 T2 , −1/3 −2/3 y¯2 = y2 T1 T2 , see (15). Then the Miura oper associated with y has the form D = ∂ + F1 + F2 + log′ (¯ y1 )H1 + log′ (¯ y2 )H2 . [1]
[1,2]
Let y¯1 , y¯2 be the reduced diagonal sequence of functions associated with y and the sequence of indices [1, 2], in other words, [1]
W (¯ y1 , y¯1 ) = y¯2 ,
[1,2]
W (¯ y2 , y¯2
[1]
) = y¯1 .
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Let y¯2 be the reduced diagonal sequence of functions associated with y and the sequence [2] of indices [2], in other words, W (¯ y2 , y¯2 ) = y¯1 . Then Y = y¯1
−H1
y¯2
−H2
e
[1] y ¯ 1 F 1 y ¯1
e
[1,2] y ¯ 2 [F2 ,F1 ] y ¯2
e
[2] y ¯ 2 F 2 y ¯2
is a solution of the differential equation DY = 0 with values in SL (3, C). Indeed, by Lemma 6.4 it suffices to show that [1] y ¯
1 Y¯ = e y¯1
F1
e
[1,2] y ¯ 2 [F2 ,F1 ] y ¯2
e
[2] y ¯ 2 F 2 y ¯2
¯ = 0 where is a solution of the differential equation DY ¯ = ∂ + y¯2 F1 + y¯1 F2 . D (¯ y 1 )2 (¯ y 2 )2 Indeed, ¯ Y¯ = e D
[1] y ¯ 1 F 1 y ¯1
[1] y ¯1
[1]
y¯ (∂ + ( 1 y¯1
!′
[1]
[1,2]
y ¯ 2 1 y¯1 y¯1 y ¯2 )F + [F , F ] + F ) e + 1 2 1 2 2 2 2 (¯ y1 ) (¯ y2 ) (¯ y2 )
[1]
[1,2]
[F2 ,F1 ]
e
[2] y ¯ 2 F 2 y ¯2
[2]
y ¯2 y ¯2 y¯1 y¯1 [F2 ,F1 ] F2 y ¯2 y ¯2 [F , F ] + F ) e e 2 1 2 (¯ y 2 )2 (¯ y 2 )2 ! [1,2] [1] [2] [1,2] ′ [1] y ¯ y ¯ y ¯ 1 F 2 2 F y ¯ y¯ y¯1 [F ,F ] 2 y ¯2 = e y¯1 1 e y¯2 2 1 (∂ + ( 2 + 1 2 )[F2 , F1 ] + F ) e 2 y¯2 (¯ y2 ) (¯ y 2 )2
= e y¯1
F1
(∂ +
[1] y ¯1 F1 y ¯1
[1,2] y ¯2 [F2 ,F1 ] y ¯2
[2]
y ¯2 y¯1 F2 y ¯2 =e e (∂ + F ) e 2 2 (¯ y2 ) ! [1] [1,2] [2] [2] ′ y ¯1 y ¯2 y ¯2 y ¯ y¯1 F [F ,F ] F = e y¯1 1 e y¯2 2 1 e y¯2 2 (∂ + ( 2 + )F2 ) id y¯2 (¯ y 2 )2 [1] y ¯1
= e y¯1
F1
e
[1,2] y ¯2 [F2 ,F1 ] y ¯2
[2] y ¯2
e y¯2
F2
∂ id = 0 .
Now consider the general case. Let g = slr+1 . Let y = (y1 , . . . , yr ) represent a critical point of the slr+1 master function (1) associated with parameters z, Λ, Λ∞ . Introduce the functions y¯1 , . . . , y¯r by formula (15), where B = (bi,j ) is the matrix inverse to the Cartan matrix of slr+1 . Then the Miura oper associated with y has the form D = ∂ +
r X
Fj +
j=1
[i]
[i,i+1]
r X
log′ (¯ y j ) Hj .
(20)
j=1
[i,...,r]
For i = 1, . . . , r, let yi , yi+1 , . . . , yr be the diagonal sequence of polynomials associated with y and the sequence of indices [i, i + 1, . . . , r], in other words, [i]
[i,i+1]
W (yi, yi ) = Ti yi−1 yi+1 , [i,...,r−1]
W (yr−1, yr−1
[i]
W (yi+1, yi+1 ) = Ti+1 yi y¯i+2 , . . . , [i,...,r−2]
) = Tr−1 yr−2
yr ,
[i,...,r−1]
W (yr , yr[i,...,r] ) = Tr yr−1
.
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177
Define r + 1 functions Y0 , Y1 , . . . , Yr of x with values in SL (r + 1, C) by the formulas Y0 =
r Y
yj
−Hj
j=1
Tj
wj
,
Yi =
r Y
y
e
[i,...,j] j yj
[Fj ,[Fj−1 ,[...,[Fi+1,Fi ]...]]]
,
for i > 0.
j=i
Note that inside each product the factors commute. Theorem 6.5. The product Y0 Y1 . . . Yr is a solution of the differential equation DY = 0 with values in SL (r + 1, C) where D is given by (20). Note that if Y (x) is a solution of the equation DY = 0 and g ∈ SL (r + 1, C), then Y (x)g is a solution too. The proof of the theorem is straightforward. One uses Lemma 6.4 and then shows that r r X X y¯j−1 y¯j+1 y¯j−1y¯j+1 Fj ) Yi = Yi (∂ + Fj ) (∂ + 2 y¯j y¯j 2 j=i+1 j=i
for i = 1, . . . , r. In this formula we set y¯0 = y¯r+1 = 1.
6.3 The Br critical points and Cr opers Consider the root system of type Br . Let α1 , . . . , αr−1 be the long simple roots and αr the short one. We have (αr , αr ) = 2,
(αi , αi ) = 4,
(αi , αi+1 ) = −2,
i = 1, . . . , r − 1,
and all other scalar products are equal to zero. The root system Br corresponds to the Lie algebra so2r+1 . Let h B be its Cartan subalgebra. Consider the root system of type Cr . The root system Cr corresponds to the Lie algebra sp2r . Let F1 , . . . , Fr , H1 , . . . , Hr , E1 , . . . , Er be its Chevalley generators and w1 , . . . , wr the fundamental coweights. The symplectic group Sp(2r, C) is the simply connected group with Lie algebra sp2r . The Lie algebras so2r+1 and sp2r are Langlands dual. A We consider also the root system of type A2r−1 with simple roots α1A , . . . , α2r−1 . The root system A2r−1 corresponds to the Lie algebra sl2r . We denote h A its Cartan subalgebra. We have a map h ∗B → h ∗A , Λ 7→ ΛA , where ΛA is defined by A hΛA , (αiA )∨ i = hΛA , (α2r−i )∨ i = hΛ , (αi )∨ i,
i = 1, . . . , r.
Let Λ1 , . . . , Λn ∈ h ∗B be dominant integral so2r+1 -weights, z1 , . . . , zn complex numbers. Let the polynomials T1 , . . . , Tr be given by (4). Remind that an r-tuple of polynomials
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y represents a critical point of a master function associated with so2r+1 , Λ1 , . . . , Λn , z1 , . . . , zn , if and only if y is generic with respect to weights Λ1 , . . . , Λn of so2r+1 , and points z1 , . . . zn and there exist polynomials y˜i , i = 1, . . . , r, such that W (yi , y˜i ) = Ti yi−1 yi+1 , W (yr , y˜r ) =
i = 1, . . . , r − 1,
2 Tr yr−1 .
For an r-tuple of polynomials y = (y1 , . . . , yr ), let u be the 2r −1-tuple of polynomials (u1 , . . . , u2r−1) = (y1 , . . . , yr−1 , yr , yr−1, . . . , y1 ). Lemma 6.6 ([11]). An r-tuple y represents a critical point of the so2r+1 master function associated with Λ1 , . . . , Λn , z1 , . . . , zn , if and only if the 2r − 1-tuple of polynomials A u represents a critical point of the sl2r master function associated with ΛA 1 , . . . , Λn , z1 , . . . , zn . We start with an example. Let y = (y1 , y2 ) represent a critical point of the so3 master −1/2 function (1) associated with parameters z, Λ, Λ∞ . Set y¯1 = y1 T1 −1 T2 , −1 −1 y¯2 = y2 T1 T2 , see (15). Let u = (u1 , u2 , u3 ) = (y1 , y2 , y1 ) be the tuple representing the corresponding sl4 critical point. Set u ¯ = (¯ u1 , u¯2 , u¯3 ) = (¯ y1 , y¯2, y¯1 ). [1] [1,2] Let y¯1 , y¯2 be the so3 reduced diagonal sequence of functions associated with y and the sequence of indices [1, 2], in other words, [1]
[1,2]
W (¯ y1 , y¯1 ) = y¯2 ,
W (¯ y2 , y¯2
[1]
) = (¯ y 1 )2 .
[2]
Let y¯2 be the so3 reduced diagonal sequence of functions associated with y and the sequence of indices [2], in other words, [2]
W (¯ y2, y¯2 ) = (¯ y 1 )2 . [1]
[1]
[1,2]
Let u¯1 = y¯1 , u¯2 be the sl4 reduced diagonal sequence of functions associated with u and the sequence of indices [1, 2], in other words, [1]
[1,2]
W (¯ y1 , y¯1 ) = y¯2 ,
[1]
W (¯ y2 , u¯2 ) = y¯1 y¯1 .
Then [1] y ¯ 1 F
Y = y¯1−H1 y¯2−H2 e y¯1
1
[1,2] ¯ 1y 2 [[F2 ,F1 ],F1 ] y ¯2
e2
u ¯
e
[1,2] 2 [F2 ,F1 ] y ¯2
[2] y ¯ 2 F
e y¯2
2
is an Sp (4, C)-valued solution of the differential equation DY = 0 where D = ∂ + F1 + F2 + log′ (¯ y1 ) H1 + log′ (¯ y 2 ) H2 . Indeed, denote the factors of Y by P1 , . . . , P6 counting from the left. By Lemma 6.4 ¯ = 0 where it suffices to show that the product P3 P4 P5 P6 is a solution of the equation DY 2 ¯ = ∂ + y¯2 F1 + y¯1 F2 . D y¯22 y¯22
E. Mukhin, A. Varchenko / Central European Journal of Mathematics 3(2) 2005 155–182
179
We have [1]
[1]
y 1 )2 y¯12 ¯ P3 P4 P5 P6 = P3 (∂ + y¯1 y¯1 [F2 , F1 ] + 1 (¯ D [[F , F ], F ] + F2 ) P4 P5 P6 2 1 1 y¯22 2 y¯22 y¯22 [1]
y¯12 y¯1 y¯1 = P3 P4 (∂ + [F , F ] + F2 ) P5 P6 2 1 y¯22 y¯22 y¯ 2 = P3 P4 P5 (∂ + 12 F2 ) P6 = P3 P4 P5 P6 ∂ id = 0 . y¯2 Now consider the general case. Let y = (y1, . . . , yr ) represent a critical point of the so2r+1 master function (1) associated with parameters z, Λ, Λ∞ . Introduce the functions y¯1 , . . . , y¯r by formula (15), where B = (bi,j ) is the matrix inverse to the Cartan matrix of so2r+1 . Then the sp2r Miura oper associated with y has the form r X
D = ∂ +
r X
Fj +
j=1
log′ (¯ y j ) Hj .
(21)
j=1
Let u = (u1 , . . . , u2r−1 ) = (y1 , . . . , yr , . . . , y1) be the tuple representing the corresponding sl2r critical point. [i] [i,i+1] [i,...,r] For i = 1, . . . , r, let yi , yi+1 , . . . , yr be the so2r+1 diagonal sequence of polynomials associated with y and the sequence of indices [i, i + 1, . . . , r], in other words, [i]
[i,i+1]
W (yi, yi ) = Ti yi−1 yi+1 , [i,...,r−1]
W (yr−1, yr−1
[i]
W (yi+1, yi+1 ) = Ti+1 yi yi+2 , . . . ,
[i,...,r−2]
) = Tr−1 yr−2
[i,...,r−1] 2
W (yr , yr[i,...,r] ) = Tr (yr−1
yr ,
).
For i = 1, . . . , r − 1, let [i]
[i]
[i,i+1]
ui = yi ,
ui+1
ur[i,...,r] ,
= yi+1 ,
[i,i+1]
... ,
ur−1
[i,...,r+1]
... ,
u2r−i−1
ur+1
,
[i,...,r−1]
[i,...,r−1]
= yr−1
,
[i,...,2r−i−1]
be the sl2r diagonal sequence of polynomials associated with u and the sequence of indices [i, i + 1, . . . , 2r − i − 1], in other words, [i,...,r−1]
W (yr , ur[i,...,r] ) = Tr yr−1 [i,...,r+l]
W (yr−l , ur+l
yr−1 , [i,...,r+l−1]
) = Tr−l ur+l−1
[i,...,r+1]
W (yr−1 , ur+1 yr−l−1 ,
) = Tr−1 u[i,...,r] yr−2 , r
for l = 2, . . . , r − i − 1.
For i ∈ {1, . . . , r} set Fi,i = Fi . For 1 ≤ i < j < r set Fi,j = [Fj , [Fj−1, [..., [Fi+1 , Fi ]...]]] . ∗ Set Fi,r = [Fr , Fi,r−1 ] and for 1 ≤ i < j < r set ∗ ∗ Fi,j = [Fj , [Fj+1 , [...[Fr−2 , [Fr−1 , Fi,r ]]...]]] .
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Define r + 1 functions Y0 , Y1 , . . . , Yr of x with values in Sp (2r, C) by the formulas Y0 =
r Y
yj
−Hj
Tj
wj
,
j=1
Yi =
r−1 Y
y
e
j=i
[i,...,j] j yj
Fi,j
!
e
[i,...,r] 1 yr 2 yr
[[Fr ,Fi,r−1 ],Fi,r−1 ]
2r−i−1 Y
[i,...,j] j y2r−j
u
e
∗ Fi,2r−j
j=r
!
[r] yr
for i ∈ {1, . . . , r − 1}, and Yr = e yr Fr . Note that inside each product the factors commute. Theorem 6.7. The product Y0 Y1 . . . Yr is a solution of the differential equation DY = 0 with values in Sp (2r, C) where D is given by (21). The proof is straightforward. One uses Lemma 6.4 and then shows that r−1 r−1 2 2 X X y¯r−1 y¯r−1 y¯j−1 y¯j+1 y¯j−1y¯j+1 (∂ + Fr + Fj ) Yi = Yi (∂ + Fr + Fj ) 2 2 2 2 y¯r y ¯ y ¯ y ¯ r j j j=i j=i+1
for i = 1, . . . , r − 1. Remark. Theorems 6.5 and 6.7 give explicit formulas for solutions of the differential equation DY = 0 where D is the Miura oper associated to a critical point of type Ar or Br . In a similar way one can construct explicit formulas for solutions in the case of the Miura oper associated to a critical point of type Cr , cf. Section 7 in [11].
6.4 General formulas for solutions Let g be a simple Lie algebra with Cartan matrix A. Let t g be its Langlands dual with Chevalley generators F1 , . . . , Fr , H1 , . . . , Hr , t t E1 , . . . , Er . Let w1 , . . . , wr be the fundamental coweights of g . Let G be the complex simply connected Lie group with Lie algebra t g . Let V be a complex finite dimensional representation of t G. Let vlow be a lowest weight vector of V , t n− vlow = 0. Let y = (y1 , . . . , yr ) represent a critical point of the g master function (1) associated with parameters z, Λ, Λ∞ . Let Dy be the t g Miura oper associated with y. Let i = [i1 , . . . , ik ], ij ∈ {1, . . . , r}, be a sequence of natural numbers. Let y [i1 ] = [i ] [i ] [i ,i ] [i ,i ] [i ,...,i ] [i ,...,i ] (y1 1 , . . . , yr 1 ), y [i1 ,i2 ] = (y1 1 2 , . . . , yr 1 2 ), . . . , y [i1 ,...,ik ] = (y1 1 k , . . . , yr 1 k ) be a sequence of tuples associated with the critical point y and the sequence of indices i, see Section 2.5.
E. Mukhin, A. Varchenko / Central European Journal of Mathematics 3(2) 2005 155–182
Theorem 6.8. The V -valued function ! ! [i ] yi11 ′ Y = exp −log Ei1 exp −log′ yi1 [i ,...,ik ]
exp −log′
yik1
[i ,...,i ] yik1 k−1
!
Eik
!
r Y
[i ,i2 ]
yi21
[i ]
yi21
!
Ei2
[i ,...,ik ] −Hj
( yj 1
)
!
Tj
181
...
wj
vlow
j=1
is a solution of the differential equation Dy Y = 0 . The proof is straightforward and follows from the identity ! [i1 ,...,ij ] ! y i j Dy[i1 ,...,ij ] = exp ad log′ Eij · Dy[i1 ,...,ij−1 ] , [i1 ,...,ij−1 ] yij see Theorem 3.3. Let d be the determinant of the Cartan matrix of g . Corollary 6.9. Every coordinate of every solution of the equation Dy Y = 0 with values in a finite dimensional representation of t G can be written as a rational func1/d 1/d 1/d 1/d tion R(f1 , . . . , fN ; T1 , . . . , Tr ) of functions T1 , . . . , Tr and suitable polynomials f1 , . . . , fN which appear as coordinates of tuples in the g population Py originated at y. Since t G has a faithful finite dimensional representation, the solutions of the differential equation Dy Y = 0 with values in t G also can be written as rational functions of 1/d 1/d functions T1 , . . . , Tr and coordinates of tuples of Py , cf. Sections 6.2 and 6.3.
References [1] A. Beilinson and V. Drinfeld: Opers, preprint. [2] I.N. Bernshtein, I.M. Gel’fand and S.I. Gel’fand: “Structure of representations generated by vectors of highest weight”, Funct. Anal. Appl., Vol. 5, (1971), pp. 1–8. [3] A. Borel: Linear algebraic groups, New York, W.A. Benjamin, 1969. [4] L. Borisov and E. Mukhin: “Self-self-dual spaces of polynomials”, math. QA/0308128, (2003), pp. 1–38. [5] V. Drinfeld and V. Sokolov: “Lie algebras and KdV type equations”, J. Sov. Math., Vol. 30, (1985), pp. 1975–2036. [6] B. Feigin, E. Frenkel and N. Reshetikhin: “Gaudin model, Bethe Ansatz and Critical Level”, Commun. Math. Phys., Vol. 166, (1994), pp. 29–62. [7] E. Frenkel: “Affine Algebras, Langlands math.QA/9506003, (1999), pp. 1–34.
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Ansatz”,
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[8] E. Frenkel: “Opers on the projective line, flag manifolds and Bethe anzatz”, math.QA/0308269, (2003), pp. 1–48. [9] J. Humphreys: Linear algebraic groups, Springer-Verlag, 1975. [10] V. Kac: Infinite-dimensional Lie algebras, Cambridge University Press, 1990. [11] E. Mukhin and A. Varchenko: “Critical Points of Master Functions and Flag Varieties”, math.QA/0209017, (2002), pp. 1–49. [12] E. Mukhin and A. Varchenko: “Populations of solutions of the XXX Bethe equations associated to Kac-Moody algebras”, math.QA/0212092, (2002), pp.1–8. [13] E. Mukhin and A. Varchenko: “Solutions to the XXX type Bethe Ansatz equations and flag varieties”, math.QA/0211321, (2002), pp. 1–32. [14] E. Mukhin and A. Varchenko: “Discrete Miura Opers and Solutions of the Bethe Ansatz Equations”, math.QA/0401137, (2004), pp. 1–26. [15] E. Mukhin and A. Varchenko: “Miura Opers and Critical Points of Master Functions”, math.QA/0312406, (2003), pp. 1–27. [16] E. Mukhin and A. Varchenko: “Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture”, math.QA/0501144, (2005), pp. 1–40. [17] N. Reshetikhin and A. Varchenko: “Quasiclassical asymptotics of solutions to the KZ equations”, In: Geometry, topology & physics. Conf. Proc. Lecture Notes Geom. Topology, VI, Internat. Press, Cambridge, MA, 1995, pp. 293–322. [18] I. Scherbak and A. Varchenko: “Critical point of functions, sl2 representations and Fuchsian differential equations with only univalued solutions”, math. QA/0112269, (2001) pp. 1–25. [19] V. Schechtman and A. Varchenko: “Arrangements of hyperplanes and Lie algebra homology”, Invent. Math., Vol. 106, (1991), pp. 139–194.
CEJM 3(2) 2005 183–187
On the weak non-defectivity of Veronese embeddings of projective spaces∗ Edoardo Ballico† Department of Mathematics, University of Trento, 38050 Povo (TN), Italy
Received 24 November 2004; accepted 10 January 2005 Abstract: Fix integers n, x, k such that n ≥ 3, k > 0, x ≥ 4, (n, x) 6= (3, 4) and k(n + 1) < n+x . Here we prove that the order x Veronese embedding of Pn is not weakly (k −1)-defective, n i.e. for a general S ⊂ Pn such that ♯(S) = k + 1 the projective space |I2S (x)| of all degree t − 1 − k(n + 1) (proved by hypersurfaces of Pn singular at each point of S has dimension n+x n Alexander and Hirschowitz) and a general F ∈ |I2S (x)| has an ordinary double point at each P ∈ S and Sing(F ) = S. c Central European Science Journals. All rights reserved.
Keywords: Veronese variety, weakly defective variety, zero-dimensional scheme, double point, fat point, Veronese embedding MSC (2000): 14N05
1
Introduction
The main aim of this paper is to use the so-called Horace Method introduced by A. Hirschowitz to prove the following result. Theorem 1.1. Fix integers n ≥ 3, x ≥ 4, (n, x) 6= (3, 4) and k ≥ 0 such that k(n + 1) <
n+x n
(1)
and a general S ⊂ Pn such that ♯(S) = k. Let |I2S (x)| denote the projective space of all degree x hypersurfaces of Pn singular at each point of S. Then dim(|I2S (t)|) = ∗ †
The author was partially supported by MIUR and GNSAGA of INdAM (Italy). E-mail:
[email protected]
184
E. Ballico / Central European Journal of Mathematics 3(2) 2005 183–187
n+x n
− k(n + 1) − 1. A general F ∈ |I2S (x)| satisfies Sing(F ) = S and it has an ordinary node at each point of S. The computation of dim(|I2S (x)|) is an important theorem due to J. Alexander and A. Hirschowitz [1, 2, 3, 4, 6]. With the classical terminology their theorem means that the degree x Veronese embedding of Pn is not (k − 1)-defective. They also give the list of all triples (n, x, k) such that the degree x ≥ 3 Veronese embedding of Pn is (k − 1)defective: the triples (n, x, k) ∈ {(2, 4, 5), (3, 4, 9), (4, 3, 7), (4, 4, 14)} ([6], Th. 1). The new (I hope) part is that Sing(F ) = S and that F has an ordinary node at each point of S for a general F ∈ |I2S (x)|. For each fixed pair (n, x) there is at most one integer − 1)/(n + 1)]) satisfying (1) and for which Theorem 1.1 is not known k (k = [( n+x n to be true by [10], Cor. 4.5. As obvious to everybody working on this topic the case k = [( n+x − 1)/(n + 1)] is by far the most difficult. To prove it we will use an idea n due to M. Mella and used in [10]. For related examples in which the singular locus has positive dimension, see [9] and [8], Remark 6.2, which quotes [11], and [10], Remark 4.4. We work over an algebraically closed field K with char(K) = 0. Our proof of Theorem 1.1 heavily depends on the characteristic zero assumption: a key tool will be [7], Th. 1.4. We borrowed a key idea from [10].
2
The proof
For any scheme A and any P ∈ Areg let 2P (or 2{P, A} if there is any danger of misunderstanding) denote the first infinitesimal neighborhood of P in A, i.e. the closed subscheme of A with (IP )2 as its ideal sheaf. Hence 2P is a zero-dimensional subscheme of A and length(2P ) = dim(A)+1. For any finite subset S ⊂ Areg , set 2S := ∪P ∈S 2P and 2{S, A} := ∪P ∈S 2{P, A}. For any closed subscheme Z of A and every effective Cartier divisor D of A let ResD (Z) denote the residual scheme of of Z with respect to D, i.e. the closed subscheme of A with IZ,A : ID,A as its ideal sheaf. For any effective Cartier divisor D of A such that P ∈ Dreg we have 2{P, A} ∩ D = 2{P, D} and ResD (2{P, A}) = {P }. We will often use the following elementary form of the so-called Horace Lemma. Lemma 2.1. Let H ⊂ Pn be a hyperplane and Z ⊂ Pn a closed subscheme. Then: (a) h0 (Pn , IZ (d)) ≤ h0 (Pn , IResH (Z) (d − 1)) + h0 (H, IZ∩H (d)); (b) h1 (Pn , IZ (d)) ≤ h1 (Pn , IResH (Z) (d − 1)) + h1 (H, IZ∩H (d)). Proof. By the very definition of a residual scheme with respect to H, there is the following exact sequence 0 → IResH (Z) (d − 1) → IZ (d) → IZ∩H (d) → 0 (2) whose long cohomology exact sequence proves the lemma.
The following result is a very particular case of [5], Lemma 2.3 (see in particular Fig. 1 on p. 308). Lemma 2.2. Let H ⊂ Pn be hyperplane, Z ⊂ Pn a closed subscheme not containing H and s a positive integer. Let U be the union of Z and s general double points of
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Pn . Let S be the union of s general points of H. Let E ⊂ H be the union of s general double points of H (not double points of Pn , i.e. each of them has length n). To prove h1 (Pn , IU (d)) = 0 (resp. h0 (Pn , IU (d)) = 0) it is sufficient to prove h1 (H, I(Z∩H)∪S (d)) = h1 (Pn , IResH (Z)∪E (d − 1)) = 0 (resp. h0 (H, I(Z∩H)∪S (d)) = h0 (Pn , IResH (Z)∪E (d − 1)) = 0). For all integers n > 0, t > 0 define the integers an,t , bn,t , cn,t and dn,t using the following relations: n+t (n + 1)an,t + bn,t = , 0 ≤ bn,t ≤ n n n+t (n + 1)cn,t + dn,t + 1 = , 0 ≤ dn,t ≤ n n
(3)
(4)
Notice that cn,t = an,t and dn,t = bn,t − 1 if bn,t > 0, while cn,t = an,t − 1 and dn,t = n if bn,t = 0. Subtracting (3) from the same equation for the integers n, t−1 we obtain the following relation: n+t−1 (n + 1)(an,t − an,t−1 ) + bn,t − bn−1,t = (5) n−1 Using (3) for the integers n − 1, t and (5), we obtain the following relation: (n + 1)(an,t − an,t−1 ) + bn,t − bn−1,t = nan−1,t + bn−1,t
(6)
Lemma 2.3. Fix a hyperplane H ∈ Pn and an integer y > 0. Let D be an irreducible ydimensional family of hypersurfaces of Pn . Then for a general B ⊂ H such that ♯(B) = y there is Y ∈ D such that B ⊂ Y . Proof. Take a general P ∈ H. Since y > 0 there are infinitely many hypersurfaces parametrized by D. Hence the set of all Y ∈ D containing P is non-empty and contains an irreducible subfamily DP ⊂ D of dimension y − 1. If y = 1, we are done. If y ≥ 2 we use induction on y and the family DP for the integer y ′ := y − 1. Remark 2.4. If x ≥ 2, ♯(A) = cn,x − an−1,x − bn−1,x and h1 (Pn , I2A (x − 1)) = 0, then n+x n+x−1 −(n+1)c +(n+1)(a +b ) = − n +1+ h0 (Pn , I2A (x−1)) = n+x−1 n,x n−1,x n−1,x n n n+x−1 dn,x +an−1,x +(n+1)bn−1,x n−1 −bn−1,x = 1+dn,x +nbn−1,x +an−1,x ≥ 1+an−1,x +bn−1,x .
Lemma 2.5. Assume either x = 3 and n ≥ 10 or x ≥ 4 and n ≥ 4. Then n+x−1 (n + 1)(an,x − an−1,x ) + n + 1 ≤ n Proof. Since (n + 1)an,x ≤ 2n − 1 ≥ an−1,x , i.e. if
n+x n
and nan−1,x ≥
(2n − 1)(n + 1) ≤
(7)
n+x−1 n−1
, the inequality (7) is satisfied if
n+x−1 n−1
(8)
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which is satisfied if either x = 3 and n ≥ 10 or x ≥ 4 and n ≥ 4. Lemma 2.6. We have
n+x−1 n
Proof. Since (n + 1)cn,x < (n + x). Lemma 2.7. We have n ≥ 3 and x ≥ 3.
n+x n
n+x−1 n
≥ cn,x for all n ≥ 2 and x ≥ 3
, it is sufficient to use the obvious inequality (n + 1)x ≥
− (n + 1)cn,x + (n + 1)(an−1,x + bn−1,x ) ≥ an−1,x for all
Proof. By (3) and (4) this inequality is equivalent to the inequality + nbn−1,x , which is obviously satisfied. 1 + dn,x + n+x−1 n−1
n+x−1 n
−
n+x n
+
Proof (of Theorem 1.1). Fix a general S ⊂ Pn such that ♯(S) = cn,x − an−1,x . By Lemma 2.5 and the inequality cn,x ≤ an,x we have (n + 1)♯(S) + n + 1 ≤ n+x−1 . Hence n 1 n h (P , I2S (x − 1)) = 0 and a general Y ∈ |I2S (x − 1)| has an ordinary node at each point of S and Sing(Y ) = S ([10], Cor. 4.5). Let H ⊂ Pn be a general hyperplane. Hence H ∩ S = ∅ and H is transversal to Y . Now we fix H and S and move Y . Since n ≥ 3, Y ∩ H is irreducible. We fix a general B ⊂ H ∩ Y such that ♯(B) = an−1,x . Claim 2.8. We have h1 (Pn , I2S∪2B (x)) = 0 Proof (of the Claim). Since (n−1, x) ∈ / {(2, 4), (3, 4), (4, 3), (4, 4)} and B is general in an integral degree x − 1 hypersurface of H, we have h1 (H, I2B (x)) = 0, i.e. h0 (H, I2B (x)) = bn−1,x . Notice that ♯(S) ≥ bn−1,x (Lemma 2.3). Fix S ′′ ⊆ S such that ♯(S ′′ ) = bn−1,x and set S ′ := S\S ′′ . S ′′ may be seen as a set of bn−1,x general points of Pn . We degenerate it (keeping fixed S ′ ∪ B) into a union E of bn−1,x general points of H. Hence h0 (H, I2B∪E (x)) = h1 (H, I2B∪E (x)) = 0. Since S ∩ H = ∅, then Res2S∪2B = 2S ∪ B. The local deformation space of an ordinary nodal hypersurface singularity is one-dimensional. Since h1 (Pn , I2S (x − 1)) = 0, we obtain that, moving S, the set of the possible nodal hypersurfaces Y is (near Y ) at least of dimension n+x−1 −1−♯(S) ≥ an−1,x (Lemma 2.6). n Hence by Lemma 2.2 to prove the Claim it is sufficient to prove h1 (Pn , I2S ′ ∪2{E,H}∪B (x − 1)) = 0. First, we will check that h1 (Pn , I2S ′ ∪2{E,H} (x − 1)) = 0. To prove this vasnishing it is sufficient to prove h1 (Pn , I2S ′ ∪2E (x − 1)) = 0. Since ♯(E) ≤ n, any n points of Pn are contained in a hyperplane and S ′ is chosen independently from H and E, we may consider S ′ ∪ E as a general union of cn,x − an−1,x points of Pn . Hence h1 (Pn , I2S ′ ∪2E (x − 1)) = 0 and thus h1 (Pn , I2S ′ ∪2{E,H} (x − 1)) = 0. By Remark 2.4 and Lemma 2.3 for fixed S ′ we may take B ∪ E general in H. Hence for fixed S ′ ∪ E we may take B general in H. By the generality of B, S ′ ∩ H = ∅ and h1 (Pn , I2S ′ ∪2{E,H} (x − 1)) = 0, to prove h1 (Pn , I2S ′ ∪2{E,H}∪B (x−1)) = 0 and hence the Claim it is sufficient to prove the inequality h0 (Pn , I2S ′ ∪2{E,H} (x − 1)) − h0 (Pn , I2S ′ (x − 2)) ≥ ♯(B) (see e.g. [6], Lemma 3), i.e. the
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inequality n+x−1 − (n + 1)cn,x + (n + 1)(an−1,x + bn−1,x ) ≥ h0 (Pn , I2S ′ (x − 2)) + an−1,x (9) n First assume cn,x − an−1,x − bn−1,x ≤ an,x−2 . Then h1 (Pn , I2S ′ (x − 2)) = 0 and hence h0 (Pn , I2S ′ ∪2{E,H} (x − 1)) − h0 (Pn , I2S ′ (x − 2)) = n+x−2 − nbn−1,x . Hence the inequn−1 ality (9) is satisfied in this case. Now assume cn,x − an−1,x − bn−1,x > an,x−2. Hence h0 (Pn , I2S ′ (x − 2)) = 0. Hence the inequality (9) is satisfied by Lemma 2.7, proving the Claim. Set A := Y ∪ H. Since S ∩ H = ∅ and cn,x > an−1,x , at least one of the points of S is an isolated singular point of A. Hence a general E ∈ |I2S∪2B (x)| has at least one isolated singular point and this point is contained in S ∪ B, i.e. in a set T such that h1 (Pn , I2T (x)) = 0 and ♯(T ) = cn,x . Hence we may apply the semicontinuity theorem for cohomology and the openness of smoothness to obtain that for a general G ⊂ Pn such that ♯(G) = cn,x , the linear system |I2G (x)| has the expected dimension and a general member of it has an isolated singularity at one point of G. We conclude by [7], Th. 1.4.
References [1] J. Alexander: “Singularit´es imposables en position g´en´erale aux hypersurfaces de Pn ”, Compositio Math., Vol. 68, (1988), pp. 305–354. [2] J. Alexander and A. Hirschowitz: “Un lemme d’Horace diff´erentiel: application aux singularit´e hyperquartiques de P5 ”, J. Algebraic Geom., Vol. 1, (1992), pp. 411–426. [3] J. Alexander and A. Hirschowitz: “La m´ethode d’Horace ´eclat´e: application `a l’interpolation en degr´e quatre”, Invent. Math., Vol. 107, (1992), pp. 585–602. [4] J. Alexander and A. Hirschowitz: “Polynomial interpolation in several variables”, J. Algebraic Geom., Vol. 4, (1995), pp. 201–222. [5] J. Alexander and A. Hirschowitz: “An asymptotic vanishing theorem for generic unions of multiple points”, Invent. Math., Vol. 140, (2000), pp. 303–325. [6] K. Chandler: “A brief proof of a maximal rank theorem for generic double points in projective space”, Trans. Amer. Math. Soc., Vol. 353(5), (2000), pp. 1907–1920. [7] L. Chiantini and C. Ciliberto: “Weakly defective varieties”, Trans. Amer. Math. Soc., Vol. 454(1), (2002), pp. 151–178. [8] C. Ciliberto: “Geometric aspects of polynomial interpolation in more variables and of Waring’s problem”, In: European Congress of Mathematics (Barcelona, 2000), Progress in Math., Vol. 201, Birkh¨auser, Basel, 2001, pp. 289–316. [9] C. Ciliberto and A. Hirschowitz: “Hypercubique de P4 avec sept points singulieres g´en´eriques”, C. R. Acad. Sci. Paris, Vol. 313(I), (1991), pp. 135–137. [10] M. Mella: Singularities of linear systems and the Waring problem, arXiv mathAG/0406288. [11] A. Terracini: “Sulla rappresentazione delle coppie di forme ternarie mediante somme di potenze di forme lineari”, Ann. Mat. Pura e Appl., Vol. 24, (1915), pp. 91–100.
CEJM 3(2) 2005 188–202
Banach manifolds of algebraic elements in the algebra L(H) of bounded linear operators∗ Jos´e M. Isidro† Facultad de Matem´aticas, Universidad de Santiago, Santiago de Compostela, Spain
Received 17 November 2004; accepted 12 January 2005 Abstract: Given a complex Hilbert space H, we study the manifold A of algebraic elements in Z = L(H). We represent A as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C∗ -algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0 < r < ∞) are real-analytic direct submanifolds of Z. Using the C∗ -algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M , and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection. c Central European Science Journals. All rights reserved. ° Keywords: Jordan-Banach algebras, JB∗ -triples, algebraic elements, Grassmann manifolds, Riemann manifolds MSC (2000): 17C27, 17C36, 17B65
1
Introduction
In this paper we are concerned with certain infinite-dimensional Grassmann manifolds in Z : = L(H), the space of bounded linear operators z : H → H in a complex Hilbert space H. Grassmann manifolds are a classical object in Differential Geometry and in recent years several authors have considered them in the Banach space setting. Besides the Grassmann structure, a Riemann and a K¨ahler structure has sometimes been defined even in the infinite-dimensional setting. Let us recall some aspects of the topic that are relevant for our purpose. ∗ †
Supported by Ministerio de Educaci´on y Cultura of Spain, Research Project BFM2002-01529. E-mail:
[email protected]
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The study of the manifold of minimal projections in a finite-dimensional simple formally real Jordan algebra was made by U. Hirzebruch in [4], who proved that such a manifold is a compact symmetric Riemann space of rank 1, and that every such a space arises in this way. Later on, Nomura in [13, 14] established similar results for the manifold of fixed finite rank projections in a topologically simple real Jordan-Hilbert algebra. In [7], the authors studied the Riemann structure of the manifold of finite rank projections in Z without the use of any global scalar product. As pointed out there, the Jordan-Banach structure of Z encodes information about the differential geometry of some manifolds naturally associated to it, one of which is the manifold of algebraic elements in Z. On the other hand, the Grassmann manifold of all projections in Z has been discussed by Kaup in [11]. See also [1, 8] for related results. It is therefore reasonable to study the manifold of algebraic elements in Z. We restrict our considerations to the set A of all normal algebraic elements in Z that have finite rank. Normality allows us to use spectral theory which is an essential tool. In the case H = Cn all elements in Z are algebraic (as any square matrix is a root of its characteristic polynomial) and have finite rank, whereas for arbitrary H the set of all (finite and non finite rank) algebraic elements is norm total in Z, see [5] (Lemma 3.11). Under the above restrictions A is represented as a disjoint union of closed connected subsets M of Z, each of which is homogeneous and invariant under the natural action of G, the group of all C∗ -automorphisms of Z. Actually these sets are the orbits of G in A. The family of these orbits is quite plentiful and different orbits may have quite different properties. If an orbit M contains a hermitian element then all elements in M are hermitian and M turns out to be a closed real-analytic direct submanifold of Z. Using algebraic tools, a real-analytic Banach-manifold structure and a G-invariant affine connection ∇ are defined on M in that case, and the ∇-geodesics are computed. For a ∈ M , the restriction to M of the Peirce reflection Sa on Z around the projection a := supp(a) is a real-analytic involution of M for which a is a fixed point. The set FixM (Sa ) of the fixed points of such involution is a direct real-analytic submanifold of Z. If a is a finite rank projection then M is a symmetric manifold. For an orbit M and a point a ∈ M , the following conditions on Ta M are known to be equivalent: (1) Ta M is linearly homeomorphic to a Hilbert space, (2) Ta M is a reflexive Banach space, (3) the rank of a is finite. If these conditions hold for some a ∈ M , then this occurs for all a ∈ M . If in addition a is a finite rank projection, then a G-invariant Riemann structure can be defined on M . We take a JB∗ -triple system approach instead of the Jordan-algebra approach of [13, 14]. As noted in [1] and [6], within this context the algebraic structure of JB∗ -triple acts as a substitute for the Jordan algebra structure. Since M consists of elements with a fixed finite rank r, (0 < r < ∞), the JB∗ -triple structure provides a local scalar product known as the algebraic metric of Harris ([2], prop. 9.12). Although Z is not a Hilbert space, the use of the algebraic scalar product allows us to define a G-invariant Riemann structure on M for which ∇ is the Levi-Civita connection.
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Algebraic preliminaries.
For a complex Banach space X denote by XR the underlying real Banach space, and let L(X) and LR (X) respectively be the Banach algebra of all bounded complex-linear operators on X and the Banach algebra of all bounded real-linear operators on XR . A complex Banach space Z with a continuous mapping (a, b, c) 7→ {abc} from Z × Z × Z to Z is called a JB*-triple if the following conditions are satisfied for all a, b, c, d ∈ Z, where the operator a¤b ∈ L(Z) is defined by z 7→ {abz} and [ , ] is the commutator product: (1) (2) (3) (4)
{abc} is symmetric complex linear in a, c and conjugate linear in b. [a¤b, c¤d] = {abc}¤d − c¤{dab}. a¤a is hermitian and has spectrum ≥ 0. k{aaa}k = kak3 .
If a complex vector space Z admits a JB*-triple structure, then the norm and the triple product determine each other. For x, y, z ∈ Z we write L(x, y)(z) = (x2y)(z) and Q(x, y)(z) := {xzy}. Note that L(x, y) ∈ L(Z) whereas Q(x, y) ∈ LR (Z), and that the operators La = L(a, a) and Qa = Q(a, a) commute. A derivation of a JB*-triple Z is an element δ ∈ L(Z) such that δ{zzz} = {(δz)zz} + {z(δz)z} + {zz(δz)} and an automorphism is a bijection φ ∈ L(Z) such that φ{zzz} = {(φz)(φz)(φz)} for z ∈ Z. The latter occurs if and only if φ is a surjective linear isometry of Z. The group Aut(Z) of automorphisms of Z is a real Banach-Lie group whose Banach-Lie algebra is the set Der(Z) of all derivations of Z. The connected component of the identity in Aut(Z) is denoted by Aut◦ (Z). Two elements x, y ∈ Z are orthogonal if x¤y = 0 and e ∈ Z is called a tripotent if {eee} = e, the set of which is denoted by Tri(Z). For e ∈ Tri(Z), the set of eigenvalues of e2e ∈ L(Z) is contained in {0, 12 , 1} and the topological direct sum decomposition, called the Peirce decomposition of Z,
Z = Z1 (e) ⊕ Z1/2 (e) ⊕ Z0 (e)
(1)
holds. Here Zk (e) is the k- eigenspace of e2e and the Peirce projections are
P1 (e) = Q2 (e),
P1/2 (e) = 2(e2e − Q2 (e)),
P0 (e) = Id − 2e2e + Q2 (e).
We will use the Peirce rules {Zi (e) Zj (e) Zk (e)} ⊂ Zi−j+k (e) where Zl (e) = {0} for l 6= 0, 1/2, 1. In particular, every Peirce space is a JB∗ -subtriple of Z and Z1 (e)2Z0 (e) = {0} = Z0 (e)2Z1 (e). A JB∗ -triple Z may have no non-zero tripotents however the set of them is plentiful if Z is a dual Banach space.
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Let e = (e1 , · · · , en ) be a finite sequence of non-zero mutually orthogonal tripotents ej ∈ Z, and define for all integers 0 ≤ j, k ≤ n the linear subspaces Zjj (e) = Z1 (ej )
1 ≤ j ≤ n,
Zjk (e) = Zkj (e) = Z1/2 (ej ) ∩ Z1/2 (ek ) \ Z0j (e) = Zj0 (e) = Z1/2 (ej ) ∩ Z0 (ek )
1 ≤ j, k ≤ n, j 6= k, 1 ≤ j ≤ n,
(2)
k6=j
Z00 (e) =
\
Z0 (ej ).
1≤j≤n
Then the following topologically direct sum decomposition, called the joint Peirce decomposition relative to the family e, holds ¡ M ¢ ¡ M ¢ ¡M ¢ Z= Zk0 (e) ⊕ Zkj (e) ⊕ Zkk (e) . (3) 0≤k≤n
1≤k<j≤n
1≤k≤n
The Peirce spaces multiply according to the rules {Zjm Zmn Znk } ⊂ Zjk , and all products that cannot be brought to this form (after reflecting pairs of indices if necessary) vanish. The projectors Pkj (e) : Z → Zkj (e), called joint Peirce projectors, are invariant under the group Aut(Z), that is, they satisfy Pkj (h(e)) = hPkj (e)h−1 ,
h ∈ Aut(Z),
where h(e) := (h(e1 ), · · · h(en )), and the explicit formula for the Pkj (e) can be found in [5] (Lemma 3.15). If W is a complex Banach space with an involution ∗ , then its selfadjoint part Ws := {w ∈ W : w∗ = w} is a purely real Banach space. In the joint Peirce decomposition of Z relative to the orthogonal family e := (e1 , · · · , en ) every Peirce space Zjk (e), (0 ≤ j ≤ k ≤ n), is invariant under the natural involution ∗ of Z, hence they are complex Banach spaces with involution too. Recall that every C*-algebra Z is a JB*-triple with respect to the triple product 2{abc} := (ab∗ c + cb∗ a). In that case, every projection in Z is a tripotent and more generally the tripotents are precisely the partial isometries in Z. C∗ -algebra derivations and C∗ -automorphisms are derivations and automorphisms of Z as a JB∗ -triple though the converse is not true. More precisely, for Z = L(H), the group of C∗ algebra automorphisms consists of those elements in Aut(Z) that fix the unit of Z, i.e., G = {g ∈ Aut(Z) : g(1) = 1}. We refer to [9], [11], [15] and the references therein for the background of JB∗ -triple theory, and to [12] for the finite dimensional case.
3
Banach manifolds of algebraic elements in L(H).
From now on, Z will denote the C∗ -algebra L(H). An element a ∈ Z is said to be algebraic if it satisfies the equation p(a) = 0 for some non identically null polynomial p ∈ C[X]. By elementary spectral theory σ(a), the spectrum of a in Z, is a finite set
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whose elements are roots of the algebraic equation p(λ) = 0. In case a is normal we have X a= λ eλ λ∈σ(a)
where λ and eλ are, respectively, the spectral values and the corresponding spectral projections of a. If 0 ∈ σ(a) then e0 , the projection onto ker(a), satisfies e0 6= 0 but in the above representation the summand 0 e0 is null and will be omitted. Thus for normal algebraic elements a ∈ Z we have X a= λ eλ (4) λ∈σ(a)\{0}
In particular, in (4) the numbers λ are non-zero pairwise distinct complex numbers and the eλ are pairwise orthogonal non-zero projections. We say that a has finite rank if dim a(H) < ∞, which always occurs if dim(H) < ∞. Set rλ := rank(eλ ). Then a has finite rank if and only if rλ < ∞ for all λ ∈ σ(a)\{0} (the case 0 ∈ σ(a) and dim ker a = ∞ may occur and still a has finite rank). Hence every finite rank normal algebraic element a ∈ Z gives rise to: (i) a positive integer n which is the cardinal of σ(a)\{0}, (ii) an ordered n-tuple (λ1 , · · · , λn ) of numbers in C\{0}, which is the set of the pairwise distinct non-zero spectral values of a, (iii) an ordered n-tuple (e1 , · · · , en ) of non-zero pairwise orthogonal projections, and (iii) an ordered n-tuple (r1 , · · · , rn ) where rk ∈ N\{0} is the rank of the spectral projection ek . The spectral resolution of a is unique except for the order of the summands in (4), therefore these three n-tuples are uniquely determined up to a permutation of the indices (1, · · · , n). The operator a can be recovered from the set of the first two ordered n-tuples, a being given by (4). Given the n-tuples Λ := (λ1 , · · · , λn ) and R := (r1 , · · · , rn ) in the above conditions, we let X M (n, Λ, R) := { λk ek : ej ek = 0 for j 6= k, rank(ek ) = rk , 1 ≤ j, k ≤ n } (5) k
be the set of the elements (4) where the coefficients λk and ranks rk are given and the ek range over non-zero, pairwise orthogonal projections of rank rk . For instance, for n = 1, Λ = {1} and R = {r} we obtain the manifold of projections with a given finite rank r, that was studied in [7]. The involution z 7→ z ∗ on Z is a C∗ -algebra antiautomorphism that fixes every projection, preserves normality, orthogonality and ranks, hence it maps the set A onto itself. ¯1, · · · , λ ¯ n ). Then z 7→ z ∗ induces a For the n-tuple Λ = (λ1 , · · · , λn ) we set Λ∗ := (λ map M (n, Λ, R) → M (n, Λ, R)∗ where M (n, Λ, R)∗ = {z ∗ : z ∈ M } = M (n, Λ∗ , R), and Λ ⊂ Rn if and only if M (n, Λ, R) consists of hermitian elements. P To a normal algebraic element a = λ∈σ(a)\{0} λeλ we associate a, called the support of a, and e where X a = supp(a) := eλ = e1 + · · · + en , e := e(a) := (e1 , · · · , en ). λ∈σ(a)\{0}
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Proposition 3.1. Let A and H be the set of all normal (respectively, hermitian) algebraic elements of finite rank in Z, and let M (n, Λ, R) be defined as in (5). Then [ [ A= M (n, Λ, R), H= M (n, Λ, R) (6) n, Λ, R
n, Λ=Λ∗ , R
is a disjoint union of G-invariant closed connected subsets of Z on each of which the group G acts transitively. The sets M = M (n, Λ, R) are the orbits of G in A (respectively, in H). Proof. It suffices to prove the statements concerning A. We have seen that A ⊂ S M (n, Λ, R). Conversely, let a belong to some M (n, Λ, R) hence we have a = Pn, Λ, R k λk ek for some orthogonal projections ek . Then Id = (e1 + · · · + en ) + f where f is the projection onto ker(a) if 0 ∈ σ(a) and f = 0 otherwise. The above properties of the ek , f yield easily ap(a) = 0 or p(a) = 0 according to the cases, where p ∈ C[X] is the polynomial p(z) = (z − λ1 ). · · · .(z − λn ). Hence a ∈ A. Clearly (6) is union of disjoint subsets. Fix one of the sets M := M (n, Λ, R) and take any pair a, b ∈ M . Then a = λ1 p 1 + · · · + λ n p n ,
b = λ 1 q1 + · · · + λ n qn .
P P In case 0 ∈ σ(a), set p0 := Id − k pk and q0 := Id − k qk . Since rank pk =rank qk < ∞, the projections pk and qk are unitarily equivalent and so are p0 and q0 . Let us choose orthonormal basis Bkp and Bkq in the ranges pk (H) and qk (H) for k = 0, 1, · · · , n. Then S p S q k Bk and k Bk are two orthonormal basis in H. The unitary operator U ∈ Z that exchanges these basis satisfies U a = b. In particular, M is the orbit of any of its points under the action of the unitary group of H. Since this group is connected and its action on Z is continuous, M is connected. By the orthogonality properties of the ek , the successive powers of a have the expression al = λl1 e1 + · · · + λln en , 1 ≤ l ≤ n, where the determinant det(λlk ) 6= 0 does not vanish since it is a Vandermonde determinant and the λk are pairwise distinct. Thus the ek are polynomials in a whose coefficients are rational functions of the λk . Now we show that M is a closed subset of Z. Let w ∈ M and let (zµ )µ∈N be a sequence in M such that limµ→∞ zµ = w. We have to show that w ∈ M . Each point zµ has a spectral resolution of the form zµ = λ1 e1µ + · · · + λn enµ ,
µ ∈ N,
(7)
where the spectral values Λ = (λ1 , · · · , λn ) are fixed. By the above, each projection ekµ , (1 ≤ k ≤ n), is a polynomial in zµ , say ekµ = f1k (Λ)zµ + f2k (Λ)zµ2 + · · · + fnk (Λ)zµn ,
1 ≤ k ≤ n,
µ ∈ N,
(8)
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where the coefficients fkj (Λ) are rational functions of the spectral values Λ = (λ1 , · · · , λn ) and do not depend on the index µ ∈ N. Since limµ→∞ zµ = w and the power operation in Z is continuous, the expression (8) yields the existence of the limit ek := lim ekµ = f1k (Λ)w + f2k (Λ)w2 + · · · + fnk (Λ)wn , µ→∞
1 ≤ k ≤ n.
In particular, each of the sequences (ekµ )µ∈N , (1 ≤ k ≤ n), is a Cauchy sequence in Z and more precisely in the subset of Z that consists of the projections that have a fixed given finite rank rk . Since the latter set is closed, we have rank(ek ) = rk . Taking the limit for µ → ∞ in (7) we get w = λ1 e1 + · · · + λn en which shows w ∈ M . This completes the proof. ¤ To establish our main result [Theorem (3.4) below] we need some notation and techP nical results. To a normal algebraic element a ∈ Z with spectral resolution a = k λk ek , we associate the Peirce space Ξ(e) := Z1/2 (e1 ) + · · · + Z1/2 (en ) ⊂ Z.
(9)
Remark that Ξ(e) is linearly homeomorphic to a closed subspace of the product Z1/2 (e1 )× · · ·×Z1/2 (en ). Indeed, the spaces Z1/2 (ek ), (1 ≤ k ≤ n), are not direct summands in Ξ(e), however by ([12], th. 3.14 (3)) and ([5], lemma 3.15), Ξ(e) is a topologically complemented subspace of Z and we have ¢ ¢ ¡M ¡ M Zk0 . (10) Zrs ⊕ Ξ(e) = 1≤r<s≤n
1≤k≤n
Hence each u ∈ Ξ(e) determines in a unique way the projections ur,s and uk,0 of u onto the subspaces Zr,s (e) and Zk,0 (e), which in turn give in a unique way vectors uk := uk,0 + P P u satisfying u ∈ Z (e ) and u = r,k k k 1/2 r6=k 1≤k≤n uk . The map φ : u 7→ (u1 , · · · , un ), (u ∈ Ξ(e)), where the uk have been just defined, is injective since (10) is a direct sum, ¡ ¢ Q hence it is an isomorphism onto the image φ Ξ(e) ⊂ nk=1 Z1/2 (ek ). When this product space is endowed with the norm of the supremum, φ is continuous by the continuity of P the Peirce projectors and the inverse φ−1 : (u1 , · · · , un ) 7→ u = k uk is also continuous. ¡ ¢ Q In particular φ Ξ(e) is closed in nk=1 Z1/2 (ek ) and we shall always identify Ξ(e) with ¡ ¢ Q its image φ Ξ(e) ⊂ nk=1 Z1/2 (ek ). We define JB∗ -triple inner derivation valued map Φa : Ξ(e) → Der(Z) by X Φa (u) := (ek ¤uk − uk ¤ek ) u = (u1 , · · · , un ) ∈ Ξ(e).
(11)
1≤k≤n
Remark that all Peirce spaces Zk,j (e) as well as Ξ(e) are invariant under the canonical adjoint operation of Z = L(Z). By ([5], lemma 3.15) for 1 ≤ k 6= j ≤ n the Peirce projector onto the space Zkj (e) = Z1/2 (ek ) ∩ Z1/2 (ej ) is the operator Pkj (e) = 4Q(ek , ej )2 . Therefore the map Zkj (e) → Zkj (e) defined by w 7→ w# := 2Q(ek , ej )w
(w ∈ Zkj (e))
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is a conjugate-linear involution on Zkj (e) which induces a decomposition of this space into the direct sum of the ±-eigensubspaces of Q(ek , ej ). Finally by ([1] th. 3.1), for u = (u1 , · · · , un ) in the selfadjoint part Ξ(e)s of Ξ(e) the triple derivation Φa (u) is actually a C∗ -algebra derivation and we define the space Dera (Z) := {Φa (u) : u = u∗ ∈ Ξ(e)s }. P Lemma 3.2. Let a = k λk ek be the spectral resolution of a normal algebraic element P in Z. Let u = uk ∈ Z1/2 (ek ) and uk = u∗k are selfadjoint elements for k uk where P k = 1, · · · , n. Let uk = uk0 + r6=k urk be the joint Peirce decomposition of uk relative to e. Then 1 X 1 X (λj − λk )ukj − λk uk0 . (12) [Φa (u)]a = 2 1≤j, k≤n 2 1≤k≤n j6=k
Proof. First we check that 1 {ek uk ej } = Q(ek , ej )(uk ) = ukj 2
for k 6= j
and {ek uk ek } = 0 (1 ≤ j, k ≤ n). (13)
Clearly {ek uk ek } = Q(ek )uk ∈ Q(ek )Z1/2 (ek ) = 0 by the Peirce rules. For k 6= j we have T uk,0 ∈ Z1/2 (ek ) ∩ r6=k Z0 (er ) ⊂ Z0 (ej ) hence {ek uk0 ej } = 0. By ([5], lemma 3.15) for 1 ≤ k 6= j ≤ n we have Pkj (e) = 4Q(ek , ej )2 . Since the uk are ∗-selfadjoint (hence also #-selfadjoint), we have by the Peirce rules {ek
X
ukr ej } =
1≤r≤n r6=k
1 1 {ek ukr ej } = {ek ukj ej } = Q(ek , ej )ukj = u# kj = ukj . 2 2 1≤r≤n X r6=k
As a consequence X X ¢ 1 X λj ukj . ek ¤uk a = λj {ek ukj ej } = 2 1≤k,j≤n 1≤k≤n 1≤k≤n 1≤j≤n
¡ X
(14)
j6=k
Next we use uk ∈ Z1/2 (ek ) to compute X X X X ¢ 1 X uk ¤ek a = λj {uk ek ej } = {uj , ej , ej } = λj uj0 + λj ujk . 2 1≤j≤n 1≤j≤n 1≤k≤n 1≤k≤n1≤j≤n 1≤j,k≤n
¡ X
Collecting the results in (14) and (15) and using ukj = ujk one gets (12).
j6=k
(15) ¤
Corollary 3.3. Assume that in lemma (3.2) the algebraic element a is hermitian. Then the map Φa : u 7→ [Φa (u)](·) is a real-linear isomorphism of the Banach space Ξ(e)s onto Dera (Z). Proof. If a is hermitian then the λk are real numbers, hence [Φa (u)](·) ∈ Dera (Z). Clearly u 7→ Φa (u) is a real-linear map. By (12) the relation Φa (u) = 0 implies uk,j = 0 = uk,0
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since λk 6= λj and λj 6= 0, therefore u = 0. Moreover Φa is surjective. Indeed, let P δ ∈ Dera (Z) be arbitrarily given. Then δ = k (vk ¤ek − ek ¤vk ) for some v = (v1 , · · · , vn ) in Ξ(e)s , and by (12) we can recover v from the value δ(a) that the derivation δ takes at the point a ∈ Z. If we let πk,j (e) : Z → Zk,j (e) denote the Peirce joint projection relative to the family e, then (12) reads vk,j =
¡ ¢ 2 πk,j (e) δ(a) , λj − λk
vk,0 =
¡ ¢ −2 πk,0 (e) δ(a) . λk
Since the evaluation at a and the Peirce projections are continuous, so is Φ−1 a .
¤
Recall that a subset M ⊂ Z is called a real analytic submanifold if to every a ∈ M there are open subsets P, Q ⊂ Z and a closed real-linear subspace X ⊂ Z with a ∈ P and φ(P ∩ M ) = Q ∩ X for some bianalytic map φ : P → Q. If to every a ∈ M the linear subspace X = Ta M , called the tangent space to M at a, can be chosen to be topologically complemented in Z then M is called a direct submanifold of Z. Theorem 3.4. The selfadjoint orbits M = M (n, Λ, R) defined in (5) are closed real analytic direct submanifolds of Z, the tangent space at the point a ∈ M is the selfadjoint part of space Ξ(e) defined in (9) and a local chart at a is given by X u= uk 7→ [exp Φa (u)](a), u ∈ Ξ(e)s , (16) k
with Φa (u) =
P
k (ek ¤uk
− uk ¤ek ).
Proof. Fix one of the sets M = M (n, Λ, R) with M = M ∗ and a point a ∈ M with P spectral resolution a = k λk ek . We know by (3.1) that M is closed in Z. By the orthogonality properties of the ek , the successive powers of a have the expression al = λl1 e1 + · · · + λln en ,
1 ≤ l ≤ n,
where the determinant det(λlk ) 6= 0 does not vanish since it is a Vandermonde determinant and the λk are pairwise distinct. Thus the ek are polynomials in a whose coefficients are rational functions of the λk . Next we show that the tangent space Ta M to M at a can be identified with a real vector subspace of Ξ(e)s . Consider a smooth curve t 7→ a(t), t ∈ I, through a ∈ M where I is a neighbourhood of 0 ∈ R and a(0) = a. Each a(t) has a spectral resolution a(t) = λ1 e1 (t) + · · · + λn en (t), therefore the maps t 7→ ek (t), (1 ≤ k ≤ n), are smooth curves in the manifolds Mk of the projections in Z that have fixed finite rank rk = rank(ek ), whose tangent spaces at ek = ek (0) are the real spaces Z1/2 (ek )s (see [1] or [7]). Therefore uk :=
d |t=0 ek (t) ∈ Z1/2 (ek )s , dt
1 ≤ k ≤ n.
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By assumption a is hermitian, hence σ(a) ⊂ R and the tangent vector to t 7→ a(t) at P t = 0 then satisfies u = dtd |t=0 a(t) = k λk uk ∈ Ξ(e)s , thus Ta M can be identified with a vector subspace of Ξ(e)s . In fact Ta N coincides with that space as it easily follows from the following result that should be compared with ([1] th. 3.3) Indeed, as shown above we have Z = Ξ(e)s ⊕ Y for a certain direct subspace Y . The mapping Ξ(e)s ⊕ Y → Z defined by (x, y) 7→ F (x, y) := (exp Φa (x))y is a real-analytic and its Fr´echet derivative at (0, a) is ∂F |(0,a) (u, v) = [Φa (u)]a, ∂x ¡ ¢ ∂F |(0,a) (u, v) = exp Φa (0) v = v, ∂y
which is invertible according to (3.3). By the implicit function theorem there are open sets U, V with 0 ∈ U ⊂ X and a ∈ V ⊂ Y such that W := F (U × V ) is open in Z and F : U × V → W is bianalytic and the image F (U ) is a direct real analytic submanifold of Z. So it remains to show that F (U ) = W ∩ M . P The operator Φa (u) = k (uk ¤ek − ek ¤uk ), u ∈ Ξs (a), is an inner C∗ -algebra derivation of Z, hence h := exp Φa (z) is a C∗ -algebra automorphism of Z. Actually h lies in Aut◦ (Z), the identity connected component. In particular h preserves the algebraic character and the spectral decomposition, hence it preserves M and so F (U ) = {(exp Φa (u))a : z ∈ U } ⊂ M. To complete the proof, let x ∈ Ξs (e) be given. By (3.3) the operator Φa (·) is a surjective real linear homeomorphism of Ξs (e) hence u := Φ−1 a (x) ∈ Ξs (e), and by the above paragraph t 7→ (exp Φa (tw))a, |t| < δ for some δ > 0, is a curve in M whose tangent vector at a is Φa (u) = x. Thus Ξs (e) ⊂ Ta M . ¤ The proof of (3.4) has the following corollaries Corollary 3.5. The action of the Banach Lie group G = Aut(Z) on M admits local realanalytic cross sections, more precisely: To every a ∈ M , there is an open neighbourhood Na of a in M and a real-analytic function χ : Na → G such that [χ(b)](a) = b for all b ∈ Na . Proof. According to the proof of theorem (3.4), for each element b in a neighbourhood ¡ ¢ Na of a there is a unique u ∈ Ξ(e), say u = u(b), such that [exp Φa u(b) ](a) = b. Set ¡ ¢ χ(b) := exp Φa u(b) ∈ G. Then b 7→ χ(b) satisfies the requirements. ¤
Corollary 3.6. If dim Z < ∞ then the selfadjoint sets M = M (n, Λ, R) are compact real analytic direct submanifolds of Z.
Let M be a real analytic manifold and T M its corresponding tangent bundle. Recall that a norm on T M is a lower semicontinuous function α : T M → R such that the
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restriction of α to to every tangent space Tx M , x ∈ M , is a norm on Tx M with the following property: there is a neighbourhood N of x in M which can be realized as a domain in a real Banach space E such that ckak ≤ α(u, a) ≤ Ckak for all (u, a) ∈ T N ≈ N × E and suitable constants 0 < c ≤ C. The manifold M together f, α with a fixed norm α on T M is called a real Banach manifold. If (M ˜ ) is another real f Banach manifold, then we say that a real analytic mapping φ : M → M is a contraction if α ˜ ◦ Tφ ≤ α and we say that φ is an isometry if α ˜ ◦ Tφ = α. Let M be a connected real analytic Banach manifold with a norm α and denote by L the group of all real analytic surjective isometries of g : M → M . An element s in L is called an involution of M if s2 = IdM and an involution s is called a symmetry at the point x ∈ M if x is an isolated fixed point of s. Such an involution is unique if it exists. A connected real analytic Banach manifold M is said to be symmetric if there exists a f is said to be a morphism of symmetry at every point x ∈ M . A mapping h : M → M f if h is real analytic and h ◦ sx = sh(x) ◦ h holds for all the symmetric manifolds M and M x ∈ M. S Theorem 3.7. Let H = n,Λ,R M (n, Λ, R) be the set of all hermitian algebraic elements of finite rank in Z = L(H). Then each component M = M (n, Λ, R) is a closed realanalytic direct Banach submanifold of Z. For each a ∈ M , the Peirce reflection Sa in Z around the support a = supp(a) of a is real-analytic involution M for which a is a fixed point. The set FixM (Sa ) of fixed points of Sa in M is real-analytic direct submanifold of Z. If M is the orbit of a finite rank projection them M is a symmetric manifold. Proof. Fix any orbit M (n, Λ, R) and any point a ∈ M . Set e = (e1 , · · · , en ) where P a = k λk ek is the spectral resolution of a. Let N and E := Ta M ≈ Ξ(e)s denote the neighbourhood of a in M and the Banach space for which the tangent bundle satisfies T N ≈ N × E. Define a function α : N × E → R by α(b, u) := kuk,
b∈N
u ∈ Ξ(e)s ,
where k · k is the operator norm on Z. Since M is an orbit under the group G, we can extend α in a unique way to a G-invariant norm on M in a natural way. Thus (M, α) is a Banach manifold for which G (and in fact Aut(Z)) acts as a group of isometries. For a tripotent e ∈ Tri(Z), the Peirce reflection around e is the linear map Se : = Id − P1/2 (e) or in detail z = z1 + z1/2 + z0 7→ Se (z) = z1 − z1/2 + z0 where zk are the Peirce e-projections of z, (k = 1, 1/2, 0). Recall that Se is an involutory triple automorphism of Z with Se (e) = e, and clearly the set FixZ (Se ) of the fixed points of Se in Z is FixZ (Se ) = {z ∈ Z : P1/2 (e)z = 0}. If e is a projection in Z = L(H) (taken as a tripotent) then Se is a C∗ -algebra automorphism of Z, hence Se preserves the set of projections, the orthogonality relations and ranks as well as the hermitian character of the f := Se M of elements in Z. In particular, Se transforms each orbit M onto another orbit M
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the set A of algebraic elements. Given a ∈ M , the preceding considerations apply to the P P P projection a = supp(a). By the Peirce rules we have Q(a)a = { j ej k λk ek l el } = P k λk ee = a, hence P1 (a)a = a and Sa (a) = a, therefore Sa M = M and Sa |M is a real-analytic involution of M for which a is a fixed point. For n = 1 it is known that M is symmetric ([1], [14] prop. 4.3). Thus we analyze the the set FixM (Sa ) of the fixed points of Sa in M for n > 1. By the previous discussion FixM (Sa ) = M ∩ FixZ (Sa ) = M ∩ {z ∈ Z : P1/2 (a)z = 0} = M ∩ kerP1/2 (a),
(17)
which is a real analytic submanifold of M . The points of M in a neighbourhood U of a in M have the form z = [exp Φa (u)]a. Hence any smooth curve t 7→ z(t) in FixU (Sa ) passing through a with tangent vector u ∈ Ξ(e)s has the form z(t) = [exp Φa (tu)]a and will therefore satisfy P1/2 (a)[exp Φa (tu)]a = 0 for all t in some interval around t = 0. By taking the derivative at t = 0 we get P1/2 (a)[Φa (u)]a = 0, the tangent space to FixM (Sa ) at a being the set of solutions u ∈ Ξ(e) of the above equation. By (10) it suffices to find the solutions in the subspaces Zk,j (e) and Zk0 (e). ¡ ¢ Using the Peirce rules together with (12) and the expression P1/2 (a) = 2 a¤a − Q2 (a) it is a routine exercise to show that M {u ∈ Ξ(e)s : P1/2 (a)[Φa (u)]a = 0} = Zk0 (e)s . 1≤k≤n
Now for n ≥ 2 (and dim H ≤ 3) it is immediate to see that we have Zk0 (e) 6= {0} for some 1 ≤ k ≤ n, hence‡ FixM (Sa ) does not reduce to an isolated point and S(a) is not a symmetry of M . Note that if M is symmetric then the symmetry of M around a must be S(a). ¤
4
The Jordan connection on M (n, Λ, R)
By (3.4) the tangent space T Ma to M at the point a can be identified with the real space Ξ(e)s , a direct summand in Z, the projector onto which is denoted by PΞ (e). As any Peirce projector, PΞ (e) is Aut(Z)-invariant, that is, PΞ (h(e)) = h PΞ (e) h−1 ,
h ∈ Aut(Z).
(18)
Recall that a smooth vector field X on M is a smooth function X : M → T M such that π ◦ X = IdM , where π : T M → M is the canonical projection. Thus X(x), the value of X at x ∈ M , is a pair X(x) = (x, Xx ) where Xx ∈ Tx M . For all points x in a neighborhood of a, the tangent spaces Tx M are unambiguously identified with the Banach space E ≈ Ξ(e)s ֒→ Z, hence smooth vector fields on M will be locally identified with smooth Z-valued functions X : M → Z such that X(x) ∈ Ξ(e)s for all x ∈ M . ‡
When n = 1 the all summands Zk0 (e) reduce to 0.
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We let D(M ) be the Lie algebra of smooth vector fields on M . For Y ∈ D(M ), we let Ya′ be the Fr´echet derivative of Y at a. Thus Ya′ is a bounded linear operator Z → Z, hence Ya′ Xa ∈ Z and it makes sense to take the projection PΞ (e)Ya′ Xa ∈ Ξ(e)s ≈ Ta M . Definition 4.1. We define a connection ∇ on M by (∇X Y )a := PΞ (e) Ya′ Xa ,
X, Y ∈ D(M ),
a ∈ M.
Note that if a is a projection, then ∇ coincides with the affine connection defined in ([1] def 3.6) and [7]. It is a matter of routine to check that ∇ is an affine connection on M , that it is G- invariant and torsion-free, i. e., g (∇X Y ) = ∇g(X) g (Y ),
g ∈ G,
where (g X)a := ga′ (Xga−1 ) for all X ∈ D(M ), and T (X, Y ) := ∇X Y − ∇Y X − [XY ] = 0,
X, Y ∈ D(M ).
Since ∇ has been defined in terms of the Jordan structure of Z we refer to it as the Jordan connection on M . Theorem 4.2. Let the manifold M be defined as in (5). Then the ∇-geodesics of M through the point a ∈ M are the curves γ(t) := [exp t Φa (u)]a, (t ∈ R), where a ∈ M and u ∈ Ξ(e)s . Proof. Recall that the geodesics of ∇ are the curves t 7→ γ(t) = satisfy the second order ordinary differential equation ¡
∇γ(t) γ(t) ˙ ˙
¢
γ(t)
P
k
λk ek (t) ∈ M that
= 0.
Let u ∈ Ξ(e)s . Then Φa (u) is an inner C∗ -algebra derivation of Z and h(t) := exp t Φa (u) is an inner C∗ -automorphism of Z. Thus h(t)a ∈ M and t 7→ γ(t) is a curve in the manifold M . Clearly γ(0) = a and taking the derivative with respect to t at t = 0 we get by the Peirce rules γ(t) ˙ = Φa (u)γ(t) = h(t)[Φa (u)]a, 2
2
γ¨ (t) = [Φa (u) ]γ(t) = h(t)[Φa (u) ]a,
γ(0) ˙ = [Φa (u)]a ∈ Ξ(e)s , γ¨ (0) = Φa (u)γ(0) ˙ ∈ [Φa (u)]Ξ(e)s .
In particular PΞ (e)[Φa (u)2 ]a = 0. The definition of ∇ and the relation (18) give ³ ´ ¡ ¢ ′ ∇γ(t) γ(t) ˙ = PΞ (γ(t)) γ(t) ˙ γ(t) γ(t) ˙ = PΞ (γ(t)) γ¨ (t) = ˙ γ(t) ¡ ¢ ¡ ¢ PΞ h(t)a h(t)[Φa (u)]a = h(t)PΞ e [Φa (u)2 ]a = 0
for all t ∈ R. Recall that by (3.3) the mapping u 7→ [Φa (u)]a is a linear homeomorphism of Ξ(e)s . Since geodesics are uniquely determined by the initial point γ(0) = a and the
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initial velocity γ(0) ˙ = [Φa (u)]a, the above shows that family of curves in (4.2) with a ∈ M and u ∈ Ta M ≈ Ξ(e)s are all geodesics of the connection ∇. ¤ S Proposition 4.3. Let H = n,Λ,R M (n, Λ, R) be the set of all hermitian algebraic elements of finite rank in Z = L(H). Then each component M for which n = 1 admits a G-invariant Riemann structure for which ∇ is the Levi-Civita connection. Proof. First we assume that Ξ(e) is closed under the operation of taking triple product. Suppose that rank(a) = r < ∞ for a ∈ M . Then rank(ek ) ≤ r < ∞, (1 ≤ k ≤ n), hence the JB∗ -subtriple Z1/2 (ek ) has finite rank and so Z1/2 (ek ) is a reflexive Banach T space (see [10] or [2] prop. 9.11). The closed subspace Zk0 (e) = Z1/2 (ek ) ∩ j6=k Z0 (ej ) L is also reflexive and so is the finite ℓ∞ -direct sum Ξ(e) = 1≤k≤n Zk0 (e). But Ξ(e) is a JB∗ -triple by assumption and being reflexive is linearly homeomorphic to a Hilbert space. Thus the tangent space Ta M ≈ Ξ(e)s is linearly homeomorphic to a real Hilbert space under a suitable scalar product. We may take for instance the algebraic inner product on Ξ(e)s (denoted by h· , ·i) ([2] page 161) and we can define a Riemann metric on M by ga (X, Y ) := hXa , Ya i,
X, Y ∈ D(M ),
a ∈ M.
(19)
Remark that g has been defined in algebraic terms, hence it is G-invariant. Moreover, ∇ is compatible with the Riemann structure, i. e. X g(Y, W ) = g(∇X Y, W ) + g(Y, ∇X W ),
X, Y, W ∈ D(M ).
Therefore, ∇ is the only Levi-Civita connection on M and each symmetry of M (as induced by a Peirce reflection) is an isometry. Remark that for n = 1 the Peirce joint decomposition of Z relative to e = e reduces to (1) and so Ξ(e) = Z1/2 (e) is a subtriple of Z. Actually this is the only case in which Ξ(e) is closed under triple product. ¤
References [1] C.H. Chu and J.M. Isidro: “Manifolds of tripotents in JB∗ -triples”, Math. Z., Vol. 233, (2000), pp. 741–754. [2] S. Dineen: The Schwarz lemma, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1989. [3] L.A. Harris: “Bounded symmetric homogeneous domains in infinite dimensional spaces”, In: Proceedings on Infinite dimensional Holomorphy, Lecture Notes in Mathematics, Vol. 364, 1973, Springer-Verlag, Berlin, 1973, pp. 13–40 ¨ [4] U. Hirzebruch: “Uber Jordan-Algebren und kompakte Riemannsche symmetrische R¨aume von Rang 1”, Math. Z., Vol. 90, (1965), pp. 339–354. [5] G. Horn: “Characterization of the predual and ideal structure of a JBW∗ -triple”, Math. Scan., Vol. 61, (1987), pp. 117–133.
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[6] J.M. Isidro: The manifold of minimal partial isometries in the space L(H, K) of bounded linear operators”, Acta Sci. Math. (Szeged), Vol. 66, (2000), pp. 793–808. [7] J.M. Isidro and M. Mackey: “The manifold of finite rank projections in the algebra L(H) of bounded linear operators”, Expo. Math., Vol. 20(2), (2002), pp. 97–116. [8] J.M. Isidro and L. L. Stach´o: “On the manifold of finite rank tripotents in JB∗ -triples”, J. Math. Anal. Appl., to appear. [9] W. Kaup: “A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces”, Math. Z., Vol. 183, (1983), pp. 503–529. ¨ [10] W. Kaup: “Uber die Klassifikation der symmetrischen Hermiteschen Mannigfaltigkeiten unendlicher Dimension, I, II”, Math. Ann., Vol. 257, (1981), pp. 463–483 and Vol. 262, (1983), pp. 503–529. [11] W. Kaup: “On Grassmannians associated with JB∗ -triples”, Math. Z., Vol. 236, (2001), pp. 567–584. [12] O. Loos: Bounded symmetric domains and Jordan pairs Mathematical Lectures, University of California at Irvine, 1977. [13] T. Nomura: “Manifold of primitive idempotents in a Jordan-Hilbert algebra”, J. Math. Soc. Japan, Vol. 45, (1993), pp. 37–58. [14] T. Nomura: “Grassmann manifold of a JH-algebra”, Annals of Global Analysis and Geometry, Vol. 12, (1994), pp. 237–260. [15] H. Upmeier: Symmetric Banach manifolds and Jordan C∗ -algebras, North Holland Math. Studies, Vol. 104, Amsterdam, 1985.
CEJM 3(2) 2005 203–214
On the Riemann zeta-function and the divisor problem II Aleksandar Ivi´c∗ Katedra Matematike RGF-a, Universiteta u Beogradu, - uˇsina 7, 11000 Beograd, Serbia and Montenegro D
Received 21 December 2004; accepted 21 February 2005 Abstract: Let ∆(x) denote the error term in the Dirichlet divisor problem, and E(T ) the error term in the asymptotic formula for the mean square of |ζ( 12 + it)|. If E ∗ (t) = E(t) − 2π∆∗ (t/2π) with ∆∗ (x) = −∆(x) + 2∆(2x) − 12 ∆(4x), then we obtain Z
T
|E ∗ (t)|5 dt ≪ε T 2+ε
0
and Z
T
544
601
|E ∗ (t)| 75 dt ≪ε T 225 +ε .
0
It is also shown how bounds for moments of |E ∗ (t)| lead to bounds for moments of |ζ( 12 + it)|. c Central European Science Journals. All rights reserved.
Keywords: Dirichlet divisor problem, Riemann zeta-function, power moments of |ζ( 12 + it)|, power moments of E ∗ (t) MSC (2000): 11N37, 11M06
1
Introduction and statement of results
This work is the continuation of [8], where several aspects of the connection between the divisor problem and ζ(s), the zeta-function of Riemann, were investigated. As usual, let ∆(x) =
X
d(n) − x(log x + 2γ − 1)
n≤x
∗
E-mail:
[email protected],
[email protected]
(1.1)
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A. Ivi´c / Central European Journal of Mathematics 3(2) 2005 203–214
denote the error term in the Dirichlet divisor problem, and Z T T 2 1 E(T ) = |ζ( 2 + it)| dt − T log + 2γ − 1 , 2π 0
(1.2)
where d(n) is the number of divisors of n, γ = −Γ′ (1) = 0.577215 . . . is Euler’s constant. Instead of ∆(x) we work with the modified function ∆∗ (x) (see M. Jutila [10]), where ∆∗ (x) := −∆(x) + 2∆(2x) − 12 ∆(4x).
(1.3)
M. Jutila (op. cit.) investigated both the local and global behaviour of the difference E ∗ (t) := E(t) − 2π∆∗ and in particular he proved that Z T
t , 2π
(E ∗ (t))2 dt ≪ T 4/3 log3 T.
(1.4)
0
In [8] this bound was complemented with the new bound Z T (E ∗ (t))4 dt ≪ε T 16/9+ε ;
(1.5)
0
neither (1.4) or (1.5) seem to imply each other. Here and later ε denotes positive constants which are arbitrarily small, but are not necessarily the same ones at each occurrence. Our first aim is to obtain another bound for moments of |E ∗ (t)|. This is given by Theorem 1.1. We have
Z
T
|E ∗ (t)|5 dt ≪ε T 2+ε .
(1.6)
0
From (1.4), (1.6) and H¨older’s inequality for integrals, it follows that Z
T ∗
4
|E (t)| dt =
0
≤
Z
T
|E ∗ (t)|2/3 |E ∗ (t)|10/3 dt 0
Z
≪ε T
T
1/3 Z |E (t)| dt
0 16/9+ε
∗
2
0
,
T
2/3 |E (t)| dt ∗
5
which implies (1.5). This means that (1.6) and (1.4) together are stronger than (1.5). Another result of a more general nature (for the definition and properties of exponent pairs see e.g., [3] or [6, Chapter 2]) is contained in Theorem 1.2. Let (κ, λ) be an exponent pair such that 2λ ≤ 1 + κ, and V ≥ T
1+λ−2κ +ε 3(2−κ)
.
(1.7)
A. Ivi´c / Central European Journal of Mathematics 3(2) 2005 203–214
205
Let tr ∈ [T, 2T ] (r = 1, . . . , R) be points such that |tr − ts | ≥ V (r 6= s) and |E ∗ (tr )| ≥ V (r = 1, . . . , R). Then R ≪ε T 1+ε V −3 + T
1+4κ+λ +ε 3κ
V−
3κ+2 κ
.
(1.8)
From Theorem 1.2 we can obtain specific bounds for moments of |E ∗ (t)|, provided we choose the exponent pair (κ, λ) appropriately. The optimal choice of the exponent pair is hard to determine, since several conditions have to hold (see e.g., (5.5)). However, by trying some of the standard exponent pairs one can obtain a bound which is not far from the optimal bound that the method allows. For instance, with the exponent pair (κ, λ) = (75/197, 104/197) (this exponent pair arises, in the terminology of exponent pairs, as (75/197, 104/197) = BA3 BA3 B(0, 1) ) we can obtain Theorem 1.3. We have Z
T
601
544
|E ∗ (t)| 75 dt ≪ε T 225 +ε .
(1.9)
0
One of the main reasons for investigating power moments of |E ∗ (t)| is the possibility to use them to derive results on power moments of |ζ( 12 + it)|, which is one of the main themes in the theory of ζ(s). A result in this direction is given by Theorem 1.4. Let k ≥ 1 be a fixed real, and let c(k) be such a constant for which Z Then we have Z
T
|E ∗ (t)|k dt ≪ε T c(k)+ε .
(1.10)
|ζ( 12 + it)|2k+2 dt ≪ε T c(k)+ε .
(1.11)
0
T
0
The constant c(k) must satisfy c(k) ≥ 1.
(1.12)
This is obvious if k is an integer, as it follows from [6, Theorem 9.6]. If k is not an integer, 2k+2 then this result yields (p = 2[k]+2 > 1) T ≪
Z
0
T
|ζ( 12
+ it)|
2[k]+2
dt ≤
Z
T 0
|ζ( 21
+ it)|
2k+2
1/p dt T 1−1/p
by H¨older’s inequality for integrals. After simplification (1.12) easily follows again. Corollary 1.5. We have Z
T 0
|ζ( 12 + it)|12 dt ≪ε T 2+ε .
(1.13)
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This follows from Theorem 1.1 and Theorem 1.4 (with k = 5), and is the well-known result of D.R. Heath-Brown [2], who had log17 T in place of T ε on the right-hand side of (1.13). Corollary 1.6. We have Z
T
0
|ζ( 21 + it)|
1238 75
601
dt ≪ε T 225 +ε .
(1.14)
). The bound (1.14) This follows from Theorem 1.3 and Theorem 1.4 (with k = 544 75 does not follow from (1.13) (and the strongest pointwise estimate for |ζ( 21 + it)|), but on the other hand (1.13) does not follow from (1.14). In principle, (1.14) could be used for deriving zero-density bounds for ζ(s) (see e.g., [6, Chapter 10]), but very likely its use would lead to very small improvements (if any) of the existing bounds.
Acknowledgment I wish to thank Prof. Matti Jutila for valuable remarks.
2
The necessary lemmas
In this section we shall state the lemmas which are necessary for the proof of Theorem 1.1. Lemma 2.1. [O. Robert–P. Sargos [11]]. Let k ≥ 2 be a fixed integer and δ > 0 be given. Then the number of integers n1 , n2 , n3 , n4 such that N < n1 , n2 , n3 , n4 ≤ 2N and 1/k
|n1
1/k
+ n2
1/k
− n3
1/k
− n4 | < δN 1/k
is, for any given ε > 0, ≪ε N ε (N 4 δ + N 2 ).
(2.1)
This Lemma was crucial in obtaining the asymptotic formulas for the third and fourth moment of ∆(x) in [9]. Lemma 2.2. Let T ε ≪ G ≪ T / log T . Then we have Z ∞ 2 2 2 ∗ E (T ) ≤ √ E ∗ (T + u) e−u /G du + Oε (GT ε ), πG 0 and
2 E (T ) ≥ √ πG ∗
Z
∞
E ∗ (T − u) e−u
2 /G2
du + Oε (GT ε ).
0
Lemma 2.2 follows on combining Lemma 2.2 and Lemma 2.3 of [8].
(2.2)
(2.3)
A. Ivi´c / Central European Journal of Mathematics 3(2) 2005 203–214
207
The next lemma is F.V. Atkinson’s classical explicit formula for E(T ) (see [1, 6] or [7]). Lemma 2.3. Let 0 < A < A′ be any two fixed constants such that AT < N < A′ T , and let N ′ = N ′ (T ) = T /(2π) + N/2 − (N 2 /4 + NT /(2π))1/2 . Then E(T ) = Σ1 (T ) + Σ2 (T ) + O(log2 T ),
(2.4)
where Σ1 (T ) = 21/2 (T /(2π))1/4
X
(−1)n d(n)n−3/4 e(T, n) cos(f (T, n)),
(2.5)
n≤N
Σ2 (T ) = −2
X
d(n)n−1/2 (log(T /(2πn))−1 cos(T log(T /(2πn)) − T + π/4),
(2.6)
n≤N ′
with p √ πn/(2T ) + 2πnT + π 2 n2 − π/4 f (T, n) = 2T arsinh √ √ = − 41 π + 2 2πnT + 61 2π 3 n3/2 T −1/2 + a5 n5/2 T −3/2 + a7 n7/2 T −5/2 + . . . ,
e(T, n) = (1 + πn/(2T )) = 1 + O(n/T ) and arsinh x = log(x +
√
−1/4
n
(2.7)
(2T /πn)
1/2
(1 ≤ n < T )
o−1 p arsinh ( πn/(2T ) )
(2.8)
1 + x2 ).
Lemma 2.4. [M. Jutila [10]]. If A ∈ R is a constant, then we have Z ∞ √ √ √ √ 3/2 −1/2 1 3 8πnT + 6 2π n T +A = α(u) cos( 8πn( T + u) + A)du, cos
(2.9)
−∞
where α(u) ≪ T 1/6 for u 6= 0, α(u) ≪ T 1/6 exp(−bT 1/4 |u|3/2 )
(2.10)
for u < 0, and α(u) = T 1/8 u−1/4 d exp(ibT 1/4 u3/2 ) + d¯exp(−ibT 1/4 u3/2 ) + O(T −1/8 u−7/4 )
(2.11)
for u ≥ T −1/6 and some constants b (> 0) and d.
3
The proof of Theorem 1.1
The proof is on the lines of [8]. We seek an upper bound for R, the number of points {tr } ∈ [T, 2T ] (r = 1, . . . , R) such that |E ∗ (tr )| ≥ V ≥ T ε and |tr − ts | ≥ V for r 6= s.
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We consider separately the points where E ∗ (tr ) is positive or negative. Suppose the first case holds (the other one is treated analogously). Then from Lemma 2.2 we have Z ∞ 2 2 2 ∗ V ≤ E (tr ) ≤ √ E ∗ (tr + u) e−u /G du + Oε (GT ε ). (3.1) πG 0 The integral on the right-hand side is simplified by Atkinson’s formula (Lemma 2.3) and the truncated formula for ∆∗ (x) (see [8, eq. (6)]), as in [8]. We take G = cV T −ε (with sufficiently small c > 0) to make the O-term in (3.1) ≤ 21 V , raise everything to the fourth power and sum over r. By H¨older’s inequality we obtain Z 2T 4 −1 ε RV ≪ε V T max ϕ(t) Σ44 (X, N; u) + Σ45 (X, N; u) + Σ46 (X; u) dt, (3.2) |u|≤G log T
T /2
with the notation introduced in (2.7), (2.8) and [8]: X Σ4 (X, N; u) : = t1/4 (−1)n d(n)n−3/4 e(t + u, n) cos(f (t + u, n)),
(3.3)
X
Σ5 (X, N; u) : = t1/4
X
(−1)n d(n)n−3/4 cos(
X
Σ6 (X; u) :=
X
t−1/4 (−1)n d(n)n3/4 cos(
n≤X 1/3−ε
p
p
8πn(t + u) − π/4),
8πn(t + u) − π/4).
(3.4)
Here we have X = T , N = T G−2 log T , and ϕ(t) is a smooth, nonnegative function supported in [T /2, 5T /2] , such that ϕ(t) = 1 when T ≤ t ≤ 2T . The basic idea is that the contributions of Σ6 (X; u) and Σ5 (X, N; u) will be approximately equal at X, and the same will be true of Σ4 (X, N; u) as well. In the latter case, as was discussed in detail in [8], one has to use Lemma 2.4 to deal with the complications arising from the presence of cos(f (t + u, n)) in (3.3). The difference from [8] is that the choice G = cV T −ε leads directly to (3.2), which is in a certain sense optimal, while in [8] the choice was N = T 5/9 . Proceeding now as in [8] (here Lemma 2.1 with k = 2 was crucial) we obtain RV 4 ≪ε V −1 T ε (T 3/2 N 1/2 + T 2 X −1 + T −1/2 X 13/2 + X 5 ) ≪ε V −1 T ε (T 2 V −1 + T 5/3 ) ≪ε T
2+ε
V
−2
(3.5)
,
since V < T 1/3 in view of the best known estimates for ∆(x) and E(t). Namely with suitable C > 0 one has (see M.N. Huxley [3, 4]) ∆(x) ≪ x131/416 logC x, 131/416 = 0.3149038 . . . , E(T ) ≪ T 72/227 logC T, 72/227 = 0.3171806 . . . .
(3.6)
Therefore (3.5) yields the large values estimate R ≪ε T 2+ε V −6 , and Theorem 1.1 easily follows, as in [5] or [6, Chapter 13] for moments of ∆(x).
A. Ivi´c / Central European Journal of Mathematics 3(2) 2005 203–214
4
209
The proof of Theorem 1.2
We start again from (3.1), choosing G = cV T −ε (< 12 V ), T = tr , so that we have Z ∞ 2 ∗ ∗ −1 E (tr ) ≥ V, E (tr ) ≪ G E ∗ (tr + u)e−(u/G) du
(4.1)
0
in case E ∗ (tr ) > 0, and the case of negative values is analogous. We relabel the points for which (4.1) holds in the sense that it will hold for r = 1, . . . , R. The proof is similar to the proof of (13.52) of Theorem 13.8 of [6]. To remove the function d(n) from the sums in (3.3)–(3.4) we use the inequality (see the Appendix of [6]) X X |(ξ, φr )|2 ≤ ||ξ||2 max |(φr , φs )|, (4.2) r≤R
r≤R
s≤R
∞ where for two complex vector sequences a = {an }∞ n=1 , b = {bn }n=1 the inner product is defined as ∞ X an¯bn . (a, b) = n=1
We shall also use (3.3)–(3.4) with N = T G−2 log T . We shall consider separately the P P points where | | ≫ V when equals Σ4 (X, N; u), Σ5 (X, N; u) or Σ6 (X; u) (|u| ≤ G log T ), as the case may be. Taking the maximum over |u| ≤ G log T over the whole sum, we may relabel the points such that they are called again t = tr , r ≤ R. Moreover, let R0 denote the number of such tr ’s (in each case) lying in an interval of length T0 , where T0 is a function of V and T that will be determined later. Thus V ≤ T0 has to hold and R ≪ R0 (1 + T /T0 ). (4.3) As in the proof of Theorem 1.2, the choice of X will be X = T 1/3−ε ,
P P P when the largest term in 6 is approximately equal to the smallest term in 4 and 5 . This choice exploits the specific structure of the function E ∗ (t), and leads to a better bound than was possible for large values of ∆(x) in Chapter 13 of [6]. Namely in the latter case the maximum occurred at n = T G−2 log T , but in our case X = T 1/3−ε < T G−2 log T , since V < T 1/3−ε must hold in view of (3.6). For example, from (3.4) and (4.2) (in case P | 6 | ≫ V holds) we obtain 2 2 √ X X log T R0 V 2 ≪ √ max (−1)n d(n)n3/4 ei 8πn(tr +u) T |u|≤G log T,M ≤X/2 r≤R0 M
2
log T ≪ √ T
max
|u|≤G log T,M ≤X/2,r≤R0
M ++
X X
s≤R0 ,s6=r M
M 5/2 log4 M ·
√ √ √ i 8πn( tr +u− ts +u)
e
!
(4.4)
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A. Ivi´c / Central European Journal of Mathematics 3(2) 2005 203–214
which corresponds to (13.60) of [6]. If we set √ √ √ f (x) = 8πx( tr + u − ts + u ), then we can use the first derivative test (Lemma 2.1 of [6]) to deduce that the contribution of x = n (in the last sum in (4.4)) for which |f ′ (x)| < 1/2 is √ X M √ √ ≪ | t + u − ts + u| r s≤R0 ,s6=r X √ 1 ≪ MT |tr − ts | s≤R0 ,s6=r √ ≪ MT V −1 log T, (4.5) since |tr − ts | ≥ V if r 6= s. The contribution of |f ′ (x)| ≥ 1/2 is estimated by the theory of exponent pairs. The portion of the last sum in (4.4) is, in this case, κ ≪ R0 |tr − ts |(MT )−1/2 M λ ≪ R0 T0κ M λ−κ/2 T −κ/2 , (4.6)
since |tr − ts | ≤ T0 . Therefore from (4.4)–(4.6) it follows that
5
κ
1
κ
R0 V 2 ≪ T −1/2 X 7/2 log6 T + X 3 V −1 log7 T + R0 T0κ X 2 +λ− 2 T − 2 − 2 log6 T 1+λ−2κ
≪ T 2/3 log6 T + T V −1 log7 T + R0 T0κ T 3 log6 T. (4.7) P P The contribution of large values of | 4 | and | 5 | is estimated analogously. We proceed, similarly as in (4.7), to obtain in these cases R0 V 2 ≪ T 1/2 log2 T max × |u|≤G log T,X≤M ≤T 1+ε V −2 2 √ X X × (−1)n d(n)n−3/4 ei 8πn(tr +u) r≤R0 M
≪ T
1/2
log2 T
M +
X X
s≤R0 ,s6=r M
≪ε T
1+ε
V
−1
M −1/2 log4 M · ! √ √ √ ei 8πn( tr +u− ts +u)
max
|u|≤G log T,r≤R0 ,X<M ≤T 1+ε V −2
+ R0 T0κ T 1/2−κ/2 log5 T
max
X<M ≤T 1+ε V −2
(4.8)
M λ−κ/2−1/2 .
The hypothesis in the formulation of the theorem was that 2λ ≤ κ + 1,
(4.9)
hence by combining (4.7) and (4.8) it follows that R0 V 2 ≪ε T 1+ε V −1 + R0 T0κ T
1+λ−2κ +ε 3
,
(4.10)
since T 2/3 ≤ T V −1 because V ≤ T 1/3 has to hold. If we choose 2
T0 = V κ T
2κ−1−λ − 2ε 3κ κ
(4.11)
A. Ivi´c / Central European Journal of Mathematics 3(2) 2005 203–214
211
then (4.10) reduces to R0 V 2 ≪ε T 1+ε V −1 , and the condition T0 ≥ V becomes V ≥ T
1+λ−2κ +ε 3(2−κ)
,
(4.12)
which is (1.7). Therefore (4.10) gives R ≪ R0 (1 + T /T0 ) ≪ε T 1+ε V −3 + T
1+4κ+λ +ε 3κ
V−
3κ+2 κ
,
thereby completing the proof of Theorem 1.2.
5
The proof of Theorem 1.3
With the choice (κ, λ) = (75/197, 104/197) it is seen that (1.7) and (1.8) of Theorem 1.2 reduce to 601
619
151
R ≪ε T ε (T V −3 + T 225 V − 75 ) (V ≥ T 957 +ε , Let JV (T ) = and write
Z
2T ∗
|E (t)| T
544 75
n
151 957
= 0.157784 . . . ).
(5.1)
o t ∈ [T, 2T ] : V ≤ |E ∗ (t)| < 2V ,
dt ≪ε log T maxε V ≥T
Z
544
|E ∗ (t)| 75 dt + T 1+ε .
(5.2)
JV (T )
For V ≤ T 151/957+ε we have, on using (1.6) of Theorem 1.1, Z Z 544 169 ∗ |E (t)| 75 dt = |E ∗ (t)|5 |E ∗ (t)| 75 dt JV (T )
JV (T )
169 151
601
≪ε T 2+ 75 · 957 +ε ≤ T 225 .
(5.3)
Suppose now that V ≥ T 151/957+ε , and divide [T, 2T ] into subintervals of length V (the last of these subintervals may be shorter). Let |E ∗ (τj )| be the supremum of |E ∗ (t)| in the jth of these subintervals, and let further t1 , ..., tRV denote the τj ’s with even or odd indices such that the intersection of the jth subinterval and JV (T ) is non-empty. Then |tr − ts | ≥ V for r 6= s, and (5.1) gives 601
619
601
619
RV ≪ε T ε (T V −3 + T 225 V − 75 ) ≪ε T 225 +ε V − 75
(5.4)
for 188
V ≤ T 591 ,
188/591 = 0.3181049 . . . .
(5.5)
But in view of (3.6) it is seen that (5.5) is always satisfied (the choice of our exponent pair was made to ensure that this is indeed the case), and we obtain from (5.4) Z 544 619 544 601 619 601 |E ∗ (t)| 75 dt ≪ RV V 1+ 75 ≪ε T 225 +ε V − 75 V 75 = T 225 +ε . (5.6) JV (T )
Theorem 1.3 follows then from (5.2), (5.3) and (5.6), on replacing T by T 2−j and summing over j = 1, 2, . . . .
212
6
A. Ivi´c / Central European Journal of Mathematics 3(2) 2005 203–214
The proof of Theorem 1.4
To prove Theorem 1.4 it is enough to prove that R ≪ε T c(k)+ε V −2k−2 ,
(6.1)
where R is the number of points tr ∈ [T, 2T ] (r = 1, . . . , R), such that |ζ( 21 + itr )| ≥ V with |tr − ts | ≥ 1 for r 6= s and V ≥ T ε . We denote actually by R the number of points with even and odd indices, so that the intervals [tr − 31 , tr + 13 ] are disjoint. Then we have, using Theorem 1.2 of [7] with k = 2, δ = 13 , RV
2
≤
R X
|ζ( 21
2
+ itr )| ≪ log T
r=1
r=1
≪ log T
R Z X
J Z X j=1
τj +G τj −G
tr + 13 tr − 13
|ζ( 12 + it)|2 dt
|ζ( 12 + it)|2 dt,
(6.2)
where τj ∈ [T − G, T + G] (j = 1, . . . , J) is a system of points such that |τj − τℓ | ≥ 2G for j 6= ℓ and T ε ≤ G = G(T ) ≪ T . By the definition of E ∗ (t) we have Z τj +G |ζ( 12 + it)|2 dt = E(τj + G) − E(τj − G) + O(G log T ) τj −G τj + G τj − G ∗ ∗ ∗ ∗ = E (τj + G) − E (τj − G) + 2π∆ − 2π∆ + O(G log T ) 2π 2π = E ∗ (τj + G) − E ∗ (τj − G) + Oε (GT ε ). Here we used the fact that ∆∗ (x) − ∆∗ (y) ≪ε xε (x − y + 1)
(1 ≪ y ≤ x),
which follows from (1.1), (1.3) and d(n) ≪ε nε . This arithmetic property of d(n) is essential, since it makes it possible to connect the large values of |ζ( 12 + it)| to sums of values of E ∗ (t), and hence to exploit the special structure of the function E ∗ (t). If we worked only with E(t), we would obtain Theorem 1.4, where (1.10) has E ∗ (t) replaced by E(t). However, the existing estimates for the moments of |E(t)| (see [5] and Chapter 13 of [6]) are not as strong as the moments of |E ∗ (t)| (cf. (1.6) and (1.9)). Returning to the proof, note that (6.2) yields ( J ) X RV 2 ≪ε log T (E ∗ (τj + G) − E ∗ (τj − G)) + RGT ε , j=1
giving RV 2 ≪ε log T with
(
J X
(E ∗ (τj + G) − E ∗ (τj − G))
j=1
G = cV 2 T −ε
)
(6.3)
(6.4)
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213
P and sufficiently small c > 0. If we use Lemma 2.2 we may replace j E ∗ (τj + G) by its majorant Z ∞X J 2 2 2 √ E ∗ (τj + G + u) e−u /G du + RGT ε , πG 0 j=1 and similarly for the sum with E ∗ (τj − G). By H¨older’s inequality we have (since J ≤ R) Z
∞
Z
∞
0
≪
J X
E ∗ (τj + G + u) e−u
du
j=1
e
J X
−u2 /G2
0
≪R
2 /G2
j=1
Z
1− k1
≪ (GR)
|E ∗ (τj + G + u)|k
1− k1
∞
−u2 /G2
e 0
J X
! k1
|E ∗ (τj + G + u)|k du
j=1
Z
5T /2
! k1
|E ∗ (t)|k dt T /2
1
R1− k du
· log T,
! k1
1
G1− k
(6.5)
by breaking the system of points τj into ≪ log T subsystems with |τj − τℓ | ≥ G log T for ℓ 6= j. From (1.10) and (6.3)–(6.5) it follows that 1
RV 2 ≪ε T ε (RV 2 )1− k · T
c(k) +ε k
V −2 ,
which on simplifying yields R ≪ε T c(k)+ε V −2k−2 ,
(6.6)
and (6.6) implies easily (1.11) of Theorem 1.4. By the same method one also obtains γ ≤ c(k)/(k + 1) for every k ≥ 1, if γ : = inf{ g ≥ 0 : E ∗ (T ) ≪ T g }, but better bounds for γ can be derived from short interval results on E ∗ (t), provided they can be obtained. The existing results make it hard to even conjecture what should be the true value of γ.
References [1] F.V. Atkinson: “The mean value of the Riemann zeta-function”, Acta Math., Vol. 81, (1949), pp. 353–376. [2] D.R. Heath-Brown: “The twelfth power moment of the Riemann zeta-function”, Quart. J. Math. (Oxford), Vol. 29, (1978), pp. 443–462. [3] M.N. Huxley: Area, Lattice Points and Exponential Sums, Oxford Science Publications, Clarendon Press, Oxford, 1996. [4] M.N. Huxley: “Exponential sums and lattice points III”, Proc. London Math. Soc., Vol. 387, (2003), pp. 591–609.
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[5] A. Ivi´c: “Large values of the error term in the divisor problem”, Invent. Math., Vol. 71, (1983), pp. 513–520. [6] A. Ivi´c: The Riemann zeta-function, John Wiley & Sons, New York, 1985; 2nd Ed., Dover, Mineola, New York, 2003. [7] A. Ivi´c: The mean values of the Riemann zeta-function, LNs, Vol. 82, Tata Inst. of Fundamental Research, Bombay (distr. by Springer Verlag, Berlin etc.), 1991. [8] A. Ivi´c: “On the Riemann zeta-function and the divisor problem”, Central European J. Math., Vol. 2(4), (2004), pp. 1–15. [9] A. Ivi´c and P. Sargos: “On the higher moments of the error term in the divisor problem”, to appear. [10] M. Jutila: “Riemann’s zeta-function and the divisor problem”, Arkiv Mat., Vol. 21, (1983), pp. 75–96; “Riemann’s zeta-function and the divisor problem II”, Arkiv Mat., Vol. 31, (1993), pp. 61–70. [11] O. Robert and P. Sargos: “Three-dimensional exponential sums with monomials”, J. reine angew. Math., in press.
CEJM 3(2) 2005 215–227
Closure Lukasiewicz algebras Manuel Abad∗, Cecilia Cimadamore†, Jos´e Patricio D´ıaz Varela‡, Laura Rueda§ , Ana Mar´ıa Suard´ıaz Departamento de Matem´ atica, Universidad Nacional del Sur, 8000 Bah´ıa Blanca, Argentina
Received 14 June 2004; accepted 10 January 2005 Abstract: In this paper, the variety of closure n-valued Lukasiewicz algebras, that is, Lukasiewicz algebras of order n endowed with a closure operator, is investigated. The lattice of subvarieties in the particular case in which the open elements form a three-valued Heyting algebra is obtained. c Central European Science Journals. All rights reserved.
Keywords: Lukasiewicz algebras, Heyting algebras, closure operators MSC (2000): 06D30, 03G20, 08B15
1
Introduction and Preliminaries
The notion of Lukasiewicz algebra of order n was introduced by Gr. C. Moisil [12], and was developed and investigated further by several authors such as A. Monteiro [13], L. Monteiro [15], R. Cignoli [7] and C. Sicoe [16, 17]. An extensive monograph on Lukasiewicz algebras was written by V. Boicescu et al. in [6]. We assume that the reader is familiar with the theory of n-valued Lukasiewicz algebras. For the basic properties, the reader is referred to [3, 6] and [7]. n−1 Definition 1.1. A Lukasiewicz algebra of order n is an algebra hL, ∧, ∨, ∼, {ϕi }i=1 , 0, 1i, n integer, n ≥ 2, of type (2, 2, 1, 1, 1, . . . , 1, 0, 0), such that hL, ∧, ∨, 0, 1i is a bounded distributive lattice and ∼, ϕ1 , ϕ2 , . . . , ϕn−1 satisfy ∗ † ‡ §
E-mail: E-mail: E-mail: E-mail:
[email protected] [email protected] [email protected] [email protected]
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∼∼ x = x,
(1)
∼ (x ∧ y) =∼ x∨ ∼ y,
(2)
ϕi (x ∨ y) = ϕi x ∨ ϕi y,
(3)
ϕi x∨ ∼ ϕi x = 1,
(4)
ϕi ϕj x = ϕj x,
(5)
ϕi ∼ x =∼ ϕn−ix,
(6)
ϕ1 x ≤ ϕ2 x ≤ . . . ≤ ϕn−1 x,
(7)
if ϕi x = ϕi y, 1 ≤ i ≤ n − 1, then x = y.
(8)
In any Lukasiewicz algebra of order n the following properties hold: ϕi (x ∧ y) = ϕi x ∧ ϕi y,
(9)
ϕi x∧ ∼ ϕi x = 0,
(10)
x ≤ y if and only if ϕi x ≤ ϕi y, for all i,
(11)
x ≤ ϕn−1 x,
(12)
ϕ1 x ≤ x,
(13)
ϕi 1 = 1, ϕi 0 = 0, for all i,
(14)
∼ x ∨ ϕn−1 x = 1,
(15)
x∧ ∼ ϕi x ∧ ϕi+1 y ≤ y, ∀ i < n − 1.
(16)
The class of Lukasiewicz algebras of order n can be defined by conditions (1)−(7), (12) and (16), and hence they form a variety which we will denote Ln . For L ∈ Ln , we denote B(L) the Boolean algebra of all complemented elements in L. It is known that x ∈ B(L) if and only if ϕi x = x, for all i . Since for every i = 1, . . . , n − 1, ϕi (L) = {x ∈ L : ϕi x = x}, it follows that B(L) = ϕi (L), 1 ≤ i ≤ n. It is also known that a Boolean algebra is a Lukasiewicz algebra of order n if we define ∼ x as the Boolean complement of x and ϕi x = x for all i. The most important example of a Lukasiewicz algebra of order n is the n-element chain Cn , n ≥ 2, 0 < 1/(n − 1) < . . . < (n − 2)/(n − 1) < 1, with the natural lattice structure and the operations ∼ and ϕi defined as j j =1− ; ∼ n−1 n−1
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ϕi
j n−1
217
0 if i + j < n = . 1 if i + j ≥ n
It is known [7] that Cn and its subalgebras are the subdirectly irreducible algebras of Ln , and that they are simple, so Ln is the variety generated by Cn . Definition 1.2. A Heyting algebra is an algebra hL, ∨, ∧, →, 0, 1i of type (2, 2, 2, 0, 0) such that hL, ∨, ∧, 0, 1i is a bounded distributive lattice where for every a, b ∈ L the set of elements x ∈ L such that a ∧ x ≤ b has greatest element a → b. Definition 1.3. A closure Lukasiewicz algebra of order n is an algebra hL, Ci where L is a Lukasiewicz algebra of order n and C is a unary operator defined on L fulfilling the following properties C0 = 0, (C1) Cx ∨ x = Cx,
(C2)
C(x ∨ y) = Cx ∨ Cy,
(C3)
CCx = Cx,
(C4)
Cϕi x = ϕi Cx, 1 ≤ i ≤ n − 1.
(C5)
The equational class of closure Lukasiewicz algebras of order n will be denoted by CLn . An important subvariety of CLn is the variety MLn of monadic Lukasiewicz algebras [1, 6, 15]. This variety is characterized within CLn by the equation C(x ∧ Cy) = Cx ∧ Cy. Another important subvariety of CLn is the variety C of closure Boolean algebras [5, 9, 2]. C consists of those algebras A in CLn that satisfy that for every element x ∈ A, ∼ x is the Boolean complement of x. Example 1.4. [15] The following is an example of a closure Lukasiewicz algebra which is not a monadic Lukasiewicz algebra. x Cx 1
u @ @ @u
c r a r
@ @ @u
@ @ @u
0
b
d
0
0
a
1
b
b
c
1
d
d
1
1
We have C(a ∧ Cd) = C(a ∧ d) = 0 and Ca ∧ Cd = d. The elements of C(L) are highlighted.
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With the operators C and ∼ we can define a new unary operator Q (an interior operator) by Qx = ∼ C ∼ x, for x ∈ L. This operator satisfies the following conditions: Q1 = 1,
(Q1)
Qx ∧ x = Qx,
(Q2)
Q(x ∧ y) = Qx ∧ Qy,
(Q3)
QQx = Qx,
(Q4)
Qϕi x = ϕi Qx, 1 ≤ i ≤ n − 1.
(Q5)
Observe that closure Lukasiewicz algebras can be defined by means of equations (Q1) to (Q5), and in that case, by defining Cx =∼ Q ∼ x we obtain the closure operator satisfying equations (C1) to (C5). If L ∈ CLn , then the set of open elements of L is Q(L) = {x ∈ L : Qx = x} and the set of closed elements of L is C(L) = {x ∈ L : Cx = x}. Q(L) and C(L) are anti-isomorphic sublattices of L such that ϕi (Q(L)) ⊆ Q(L) and ϕi (C(L)) ⊆ C(L), i = 1, . . . , n − 1. Observe that x ∈ Q(L) if and only if ∼ x ∈ C(L). Recall now the notions of axes and centers of a Lukasiewicz algebra of order n. We will see that the axes and centers of an algebra L ∈ CLn belong to C(L). Definition 1.5. [6] Let n ≥ 3. An algebra L ∈ Ln is said to be axled provided there exist a1 , . . . , an−2 ∈ L, called the axes of L, such that for every i = 1, . . . , n − 1 and every j = 1, . . . , n − 2, (a) if i + j < n, ϕi aj = 0, (b) if i + j ≥ n, ϕi aj ∨ ϕ1 x ≥ ϕn−1 x, for all x ∈ L. Lemma 1.6. If a1 , . . . , an−2 are axes of L ∈ CLn , then the aj are closed for every j. Proof. We are going to prove that Caj = aj for every j, which is equivalent to proving that Cϕi aj = ϕi aj , for every 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 2. If i + j < n, ϕi aj = 0 = Cϕi aj . In particular, ϕ1 aj = ϕ1 Caj = 0, for every j. Let us prove now that Cϕn−1 aj = ϕn−1 aj . Since, n − 1 + j ≥ n, by (b), ϕn−1 aj ∨ ϕ1 Caj ≥ ϕn−1 Caj , so ϕn−1 aj ≥ ϕn−1 Caj = Cϕn−1 aj , thus, Cϕn−1 aj = ϕn−1 aj , for every j. Suppose now that i + j ≥ n. Then, by (b), ϕi aj ∨ ϕ1 aj ≥ ϕn−1 aj . So ϕi aj ≥ ϕn−1 aj , and consequently, ϕi aj = ϕn−1 aj = Cϕn−1 aj . Hence Cϕi aj = ϕi aj . Definition 1.7. An algebra L ∈ Ln is centered if there exist c1 , . . . , cn−2 , cn−1 = 1 ∈ L, called the centers of L, such that ϕi cj = 0, if i + j < n, and ϕi cj = 1, if i + j ≥ n. Observe that if c1 , . . . , cn−1 = 1 are the centers of L ∈ CLn , then c1 , . . . , cn−2 are the axes of L, and consequently ci ∈ C(L) for all i, 1 ≤ i ≤ n − 1.
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Lemma 1.8. If ϕi x is open for every i = 1, . . . , n − 1, then x is open. Proof. If Qϕi x = ϕi x for i = 1, . . . , n − 1, then ϕi Qx = ϕi x for every i = 1, . . . , n − 1 . Hence Qx = x. Corollary 1.9. If B(L) ⊆ Q(L), then Qx = x, for all x ∈ L. Observe that as an immediate consequence of Corollary 1.9 we have that in the chain Cn , Qx = x, for all x ∈ Cn . In the next lemma the restriction of the closure operator C to B(L) is also denoted by C. Lemma 1.10. [6] If L ∈ CLn , then hB(L), Ci is a closure Boolean algebra. Proof. For every element x in L, if x ∈ B(L), then Cx ∈ B(L).
Observe that in the closure Boolean algebra hB(L), Ci, the set of open elements is Q(B(L)) = Q(L) ∩ B(L) = {x ∈ L : Qϕi x = x}. It is known that the set of open elements of a closure Boolean algebra, in this case Q(B(L)), is a Heyting algebra if we define x 7→ y = Q(∼ x ∨ y), for every x, y ∈ Q(B(L)). On the other hand, in any Lukasiewicz algebra L, we can define the implication n−1 ^ x⇒y= (∼ ϕj x ∨ ϕj y) ∨ y. j=1
With this operation L becomes a Heyting algebra [10].
Lemma 1.11. For L ∈ CLn , the (0, 1)−sublattice Q(L) is a Heyting algebra if we define the open implication x ֒→ y = Q(x ⇒ y), for x, y ∈ Q(L). Proof. If x, y ∈ Q(L), then x ∧ (x ֒→ y) = x ∧ Q(x ⇒ y) ≤ x ∧ (x ⇒ y) ≤ y. In addition, if there exists z ∈ Q(L) such that x ∧ z ≤ y, then z ≤ x ⇒ y, and consequently, Qz = z ≤ Q(x ⇒ y) = x ֒→ y. As it can be seen in the example 1.4, the restriction of ⇒ to Q(L) is not equal to ֒→, as Q(L) = {0, a, c, 1}, a ⇒ 0 = d and a ֒→ 0 = 0. On the contrary, the restriction of ֒→ to Q(B(L)) coincides with 7→ since for x, y ∈ Q(B(L)), x ֒→ y = Q(x ⇒ y) = Q(∼ x ∨ y) = x 7→ y. That is, hQ(B(L)), 7→i is a Heyting subalgebra of hQ(L), ֒→i.
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Lemma 1.12. If L ∈ Ln and L1 is a (0, 1)−sublattice of L such that ϕi (L1 ) ⊆ L1 , for all i = 1, . . . , n − 1, then there exists a unique interior operator Q defined on L fulfilling the conditions (Q1) to (Q5) and such that Q(L) = L1 , if and only if for every x ∈ L, (x] ∩ L1 has a greatest element, and for every i = 1, . . . , n − 1, max((ϕi x] ∩ L1 ) ≤ ϕi max((x] ∩ L1 ). Proof. If L1 is a (0, 1)−sublattice of L fulfilling the stated conditions, then it is known that ([3]), defining for all x ∈ L, Qx = max((x]∩L1 ), Q satisfies (Q1) to (Q4) and such that L1 = Q(L). In order to prove (Q5) since Qx ≤ x, then for all i = 1, . . . , n − 1, ϕi Qx ≤ ϕi x . Since Qx ∈ Q(L) and ϕi (Q(L)) ⊆ Q(L), for all i = 1, . . . , n − 1, ϕi Qx ∈ Q(L), and consequently, ϕi Qx ≤ max((ϕi x] ∩ Q(L)) = Qϕi x.
2
Congruences and subdirectly irreducible algebras
Let F (L) denote the set of all filters of an algebra L. A filter F ∈ F (L), is a Stone filter, is for each x ∈ F there exists an element b ∈ F ∩ B(L) such that b ≤ x. Cignoli proved [7] that for Lukasiewicz algebras, the notion of Stone filter is equivalent to that of filter satisfying the property x ∈ F implies ϕ1 x ∈ F . We define an open Stone filter as a Stone filter F such that Qx ∈ F , whenever x ∈ F . If G ⊆ B(L) is a filter in B(L) that satisfies the condition Q(G) ⊆ G, we say that G is an open filter of B(L). Let Fϕ1 Q (L), FQ (B(L)) and F (Q(B(L))) respectively denote the lattices of open Stone filters of L, open filters of B(L) and filters of Q(B(L)). The demonstration of the following theorem is not difficult. Theorem 2.1. The mappings Φ1 : Fϕ1 Q (L) → FQ (B(L)) , Φ1 (F ) = F ∩ B(L), Φ2 : FQ (B(L)) → F (Q(B(L))) , Φ2 (G) = G ∩ Q(B(L)), are lattice isomorphims. Corollary 2.2. The mapping Φ = Φ2 ◦ Φ1 : Fϕ1 Q (L) 7→ F (Q(B(L))) is a lattice isomorphism. Let L ∈ CLn , and for F ∈ Fϕ1 Q (L) , let θ = θ(F ) be the relation defined on L by: (x, y) ∈ θ ⇔ there exists f ∈ F : x ∧ f = y ∧ f. We know that θ is a congruence on the Lukasiewicz algebra L. If (x, y) ∈ θ, then x ∧ f = y ∧ f and then Q(x ∧ f ) = Q(y ∧ f ), that is, Qx ∧ Qf = Qy ∧ Qf. Since F is an open Stone filter and f ∈ F , it follows that Qf ∈ F , so (Qx, Qy) ∈ θ. Consequently, θ is a congruence on the closure Lukasiewicz algebra L. It is easy to see that the lattice of congruences of L is isomorphic to the lattice of open Stone filters of L. On the other hand, it is known that the lattice of congruences
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of the Heyting algebra Q(B(L)) is isomorphic to the lattice of filters of Q(B(L)). Then we have the following theorem, where Con(L) denotes the lattice of congruences of an algebra L. Theorem 2.3. Con(L) ≃ Fϕ1 Q (L) ≃ F (Q(B(L))) ≃ Con(Q(B(L))). In particular, the variety CLn is congruence-distributive and has the congruence extension property. It is known [15] that a closure three-valued Lukasiewicz algebra hL, Ci is a monadic Lukasiewicz algebra if and only if hB(L), Ci is a monadic Boolean algebra. This result also holds in the n−valued case. Theorem 2.4. If L ∈ CLn , for all x, y ∈ L the following conditions are equivalent: C(x ∧ Cϕi y) = Cx ∧ Cϕi y, for all i = 1, . . . , n − 1.
(i)
C(x ∧ Cy) = Cx ∧ Cy.
(ii)
C(L) is a Lukasiewicz subalgebra of L.
(iii)
C ∼ Cx =∼ Cx.
(iv)
Suppose that hL, Ci ∈ CLn , and hB(L), Ci is a monadic Boolean algebra. If x ∈ L, for each i = 1, . . . , n − 1 , ϕi C ∼ Cx = Cϕi ∼ Cx = C ∼ ϕn−i Cx = C ∼ Cϕn−i x =∼ Cϕn−ix =∼ ϕn−i Cx = ϕi ∼ Cx. Hence, C ∼ Cx =∼ Cx, so hL, Ci ∈ MLn . Consequently, we have: Corollary 2.5. An algebra hL, Ci ∈ CLn , belongs to MLn if and only if hB(L), Ci is a monadic Boolean algebra. Recall (see [3]) that a Heyting algebra H is subdirecly irreducible if and only if H ≃ H1 ⊕ 1, with H1 a Heyting algebra and H1 ⊕ 1 is the lattice obtained by adjoining a new 1 to H1 . The following theorem follows immediately from Theorem 2.3 and the above remark. Theorem 2.6. An algebra L ∈ CLn is subdirectly irreducible if and only if the Heyting algebra hQ(B(L)), ֒→i is subdirectly irreducible, that is, Q(B(L)) ≃ H1 ⊕ 1, for some H1 Heyting algebra. Theorem 2.7. The simple objects of the variety CLn are the simple monadic Lukasiewicz algebras of order n. Proof. If L ∈ CLn is simple, the unique filters of Q(B(L)) are the trivial ones. Consequently, hB(L), Qi is simple, and hB(L), Ci is simple, that is C is a monadic quantifier. So hB(L), Ci is a monadic Boolean algebra. By Corollary 2.5, it follows that L is a monadic Lukasiewicz algebra of order n. The converse is immediate. Corollary 2.8. [6, 11] An algebra L ∈ CLn is semisimple if and only if L ∈ MLn .
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A particular subvariety
This section is devoted to the study of the subvariety of CLn of those closure Lukasiewicz algebras in which the Heyting algebra of open elements hQ(L), ֒→i is a three-valued Heyting algebra. Recall that a three-valued Heyting algebra is a Heyting algebra hA, →i such that ((x → z) → y) → (((y → x) → y) → y) = 1, for every x, y, z ∈ A [14]. In the case of closure algebras, a similar investigation was carried out for the subvariety CT of those closure Boolean algebras such that the set of open elements form a three-valued Heyting algebra [9]. Recall that if L is a simple algebra in CLn , then L is a simple algebra in MLn . Thus, by [1], Q(L) is a simple n-valued Lukasiewicz algebra, that is, Q(L) is a subalgebra of Cn . If, in addition, Q(L) is a three-valued Heyting algebra, then n = 3. Indeed, for n > 3, the algebra Cn , with Q x = x for every x ∈ Cn , is a simple algebra in CLn such that Q(Cn ) is not a three-valued Heyting algebra (see [14]). In what follows we will prove that if L ∈ CLn is a non-simple subdirectly irreducible algebra such that Q(L) is a three-valued Heyting algebra, then L ∈ CT . So, if L ∈ CLn is such that Q(L) is a three-valued Heyting algebra, then L ∈ CL3 . We denote this subvariety by CT L3 and we have that if L ∈ CLn , L ∈ CT L3 if and only if for every x, y, z ∈ L the following identity holds ((Qx ֒→ Qz) ֒→ Qy) ֒→ (((Qy ֒→ Qx) ֒→ Qy) ֒→ Qy) = 1. Theorem 3.1. An algebra L ∈ CT L3 is non-simple subdirectly irreducible if and only if Q(B(L)) = {0, a, 1}. Proof. It is a consequence of the fact that L is non-simple subdirectly irreducible if and only if hQ(B(L)), ֒→i is a non-simple subdirectly irreducible three-valued Heyting algebra. Lemma 3.2. If Q(B(L)) = {0, a, 1} then 1 is join-irreducible in Q(L). Proof. If 1 = d ∨ e, with d, e ∈ Q(L), then 1 = ϕ1 1 = ϕ1 d ∨ ϕ1 e. Since ϕ1 d, ϕ1 e ∈ Q(B(L)), it follows that either ϕ1 d = 1 or ϕ1 e = 1, that is, d = 1 or e = 1. Corollary 3.3. L ∈ CT L3 is non-simple subdirectly irreducible if and only if Q(L) = Q(B(L)) = {0, a, 1}. Proof. Suppose that L is non-simple subdirectly irreducible. As Q(L) is a three-valued Heyting algebra, then for every x, y ∈ Q(L), (x ֒→ y) ∨ (y ֒→ x) = 1. Since 1 is joinirreducible in Q(L), x ֒→ y = 1 or y ֒→ x = 1. Consequently, x ≤ y or y ≤ x that is, Q(L) is a chain. But Q(L) is three-valued, and thus it has at most three elements. So Q(L) = Q(B(L)) = {0, a, 1}. The converse is trivial. Let L be a Lukasiewicz algebra and suppose that there is defined on L an operation C that satisfies (C1)-(C4) of Definition 1.3. In that conditions,
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Theorem 3.4. If Q(L) = {0, a, 1}, with a ∈ B(L), then L is a CT L3 -algebra if and only if L is a closure Boolean algebra. Proof. We only have to prove that if L ∈ CT L3 , then L is a closure Boolean algebra. Suppose on the contrary that L is not a Boolean algebra. Then there exists an element c ∈ L such that c∨ ∼ c < 1. So Qϕ2 (c∨ ∼ c) = 1. On the other hand, since Q(c∨ ∼ c) = a or Q(c∨ ∼ c) = 0, it follows that ϕ2 Q(c∨ ∼ c) 6= 1 , which is a contradiction. Then we can state: Theorem 3.5. [9] L ∈ CT L3 is non-simple subdirectly irreducible if and only if L ∈ CT is non-simple subdirectly irreducible. Consequently, the subdirectly irreducible algebras in CT L3 are the simple three-valued monadic Lukasiewicz algebras and the non-simple subdirectly irreducible three-valued closure Boolean algebras, that is, Si(CT L3 ) = Si(ML3 ) ∪ Si(CT ), where Si(K) denotes the class of isomorphic types of subdirectly irreducible algebras in a given class K. In particular, CT L3 = ML3 ∨ CT . Let us see now that CT L3 is locally finite. A class K of algebras is called uniformly locally finite if for any n ∈ N there exists m(n) ∈ N such that the cardinality of every n-generated subalgebra of an algebra from K is less than or equal to m(n). The following criterion for a variety to be locally finite is proved in [4]. Theorem 3.6. A variety V is locally finite if and only if V is generated by a uniformly locally finite class of algebras in V . The varieties ML3 and CT are locally finite, and then the class K = Si(ML3 ) ∪ Si(CT ) is uniformly locally finite. Then by Theorem 3.6 we have the following result. Theorem 3.7. The variety CT L3 is locally finite. Let Bk,0 be the simple monadic Boolean algebra with k atoms, and let Tr = C3r be the simple three-valued monadic Lukasiewicz algebra (see [15]). Then we have the following lemma. Lemma 3.8. Every finite subdirectly irreducible algebra in ML3 is simple. The finite simple algebras of the variety ML3 are the algebras Bk,0, k ≥ 1 and the algebras Tr , r ≥ 1. Besides Bk,0 is a subalgebra of Tr if and only if k ≤ r. Let Bk,l be the Boolean closure algebra with k + l atoms such that Q(Bk,l ) = {0, a, 1} and there are k atoms preceding a and l atoms preceding ∼ a, k ≥ 1, l ≥ 1. The following lemma was proved in [9, 2]. Lemma 3.9. The finite simple algebras in the variety CT are the algebras Bk,0 and the finite non-simple subdirectly irreducible algebras in CT are the algebras Bk,l . Besides Bs,t is a subalgebra of Bk,l if and only if s ≤ k and t ≤ l.
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As a consequence of the above results, Sif in (CT L3 ) = {Bk,l , Tr } k≥1,l≥0, r≥1 , where Sif in (K) denotes the classes of isomorphic types of finite subdirectly irreducible algebras in a given class K. To determine the lattice Λ = Λ(CT L3 ) of subvarieties of CT L3 , by well-known results of J´onsson and Davey ([8]), it suffices to characterize the ordered set J (Λ) of its join irreducible elements. Recall that for A, B ∈ Si(CT L3 ), we may define a partial preorder: A ≤ B if and only if A ∈ H(S(B)), where H(K)and S(K) respectively denote the classes of algebras that are homomorphic images of algebras in K and subalgebras of algebras in K, so that V (A) ≤ V (B) if and only if A ≤ B. Then the lattice Λ(CT L3 ) of subvarieties of CT L3 is a distributive lattice which is isomorphic to O(Sif in (CT L3 )), the lattice of down-sets (order-ideals) of the ordered set Sif in (CT L3 ). Moreover, a finitely generated subvariety U ∈ Λ(CT L3 ) is join-irreducible if and only if U = V (L), for some finite subdirectly irreducible algebra L. Let Vk,l = V (Bk,l ), with k ≥ 1, l ≥ 0, and Vr = V (Tr ), with r ≥ 1. Each one of these subvarieties is generated by a subdirectly irreducible algebra, so that they are join-irreducible in the lattice Λ(CT L3 ). Observe that Vk,0 ≤ Vr if and only if k ≤ r. For all r ≥ 1, k ≥ 1, l ≥ 0, Vr 6≤ Vk,l , being that Tr ∈ Vr is not a Boolean algebra and the algebras in Vk,l are Boolean. In addition, if l ≥ 1, Vk,l 6≤ Vr , as Bk,l ∈ / Vr . Consider now the following subvarieties: [ _ Mi = V ( Bk,i ) = Vk,i k≥1
k≥1
and Mi = V (
[
l≥1
We have that
Bi,l ) =
_
Vi,l .
l≥1
M0 ⊂ M1 ⊂ . . . ⊂ Mi ⊂ . . . ⊂ CT and M 1 ⊂ . . . ⊂ M i ⊂ . . . ⊂ CT . The proof of the following lemma is similar to that of Lemma 2.5.1.5 and Lemma 2.5.1.6 in [9]. Lemma 3.10. The subvarieties CT , ML3 , Mi and M i belong to J (Λ). Theorem 3.11. J (Λ) = {Vr }r≥1 ∪ {Vk,l }k≥1; l≥0 ∪ {Mi }i≥0 ∪ {M i }i≥1 ∪ {ML3 } ∪ {CT }. Proof. Let V ∈ J (Λ) be finitely generated. Then it is not difficult to see that V = Vr or V = Vk,l .
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Suppose that V ∈ J (Λ) and that V is not finitely generated. Since V is a subvariety of a locally finite variety, it follows that V is locally finite, and consequently, V is generated by its subdirectly irreducible finite members Sif in (V). Then Sif in (V) is not finite. Consider the sets F = {r ∈ N : Tr ∈ V} and K = {k ∈ N : Bk,k ∈ V}. If F = ∅, then V ≤ CT , and if K = ∅ , then V ≤ ML3 . In both cases, the proof follows as in [9]. Suppose that F 6= ∅ and K 6= ∅ . As V ∈ J (Λ), V = 6 CT L3 , then F is finite or K is finite. If K is infinite, CT ≤ V . Let k = maxF and then
V=
k _
Vr ∨ CT .
r=1
As V ∈ J (Λ), and it is not finitely generated, V = CT . If K is finite, let j = maxK. Consider the subvarieties
A1 = V ({Bk,l : l ≤ j}) and A2 = V ({Bk,l : k ≤ j}).
If Bk,l ∈ V, then Bk,l ∈ A1 or Bk,l ∈ A2 since otherwise l ≥ j + 1 and k ≥ j + 1, which implies that Bj+1,j+1 ∈ V a contradiction, by the maximality of j. Consequently,
V = ML3 ∨ A1 ∨ A2 .
Then we have that V = ML3 , V = A1 or V = A2 . If V = A1 let I = {i ∈ N : Bi,j ∈ A1 }. We have that I 6= ∅ as Bj,j ∈ V = A1 . If I is finite and i0 = maxI, then Bi0 ,j ∈ V and Bi0 +1,j ∈ / V. Let A = V ({Bi,l : i ≥ i0 , l ≤ j − 1}). Then V = Vi0 ,j ∨ A. Since V is not finitely generated, V = A and hence, Bi0 ,j ∈ / V, a contradiction. Thus, I is not finite. This implies, V = Mj . If we suppose that V = A2 , we can prove in a similar way that V = M j .
The ordered set of join-irreducible elements of the lattice Λ(CT L3 ) looks like the following figure, where ij stands for Vi,j and r stands for Vr . Recall that M0 is the variety of monadic Boolean algebras, ML3 is the variety of monadic Lukasiewicz algebras of order 3 and CT is the variety of the three-valued closure Boolean algebras.
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sCT HH p p p p HH sM2 p p HH p HHsM1 p p HH p p Hs p p M0 p p p p p p p p sM 3 p HH p p p p p p p p p H M 32 p HH sH 2 s H p HH 31 p p HsH p HHs HH s22 sM 1 HH HH 30 p p HHs21 p p HH HH s12 Hs H HH 11 20 HsH HHs 10
sML3 p p p p p p
s 3 s 2 s 1
With a similar argument to that of the previous theorem, the following result can be proved. Theorem 3.12. Every subvariety of CT L3 is a finite join of elements in J (Λ).
References [1] M. Abad: Estructuras c´ıclica y mon´ adica de un ´ algebra de Lukasiewicz n-valente, Notas de L´ogica Matem´atica, Vol. 36, Instituto de Matem´atica, Universidad Nacional del Sur, Bah´ıa Blanca, 1988. [2] M. Abad and J.P. D´ıaz Varela: “Free Algebras in the Variety of Three-valued Closure Algebras”, J. Austral. Math. Soc., Vol. 72, (2002), pp. 181–197. [3] R. Balbes and P. Dwinger: Distributive Lattices, University of Missouri Press, Columbia, MO, 1974. [4] G. Bezhanishvili: ”Locally finite varieties”, Algebra Universais 46, Vol. 4, 2001, pp. 531–548. [5] W. Blok: Varieties of interior algebras, Thesis (Ph.D.), University of Amsterdam, 1976. [6] V. Boicescu, A. Filipoiu, G. Georgescu and S. Rudeanu: Lukasiewicz-Moisil Algebras, North Holland, 1991. [7] R. Cignoli: Moisil Algebras, Notas de L´ogica Matem´atica, Vol. 27, Instituto de Matem´atica, Universidad Nacional del Sur, Bah´ıa Blanca, 1970. [8] B.A. Davey: “On the lattice of subvarieties”, Houston J. Math., Vol. 5, (1979), pp. 183–192. [9] J.P. D´ıaz Varela: Algebras de Clausura y su Estructura Sim´etrica, Tesis (Ph.D.), Bah´ıa Blanca, Argentina, 1997.
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[10] L. Iturrioz: “Lukasiewicz and Symmetrical Heyting Algebras”, ZML, Vol. 23(2), (1977), pp. 131–136. [11] L. Iturrioz: “Two characteristic properties of three-valued Lukasiewicz algebras”, Rep. Math. Logic, Vol. 8, (1977), pp. 63–69. [12] Gr.C. Moisil: “Notes sur les logiques non-chrysippiennes”, Ann. Sci. Univ. Jassy, Vol. 27, (1941), pp. 86–98. [13] A. Monteiro: L’aritm´etique des filtres et les espaces topologiques, Notas de L´ogica Matem´atica, Vol. 29-30, Instituto de Matem´atica, Universidad Nacional del Sur, Bah´ıa Blanca, 1974. [14] L. Monteiro: “Alg`ebre du calcul propositionel trivalent de Heyting”, Fund. Math., Vol. 74, (1972), pp. 99–109. [15] L. Monteiro: Algebras de Lukasiewicz trivalentes mon´ adicas, Notas de L´ogica Matem´atica, Vol. 32, Instituto de Matem´atica, Universidad Nacional del Sur, Bah´ıa Blanca, 1974. [16] C.O. Sicoe: “Sur les ideaux des alg`ebres Lukasiewicziennes polivalentes”, Rev. Roum. Math. Pures et Appl., Vol. 12, (1967), pp. 391–401. [17] C.O. Sicoe: “On many-valued Lukasiewicz algebra”, Proc. Japan Acad., Vol. 43, (1967), pp. 725–728.
CEJM 3(2) 2005 228–241
Exact and stable least squares solution to the linear programming problem ¨ ∗ Evald Ubi Department of Economics, Tallinn University of Technology, Kopli 101, 11712 Tallinn, Estonia
Received 29 March 2004; accepted 20 September 2004 Abstract: A linear programming problem is transformed to the finding an element of polyhedron with the minimal norm. According to A.Cline [6], the problem is equivalent to the least squares problem on positive ortant. An orthogonal method for solving the problem is used. This method was presented earlier by the author and it is based on the highly developed least squares technique. First of all, the method is meant for solving unstable and degenerate problems. A new version of the artifical basis method (M-method) is presented. Also, the solving of linear inequality systems is considered. c Central European Science Journals. All rights reserved.
Keywords: Linear programming, method of least squares, M-method MSC (2000): 90C05, 65K05
1
Introduction
The least squares method is used in mechanics, physics, statistics, but not in linear programming. The application of this universal method in mathematical programming is the main purpose of this paper. We consider the standard linear program min{z = (c, x)}, s.t.Ax = b, x ≥ 0 ∗
E-mail:
[email protected]
(1)
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and its dual max{w = (b, y)},
(2)
s.t.AT y ≤ cT , where A is an m × n matrix, b and y are m−vectors, c and x are n−vectors. In the papers [1, 2] two finite approximate methods of solving linear programming problem are considered. In one of them the least squares method is applied to the system Ax = b ǫ(c, x) = ǫz0 x ≥ 0, where z0 ≤ zmin , ǫ > 0. Another algorithm finds the least squares solution x(ǫ) to the system Ax = b ǫx = −cT x ≥ 0. It is proved that x(ǫ) → x∗ if ǫ 7→ 0 where x∗ is a solution of the problem (1). In this paper an exact orthogonal method for solving the linear programming problem is presented. This method is based on the author’s previous work where the problem is solved by the orthogonal method [3]. This method can be applied directly if the minimum zmin of the objective function of the problem (1) is known. In this case the system Ax = b (c, x) = zmin x≥0 has to be solved using least squares method described in [3]. In Section 2 we show that by shifting coordinates the optimal solution of dual problem has minimal norm. In Section 3 we describe the solving of the linear programming problem by the method of least squares. A detailed description of algorithm VD is given. It is based on the QR decomposition of coefficients matrix D. If in the solution process of the system with a triangular matrix R some variable proves to be nonpositive, then the column corresponding to this partcular variable is eliminated from the matrix R and other columns are transformed again to the triangular form using Givens rotations. In Section 4 the similarity of the presented method with the classical M-method is described and the determining of the shift parameter is discussed. In Section 5 a solution of the system of linear inequalities as an element of polyhedron with minimal norm is found. The solving of the linear programming problem, when optimal value of the objective function is known, is considered. While computing, the linear least squares techniques which is thoroughly described in [4, 5, 6], is used.
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2
Proof of the solving method
Assumption 2.1. The right hand sides of the initial problem are non-negative: b(i) ≥ 0, i = 1, ..., m
(3)
Assumption 2.2. The constraints of the problem (1) are not contradictory. If the constraints of the dual problem (2) are not contradictory, then algorithm VD described below assumes that the optimal solution y∗ of the dual problem has minimal norm among feasible solutions {y : AT y ≤ cT }, y∗ is the normal solution. If the optimal solution to the dual problem is not unique, then denote y∗ the optimal solution which has minimal norm. This is always uniquely determined. In order to achieve that the solution y∗ to the dual problem has minimal norm, transform the coordinates by shifting ′
yi = yi + tbi , i = 1, ..., m,
(4)
where t is a sufficiently large positive number. In this case the dual problem takes the form ′ ′ ′ ′ max{w = (b, y )} s.t.(A.j , y ) ≤ cj − t(A.j , b) = cj (5) j = 1, ..., n, t > 0, where A.j denotes a column of the matrix A and constant additive terms in the objective function are dropped (see example 2.1). Let us show that if the dual problem has a bounded optimal solution then for the sufficiently large shift (t) the vector with minimal norm is a solution to this problem. For this reason it is possible to consider as the objective function the expression
′
2
′ v = y = ky − tbk2 .
(6) ′
Let yt be an optimal solution to the problem with such an objective function and y ∗ an optimal solution to the problem (5). Transfer the objective function (6), max{−v/t = −(y, y)/t + 2(b, y) − t(b, b)}. According to [7] (Theorem 1,Part 3, Ch.10) there exists a number t0 such that for ′ ′ each t ≥ t0 the equality yt = y ∗ holds. In the paper [8] the following theorem is proved. Theorem 2.3. For any matrix A and vector b 6= 0 only one of the two following statements is valid: a) there exists a non-negative solution to the system Ax = b,
(7)
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b) the system of inequalities AT y ≤ 0, (y, b) > 0
(8)
holds. Set a least squares problem with the aid of which the problems (1) and (2) are solved: Au = 0, (c, u) = −1, u ≥ 0,
(9)
where u is n− vector. In more compact way this system is written as follows: Du = f, u ≥ 0,
(10)
where D = (A, c)T , f = (0, 0, ..., 0, −1)T , r = Du − f. Due to the formula (7) the following theorem can be proved. Theorem 2.4. The system (9) has a solution (r = 0) if and only if the objective function of the initial problem is unbounded. Proof. If r = 0 then substituting an infinitely small negative number −M for the right side of the last equation of system (9),we obtain a solution which differs from the previous one M times. Therefore, the condition r = 0 is equivalent to the unboundedness of the objective function of the problem min{z = (c, x)} s.t.Ax = 0 x ≥ 0. Let x0 be an arbitary feasible basic solution. Then substituting uj = xj − x0j for the basic variables and uj = xj − x0j for the other variables we get that the objective function of the initial problem is unbounded. Remark 2.5. From the proof above it can be seen that in the system (9) the right hand side of the last equation can be taken to be an arbitrary negative number. Remark 2.6. Substituting matrix D for matrix A in the first theorem, and if (8) is valid, we get for the dual problem AT y + ym+1 cT ≤ 0 −ym+1 > 0. It means that the set of feasible solutions is not empty. Assumption 2.7. The optimal solution y∗ of the dual problem is normal (feasible solution with the minimal norm) if y∗ is unique. If the optimal solution to the problem (2) is not unique then by y∗ the normal solution is denoted.
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Theorem 2.8. Let ub be the least squares solution of the system (9) and r = Dub− f 6= 0. If the optimal solution to the dual problem is normal, then it can be expressed in the form Aub Aub yb = − = − (11) rm+1 krk2 Proof. Write the problem (10) in the form min{ϕ(u) = kDu − f k2 /2}
(12)
Find the gradient ′
ϕ = D T (Du − f ) = D T r.
(13)
Suppose that the least squares solution ub to the problem (9) is determined by the k first components ′
′
ϕi (ub) = 0, ubi ≥ 0, i = 1, ..., k, ϕi (ub ) ≥ 0, ubi = 0, i = k + 1, ..., n.
(14)
krk2 = (r, Dub) − (r, f ) = (ϕ (ub), ub) + rm+1 = rm+1 > 0.
(15)
According to the assumptions r 6= 0, due to formulas (13) and (14) we have ′
Let us show that yb determined by (11) satisfies the constraints 1, 2, ..., k of the problem (2) as equalities. The system of normal equations of (9) is j=k X
[(A.i , A.j ) + ci cj ]uj = −ci , i = 1, ..., k
(16)
j=1
or after some transformation j=k X
(A.i , A.j )uj = ci [−1 −
j=k X
cj uj ], i = 1, ..., k
j=1
j=1
Substituting yb which is determined by (11) to the ith constraint of the dual problem (2) we have due to the last equation and the formula r = Du − f
a1i ybi + ... + ami ybm = −
1 rm+1
[a1i r1 + ... + ami rm ] = −
1 × 1 + c1 u1 + ... + ck uk
×[a1i (a11 u1 + ... + ami (am1 u1 + ... + amk (am1 u1 + ... + amk uk )] = 1 [ci (−1 − c1 u1 − ... − ck uk )] = ci , i = 1, ...k. 1 + c1 u1 + ... + ck uk It means that k first constraints of the dual problem are satisfied as equations.The rest of constraints are satisfied due to the formulas (13)-(15), because =−
(cT − AT yb) krk2 = D T r = ϕ (ub) ≥ 0. ′
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The vector determined by (11) −yb is a linear combination of the rows of the matrix AT : Aub Aub (17) −yb = = rm+1 krk2 Let us consider a least squares problem min{kyk2 /2}, s.t. AT y ≤ cT
(18)
The antigradient of this function −y can be due to (17) presented as a linear combination of the rows of the matrix AT where all the coefficients of this combination are non-negative. In addition the conditions of complementary slackness are fulfilled. So the vector yb is a least squares solution to the problem (18) and at the same time under the assumption 2.2 the optimal solution to the LP problem (2). Remark 2.9. The optimal solution to the initial problem (1) can be found by solving the least squares problem A.1 x1 + ... + A.k xk = b.
(19)
′
Here xj are the variables for which ϕj (ub) = 0. Such variables xj and ubj are called active. Remark 2.10. In the problem (18) at least k first constraints are equalities, so 1 ≤ k ≤ m. If k = m then the problem (19) can be solved by the Gaussian elimination method. Remark 2.11. If the least squares solution ub is not unique, then in (11) an arbitrary solution can be used (see Example 5.2). The normal solution yb is always unique. Example 2.12. min{2x1 + 4x2 + x3 + 4x4 = z} −x1 + 2x2 + x3 + x4 = 1 x1 − x2 − 2x3 + 2x4 = 2 x ≥ 0. The optimal solution to the initial problem x∗ = (0, 0, 0, 1)T , zmin = 4. The optimal solution to the dual problem is y∗ = (4 − 2p, p)T , 4/5 ≤ p ≤ 2. In the transformed problem choose the shifting parameter t = 2, new origin of coordinates O ′ = (2, 4)T , then ′ ′ y1 = y1 + 2, y2 = y2 + 4. Consider the transformed dual problem according to the formula (5),
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′
′
′
max{y1 + 2y2 = w } ′
′
−y1 + y2 ≤ 2 − t = 0 ′
′
2y1 − y2 ≤ 4 − 0t = 4 ′
′
′
′
(20)
y1 − 2y2 ≤ 1 + 3t = 7 y1 + 2y2 ≤ 4 − 5t = −6. Consider the least squares problem (9): −u1 + 2u2 + u3 + u4 = 0 u1 − u2 − 2u3 + 2u4 = 0 4u2 + 7u3 − 6u4 = −1 u ≥ 0. The least squares solution to this problem is ub = (0, 0, 0, 6/41)T . According to the formulas (10) and (11) find the optimal solution to the transformed dual problem ′
r = Dub − f = (6/41, 12/41, 5/41)T , yb = (−6/5, −12/5)T . General form of the optimal solution to the transformed dual problem is ′ y ∗ = (2 − 2p, p − 4), 4/5 ≤ p ≤ 2. The norm of this vector achieves its minimum at ′ p = 8/5, if y ∗ = yb. Shifting all the coordinates yb′ by tb we find the optimal solution ′ y ∗ = (4/5, 8/5)T to the initial dual problem, which corresponds to yb . Using least squares method the optimal solution to the initial problem can be found from the system x4 = 1 2x4 = 2 ′
containing for k ≤ m variables for which ϕi (ub ) = 0. It follows from the conditions of complementary slackness. If in the case of the optimal solution of the dual problem exactly m constraints occur to be equalities, then optimal solution to the initial problem can be found by the Gaussian elimination method from the system (19). If the initial problem is contradictory then the system (19) has no non-negative solution. E.g., this is the case if all (Aj , b) < 0, j = 1, ..., n. In Section 5 solving LP problem is considered if the optimal value of the objective function is known.
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235
Description of the algorithm VD
Describe the algorithm VD for solving the pair of dual problems (1) and (2). Let us write the least squares problem (9) in the form Du = f, u ≥ 0, where D is an (m + 1) × n matrix. In practical computations it is convenient to write the coefficients of the transformed problem found according to (5) into the first row of ′ the matrix D. It quarantees somewhat greater stability of the computing process if cj are large. Algorithm V D(A, b, c, D, f, IJ, F, G, x, y, m, n, t). ′ 1. Choose a sufficiently large shifting parameter t and calculate cj according to (5). 2. Evaluate n− vectors F and G with coordinates ′
Fj = −cj , Gj = (D.j , D.j ), j = 1, ..., n. 3. Initiate the number of active variables k = 0 and u = 0. 4. Determine the following active variable u(j0) by solving the problem max{
2 Fj2 Fj0 = = Re, } Gj Gj0
where the maximum is found for all passive (i.e. uj = 0) variables satisfying inequality Gj > 0. 5. If Re ≤ 0, then go to Step 23. 6. Increase number of active variables, k = k + 1. 7. Write index j0 into array IJ (active variables). 8. If k = m + 1 then go to Step 10. 9. Fulfil Householder transformation with an (m + 2 − k)−vector v = D.j0 to D and f , see [6] ch.10. 10. Compute new 2 Fj = Fj − Dkj fk , Gj = Gj − Dkj , j = 1, ..., n. 11. Solve the triangular system Ru = f of order k to determine the active variables. 12. Let L = k + 1 (where L denotes the number of active variables being verified). 13. Let L = L − 1. 14. If 1 ≤ L, then go to step 16. 15. If k < m + 1, then go to step 4 else go to step 23. 16. Let j = IJ(L). 17. If uj > 0, then go to Step 13. 18. Let uj = 0 and delete index j from the set IJ. 19. Transform the active columns D into the triangular form by the Givens rotations. 20. Compute new 2 Fj = Fj + Dkj fk , Gj = Gj + Dkj , j = 1, ..., n
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21. Decrease the number of active variables, k = k − 1. 22. Go to Step 11. 23. Compute r = Du − f. 24. If r = 0 then z is unbounded and the dual problem (2) is contradictory. Stop. 25. Solve by the least squares method system Abx = b, composed for all active variables uj , j ∈ IJ. 26. If system Abx = b has a solution, satisfying all equations, then it is the solution of the problem (1). 27. If system Abx = b has no solution, then problem (1) is contradictory. Stop. 28. Compute solution of the dual problem (2), y = −r/rm+1 . 29. The problems (1) and (2) are solved. Remark 3.1. If, for example, all products (Aj , b) < 0, j = 1, ..., n, then by sufficient big shifting parameter t in (5) system Abx = b has no solution (Step 25). Remark 3.2. The advantage of the algorithm VD is the following: there is no need to calculate matrices of orthogonal transformations. Remark 3.3. In the algorithm VD the number of steps is finite as at each step the calculations according to (12) give minima of squares’ sum in a subspace, which number is finite [3, 6]. Example 3.4. min{z = −x1 − 6x2 + x3 } s.t.
8x2 + 2x3 = 16
2x1
+ 2x3 = 8 x ≥ 0.
Consider the least squares problem (9): 8u2 + 2u3 = 0 2u1
+ 2u3 = 0
−u1 − 6u2 + u3 = −1 u ≥ 0. The solution in least squares is ub = (8/58, 3/58, 0)T . According to the formulas (10) and (11) find the optimal solution to the dual problem, r = (24/58, 16/58, 32/58)T , y∗ = (−0.75, −0.50)T . In this problem shifting is not needed, because y∗ is a normal solution. On the first step the variable u2 and on the second step u1 will be active.
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4
Iteration
u1
u2
u3
f
1
0
8
2
0
2
0
2
0
-1
-6
1
-1
F
1
6
-1
G
5
100
9
u
0
0
0
2
-0,600
-10
-1
-0,600
2
0
2
0
-0,800
0
2
-0,800
F
0,640
0
-1,600
G
4,64
0
8
u
0
0,06
0
3
-0,600
-10
-1
-0,600
-2,154
0
-1,116
-0,297
0
0
2,600
-0,742
F
0
0
-1,930
G
0
0
2,600
u
0,138
0,052
0
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The artifical-basis and shifting
In the former section we shifted coordinates to minimize the norm of the optimal solution to the dual problem. Now, we demonstrate the correspondence between the shift and the transformation of the primal problem in case of classical M-method. To explain this we reconsider Example 2.12. We use the penalty coefficient vector tb and artifical variables v. Example 4.1. min{2x1 + 4x2 + x3 + 4x4 + tv1 + 2tv2 = z} −x1 + 2x2 + x3 + x4 + v1 = 1 x1 − x2 − 2x3 + 2x4 + v2 = 2 x ≥ 0, v ≥ 0. Let’s eliminate the artifical variables v from the objective function,
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z(t) = (2 − t)x1 + (4 − 2t)x2 + (1 + 3t)x3 + (4 − 5t)x4 − 5t. At each value of the parameter t the coefficients of this function equal to the right sides of the dual problem (20). Therefore the penalty coefficients tbi correspond to the coordinate shift in the dual problem. In case of penalty coefficients tbi (in contrast to equal penalty M ) the number of steps of the simplex method may be smaller as the coefficients of the objective function z(t) depend both on primal objective function and the right sides of the constraints. This notion is affirmed by the solved examples. Knowing the classical penalty coefficient M and assuming positivity of all right sides bi > 0, i = 1, ..., m, the parameter t has to be chosen large enough to satisfy the conditions tbi > M, i = 1, ..., m. There is no universal method for determining the most suitable value, neither for M nor t. When a feasible basic solution to the primal problem is known, the dual problem includes constraints y ≤ 0. As according to the assumption dual objective vector is b ≥ 0, i.e. in ”opposite direction” to the negative ortant, the shift parameter should not be very large. In most cases of random coefficients no shift in the dual problem was necessary if a feasible basic solution existed. Solution of (10) using algorithm VD is voluminous as all coefficients in the system are transforming at each step. There is a similarity to the primal simplex method. It is possible to derive an algorithm similar to the revised simplex method where orthogonal transformations are used and memorized as products [6, ch 24]. The presented algorithm VD is slower than the widely used revised simplex method, that has been perfected for over 50 years. The main disadvantage of the algorithm VD is the large amount of memory capacity needed. One can use it to solve comparatively small unstable problems. For large-size sparse problems the well-known least squares technique should be used. The biggest advantage of the algorithm VD is precision as seen in the following example 4.2. The optimal solution even to the degenerate and unstable problem could be found according to (19) using least squares method. Example 4.2. Let us consider a linear programming problem with Hilbert matrix a(i, j) = 1/(i + j), b(i) = 1/(i + 1) + 1/(i + 2) + ... + 1/(i + m), a(i, m + i) = 1, the rest of elements a(i, j) = 0, c(i) = −b(i)−1/(i+1), c(m+i) = 0, x∗(i) = 1, x∗(m+i) = 0, i, j, = 1, ..., m. Well-known programs solve this problem only for m ≤ 8. The algorithm VD found solution to this problem for m ≤ 12.
5
Solving the system of linear inequalities by the least squares method
In the system of inequalities Ey ≤ h
(21)
E is a n×m matrix, h a n−vector and y an m−vector, m ≤ n or m > n. We will consider first whether this system of inequalities holds. If the vector y = 0 is not a solution to this system apply the algorithm VD. For this purpose we set a dual problem taking the
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coefficients of the objective function of the initial problem equal to zero. min{z = (h, u)}, E T u = 0, u ≥ 0.
(22)
It is known that the system (21) does not hold if and only if the goal function of the problem (22) is unbounded. Set the least squares problem (10) Du = f, u ≥ 0,
(23)
D = (E T hT ), f = (0, 0, ..., 0, −1)T , r = Du − f. Theorem 5.1. The system of inequalities (21) does not hold if and only if the problem (23) has a solution ub satisfying all equations, Dub − f = r = 0. If r 6= 0 and ub is the solution of the problem (23) in least squares then yb = −E T ub/rm+1 is the solution with minimal norm to the system (21). Proof follows directly from the theorems 2.4 and 2.8. Example 5.2. The least squares problem u1 − 2u2 = 0 u1 − 2u2 = 0 2u1 − 8u2 = −1 u≥0 corresponds to the inequalities y1 + y2 ≤ 2 −2y1 − 2y2 ≤ −8 A solution of the least squares problem is ub = (2/4, 1/4)T , r = 0 and the system of inequalities is contradictory. If the right hand side of the second inequality is taken to be -4 then the least squares solution is not unique, ub = (c, c/2 + 1/6)T , r = (−1/3, −1/3, 1/3)T , yb = (1, 1)T is the normal solution. Finally let us consider solving the LP problem if the maximum value of the objective function is known, max{w = (b, y)} AT y ≤ cT . Set the system of inequalities AT y ≤ cT −(b, y) ≤ −wmax
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and find with the aid of the formula (11) a solution to this system. Example 5.3. max{w = 2y1 + y2 } −y1 + y2 ≤ 1 y1
≤ 1.
Add the inequality −2y1 − y2 ≤ −4 to the constraints and find the least squares solution ub = (0, 3/6, 2/6)T , r = (−1/6, −2/6, 1/6)T to the system −u1 + u2 − 2u3 = 0 u1
− u3 = 0
u1 + u2 − 4u3 = −1 u ≥ 0. Due to the formula (11) y∗ = (1, 2)T .
6
Conclusions
In this paper some relations between linear programming and least squares method are considered. The algorithm VD presented uses orthogonal transformations and there is no need to calculate the respective matrices. Based on examples solved it is clear that the number of steps of the algorithm is not significantly larger than the number of constraints and negative variables occur quite seldom. Thus it was in the solved examples for m ≤ 200. There is some need for additional consideration of multiplying the constraints of the LP problem by constants and the influence of this procedure to the stability of the least squares solution and shift parameter.
7
Acknowledgements
Author is grateful to the anonymous referees for carefully reading the first version of the manuscript and for their many constructive comments and suggestions.
References ¨ [1] E. Ubi: “An Approximate Solution to Linear and Quadratic Programming Problems by the Method of least squares”, Proc. Estonian Acad. Sci. Phys. Math., Vol. 47, (1998), pp. 19–28. ¨ [2] E. Ubi: “On Computing a Stable Least Squares Solution to the Linear Programming Problem”, Proc. Estonian Acad. Sci. Phys. Math., Vol. 47, (1998), pp. 251–259.
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¨ [3] E. Ubi: “Finding Non-negative Solution of Overdetermined or Underdetermined System of Linear Equations by Method of Least Squares”, Trans. Tallinn Tech. Univ., Vol. 738, (1994), pp. 61–68. [4] R. Cline and R. Plemmons: l2 −solutions to Underdetermined Linear Systems, SIAM Review, Vol. 10, (1976), pp. 92–105. [5] A. Cline: “An Elimination Method for the Solution of Linear Least Squares Problems”, SIAM J. Numer.Anal., Vol. 10, (1973), pp. 283–289. [6] C. Lawson and R. Hanson: Solving Least Squares Problems, Prentice-Hall, NewJersey, 1974. [7] B. Poljak: Vvedenie v optimizatsiyu, Nauka, Moscow, 1983. [8] T. Hu: Integer programming and Network flows, Addison-Wesley Publishing Company, Massachusetts, 1970.
CEJM 3(2) 2005 242–244
A simple proof of a result of Abramovich and Wickstead Zafer Ercan∗ Middle East Technical University, Department of Mathematics, 06531 Ankara, Turkey
Received 9 December 2004; accepted 24 January 2005 Abstract: The paper presents a simple proof of Proposition 8 of [2], based on a new and simple description of isometries between CD0 -spaces. c Central European Science Journals. All rights reserved.
Keywords: CD0 -spaces, Riesz isomorphisms MSC (2000): 54C35, 46E25
We refer to [3] for unexplained terminology on Banach lattice theory. The space of functions f from a set X into R with {x : |f (x)| > ǫ} is finite for each ǫ > 0 is denoted by c0 (X). As usual, the space of real valued continuous functions on a topological space X is denoted by C(X). We consider C(X) and c0 (X), which are vector subspaces of the vector space of functions from X into R, under the pointwise operations. For a topological space X we write CD0 (X) := C(X) + c0 (X). Let X be a compact Hausdorff space without isolated points. Then C(X) ∩ c0 (X) = {0} and CD0 (X) = C(X) ⊕ c0 (X) is an AM-space under the supremum norm (see [1], [2] and [4] ). In this case, for each f ∈ CD0 (X) there exists unique fc ∈ C(X) (continuous part) and fd ∈ c0 (X) (discrete part) with f = fc + fd . Throughout this paper K, M and X stand for compact Hausdorff spaces without isolated points. In [5] and [6], it is proved that the spaces CD0 (X) and C(A(X)) are isometrically Riesz isomorphic spaces under the map f −→ πX (f ) defined by πX (f )(x, r) = fc (x) + rfd (x), where A(X) is the space X × {0, 1} equipped with the compact Hausdorff topology such that (xα , rα ) −→ (x, r) if and only if f (xα )+rα d(xα ) −→ f (x) + rd(x) for each f ∈ C(X) and d ∈ c0 (X). This representation leads us to the fact ∗
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that for any Riesz homomorphism π ′ : CD0 (X) −→ R with π ′ (1) = 1 there exists a unique (x, r) ∈ A(X) such that π ′ (f ) = fc (k) + rfd (k) for each f ∈ CD0 (K). By the construction of A(X), it is easy to see that the isolated points iso(A(X)) of A(X) are {(x, 1) : x ∈ X}. Let π : C(K) −→ C(M) be a Riesz homomorphism with π(1) = 1. It is well known that there exists a continuous map σ : M −→ K such that π(f ) = f ◦ σ (see Theorem 7.22 on page 103 of [3]). In particular π is a Riesz isomorphism if and only if σ is homeomorphism (Banach-Stone type theorem). Let σ : A(M) −→ A(K) be a homeomorphism. Since a homeomorphism preserves the isolated and non-isolated points, the map σ0 : M −→ K with σ(m, 0) = (σ0 (m), 0) is also a homeomorphism and the map σ1 : M −→ K with σ(m, 1) = (σ1 (m), 1) is a bijection. Let us fix the above notations. Lemma 1. Let T : CD0 (K) −→ CD0 (M) be a Riesz isomorphism. Then [T (C(K))]c = C(M) and T (c0 (K)) = c0 (M). If T (1) = 1, then there exists a homeomorphism such that [T (h)]c (m)+r[T (h)]d (m) = πK (h)(σ(m, r)), [T (h)]c (m) = hc (σ0 (m)) and T (h)(m) = h(σ1 (m)), where m ∈ M, r ∈ {0, 1} and h ∈ CD0 (K). Proof. The proof of the second part follows immediately from the representation of CD0 -spaces and Banach-Stone type theorem stated as above. For the first part: It is clear that f is an atom in CD0 (K) if and only if f is a positive multiple of the characteristic function of a single point, and f is an atom if and only if π(f ) is an atom, so π(χk ) ∈ c0 (M) for each k ∈ K. As c0 (K) is closed and π is continuous, π(c0 (K)) ⊂ c0 (M). Since π is a Riesz isomorphism we have the equality. Let h ∈ C(M). Then h = T (T −1(h)) = T [(T −1 (h))c ] + T [(T −1 (h))d ]. As T (T −1(h)d ) ∈ c0 (M), we have T (T −1 (h)c )c = h. This shows that C(M) ⊂ T (C(K))c and completes the proof. In the above lemma, it is clear that σ1 is a homeomorphism if and only if T (C(K)) = C(M). A proof of the next lemma was given in [2] which is rather involved, we give a direct proof. Lemma 2. Let T : CD0 (K) −→ CD0 (M) be a linear surjective isometry. Then a) T (1) is a unimodular function, that is |T (1)| = 1. b) suppT (1)d = {m ∈ M : T (1)d (m) 6= 0} = {m ∈ M : |T (1)d(m)| = 2}. c) T is a Riesz isomorphism if T (1) = 1. Proof. a) Since 1 is an extreme point of the unit ball of CD0 (K), so is T (1) extreme in the unit ball of CD0 (M) and as it is well-known that such extreme points must be unimodular functions, the result follows. Another (direct) alternative proof of this is following: Let f = 12 ((T (1)2 − 1)). Then || T (1) + f || ≤ 1 and ||T (1) − f || ≤ 1. This implies that T −1 (f ) = 0; so (T (1))2 = 1, i.e. T (1) is a unimodular function. b) Since suppT (1)d is countable and T (1) is unimodular, T (1)c must be unimodular. Let
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m ∈ suppT (1)d . Then our claim follows from the following fact 1 = T (1)2 (m) = (T (1)c (m) + T (1)d (m))2 = (r + T (1)d (m))2 ,
r ∈ {1, −1}.
c) This follows immediately from ||f − ||f ||1|| ≤ ||f || ⇐⇒ 0 ≤ f.
The proof of the next Theorem follows immediately from the preceding Lemmata. Theorem 3. Let T : CD0 (K) −→ CD0 (M) be a linear surjective isometry. Then there exists a homeomorphism σ : A(M) −→ A(K) such that T (f ) = T (1)(f ◦σ1 ) and T (f )c = T (1)c (fc ◦ σ0 ) for every f ∈ CD0 (K). The above theorem yields an easier proof of the next theorem which is proposition 8 of [2]. Theorem 4. Let K be a quasi-Stonean space and T : CD0 (K) → CD0 (K) be a positive onto isometry satisfying additionally the property: T (f )c = f for each f ∈ C(K). Then there are a finite subset F ⊂ K, a bijection β : K → K such that β(k) = k for all k ∈ K \ F and for each h ∈ CD0 (K), we have T h = h ◦ β. Proof . Since T (1) is unimodular by Lemma 2(a) and positive then T (1) = 1. Thus T is a Riesz isomorphism by Lemma 2(c). From the above theorem there exist bijections σr : K −→ K such that T (f )c + rT (f )d = fc ◦ σr + rfd ◦ σr , r ∈ {0, 1}. As f = T (f )c = f ◦ σ0 for each f ∈ C(K), σ0 is identity. Let E ⊂ K be clopen. As T (χE )c = χE there exists dE ∈ c0 (K) such that χE ◦ σ1 = T (χE ) = χE + dE . This implies that −1 −1 suppdE = d−1 E ({1, −1}) and is finite. It is also obvious that σ1 (E) \ E ⊂ dE ({1}). Applying Theorem 7 of [2], we have σ1−1 is almost identity, that is, there exits a finite subset F of K such that σ1−1 (k) = k for each k 6∈ F , so σ(k) = k for each k 6∈ F . Since T (f ) = f ◦ σ1 , this completes the proof by choosing β = σ1 .
References [1] Y.A. Abramovich and A. W. Wickstead: ”Remarkable classes of unitial AM-spaces“, J. of Math. Analysis and Appl., Vol. 180, (1993), pp. 398–411. [2] Y.A. Abramovich and A.W. Wickstead: ”A Banach-Stone Theorem for a New Class of Banach Spaces“, Indiana University Mathematical Journal, Vol. 45, (1996), pp. 709–720. [3] C.D. Aliprantis and O. Burkinshaw: Positive operators, Academic Press, New York, London, 1985. [4] S. Alpay and Z. Ercan: ”CD0 (K, E) and CDw (K, E) spaces as Banach lattices“, Positivity, Vol. 3, (2000), pp. 213–225. [5] Z. Ercan: ”A concrete desription of CD0 (K)-spacesas C(X)-spaces and its applications“, Proc. Amer. Math. Soc., Vol. 132, (2004), pp. 1761–1763. [6] V.G. Troitsky: ”On CD0 (K)-spaces“, Vladikavkaz Mathematical Journal, Vo. 6(1), (2004), pp. 71–73.
CEJM 3(2) 2005 245–250
Multiples of left loops and vertex-transitive graphs Eric Mwambene∗ Department of Pure and Applied Mathematics, University of the Western Cape, Bellville, 7535, South Africa
Received 27 September 2004; accepted 10 January 2005 Abstract: Via representation of vertex-transitive graphs on groupoids, we show that left loops with units are factors of groups, i.e., left loops are transversals of left cosets on which it is possible to define a binary operation which allows left cancellation. c Central European Science Journals. All rights reserved.
Keywords: Vertex-transitive graphs, groupoids, loops MSC (2000): 05C25, 20B25
1
Introduction
Because vertex-transitive graphs have symmetric properties that are favourable to modeling interconnection networks, the search for them has been intense for some time now. While some have focused on describing vertex-transitive graphs with specific properties, others have grappled with the general problem of enlisting them [5]. In the sequel, we show that the problem of enumerating vertex-transitive graphs is ultimately linked to the classical problem of enumerating groups and their subgroups. We show that since vertex-transitive graphs can be represented on left loop graphs, and since left loops are factors of groups, the problem of enumerating these graphs translates to finding quasi-associative sets on left loops. We will consider finite graphs only. As will be shown, vertex-transitive graphs can be defined as follows: vertices are elements of the left loop and adjacency is defined by a relation that is described by multiplication of the system and some given subset. Such a subset is called a Cayley set, an extension of the one defined on groups [3]. It is a subset of a groupoid that ∗
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describes an adjacency relation that is both irreflexive and symmetric. Before we define the generalised Cayley sets, recall that a groupoid is just a set endowed with a binary operation. Cayley sets in the general groupoids are defined in the following way. Definition 1.1. Let (A, ∗) be a groupoid. A subset S ⊂ A is a Cayley set if (i) a ∈ / aS for any a ∈ A; (ii) a ∈ (as)S for any a ∈ A, s ∈ S. For a given Cayley set of a groupoid, a groupoid graph is defined in the following way. Definition 1.2. Let A be a groupoid, S ⊂ A a Cayley set. The groupoid graph G = GG(A, S) is defined by V (G) := A; E(G) := {[x, xs] : x ∈ A, s ∈ S}. The focus of the sequel is on finite left loops and their corresponding graphs. Recall that a left loop is a groupoid for which one can cancel from the left. In our context, it will additionally mean that it contains a right unit. The one without a right unit will be called a left quasi-group. A loop is a groupid for which one can cancel both from left and right and contains a unit. For a groupoid graph GG(A, S) we speak of A as the underlying groupoid. When A is a left quasi-group we speak of a left quasi-group graph. Similarly we speak of a left loop graph or a loop graph when A is a left loop or a loop. It is clear that if one can cancel to the left, as is the case in a left loop, one gets a regular graph. In the next section, we show that vertex-transitive graphs are represented by left loops with the additional property that their Cayley sets are quasi-associative.
2
Vertex-transitive graphs as left loop graphs
The results of this section have their foundation in those of Sabidussi [6], who showed that every vertex-transitive graph has a multiple that is Cayley. One of the distinguishing feature of vertex-transitive graphs is that Cayley sets that describe the adjacency are quasi-associative. A Cayley set S of a groupoid A is quasiassociative if x(yS) = (xy)S for every x, y ∈ A. (1) Note that quasi-associative sets were introduced by Gauyacq [4] who called them right associative sets but since the concept does not refer to element-wise associativity, a more neutral term has been preferred. Recall that for a fixed element a of a groupoid A, the left translation λa of A by a is a map given by λa x = ax, x ∈ A. It turns out that when Cayley sets of left loops are quasi-associative, left translations are
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automorphisms of the graph. Lemma 2.1. Let G = GG(A, S), where A is a left loop and S ⊂ A a quasi-associative Cayley set. Then any left translation λa , a ∈ A, is an automorphism of G. Proof. For any x, y ∈ A, [x, y] ∈ E(G) ⇒ y = xs, s ∈ S ⇒ ay = a(xs) = (ax)s′ , s′ ∈ S ⇒ [λa (x), λa (y)] ∈ E(G) Because we are considering finite structures, it is easy to see that λa is a bijection. We have to show that λ−1 a preserves adjacency. We have that for any x, y ∈ A, [x, y] ∈ E(G) ⇒ y = xs, s ∈ S −1 −1 ⇒ λ−1 a (y) ∈ λa [xS] = λa (x)S −1 ⇒ [λ−1 a (x), λa (y)] ∈ E(G).
Hence we have the result.
Proposition 2.2. Let A be a left loop with right unit u and S a quasi-associative Cayley set. Then the graph GG(A, S) is vertex-transitive. To show that that is the case, it suffices to show that for any a ∈ A, there exists an automorphism σ ∈ Aut G such that σ(u) = a. However, for any a, we take σ = λa because by Lemma 2.1 it is an automorphism mapping u to a. At this point, a natural question arises: are vertex-transitive graphs left loop graphs with quasi-associative Cayley sets? The answer to the question is positive and the remaining part of this section is dedicated to elaborating this concept. Theorem 2.3. Let G be a vertex-transitive graph. Then there is a left loop Q with a right unit and a quasi-associative Cayley set S ⊂ Q such that G ∼ =GG(Q, S). Proof. For the given graph G, choose an arbitrary vertex u ∈ V (G) as base point. Let A be a subgroup of Aut G which acts transitively on V (G) and consider the stabilizer of u in A : Au = {α ∈ A : α(u) = u}. Let T be a transversal of the left cosets of Au . Note that given any σ, τ ∈ T, σ = τ ⇐⇒ σ(u) = τ (u).
(2)
Define a binary operation ∗ on T as follows. Given σ, τ ∈ T , let σ ∗ τ ∈ T be the representative of the coset στ Au . Thus (σ ∗ τ )(u) = στ (u).
(3)
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Denote by ǫT the representative of Au in T. It is easy to see that QT := (T, ∗) is a left quasi-group with right unit ǫT . Now, let S := {α ∈ A : [u, α(u)] ∈ E(G)}. It is routine to show that ST ⊂ T defined by ST := S ∩ T is a Cayley set. Moreover, we have the following. Let σ, τ ∈ T, α ∈ ST . Then [u, α(u)] ∈ E(G) implies στ α(u) ∈ N((στ )(u)),
(4)
where N((στ )(u)) is the set of neighbours of (στ )(u). By (3), ((σ ∗ τ )−1 (στ ))(u) = u, hence applying (σ ∗ τ )−1 to (4) we get ((σ ∗ τ )−1 (στ α))(u) ∈ N(u). Therefore ((σ ∗ τ )−1 (στ α))(u) = α′ (u) for some unique α′ ∈ ST . Thus we have (στ α)(u) = (σ ∗ τ )(α′ (u)) = ((σ ∗ τ ) ∗ α′ )(u) (because (σ ∗ τ ), α′ ∈ T ). So using (3) twice, we have (σ ∗ (τ ∗ α))(u) = ((σ ∗ τ ) ∗ α′ )(u). Hence ST is quasi-associative in (QT , ∗).
Claim 2.4. GG(QT , ST ) ∼ =G Proof. The map f : T −→ V (G) defined by f (τ ) = τ (u),
(5)
is an isomorphism GG(QT , ST ) −→ G. Since T is a left transversal of left cosets of Au , f is clearly a bijection. (i) f preserves adjacency: for α ∈ ST , the edge [τ, τ ∗α] is mapped to [τ (u), (τ ∗α)(u)] = [τ (u), τ α(u)] = τ [u, α(u)] ∈ E(G). (ii) f −1 preserves adjacency: let [x, y] ∈ E(G). There is a unique τ ∈ T such that τ (u) = x. Let α ∈ T such that α(u) = τ −1 (y). Since τ −1 (y) ∈ N(u) we have that α ∈ ST , and hence [x, y] = [τ (u), τ α(u)] = [τ (u), τ ∗ α(u)] = f [τ, τ ∗ α] This completes the proof. If the transversal T is chosen such that ǫT = 1G (the identity permutation of V (G)), then ǫT is the two-sided identity of (QT , ∗); in other words: every vertex-transitive graph can be represented by a groupoid graph of a left loop.
3
Multiples of left loops
The elements of the left loop described in Theorem 2.3 constitute a transversal T of Au so that the map f : T × Au −→ A given by f (α, a) = αa is a bijection to the group A which acts transitively on V (G). This position motivates the following definition.
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Definition 3.1. Let H be a subgroup of a group G and T a transversal of left cosets of H. For a left loop L, we say that L is a factor of G if L is isomorphic to T as a left loop with the binary operation defined by (3). In this section, we show that any left loop L with |L| = n is a factor of Sn . The possibility of such a fact lies in the resemblance of Cayley sets of left loops to those which one encounters in groups; unlike in the general groupoid case where there is not any resemblance at all. We shall explore such a resemblance before considering the question of left loops as factors of groups. Let L be a left loop with unity u. Given a ∈ L, define fa : L −→ L by fa (x) = λ−1 ax (a), x ∈ L.
(6)
−1 Taking x = u in (6), we get that fa (u) = λ−1 au (a) = λa (a) = u. Conversely, if fa (u) = u then a = (ax)u = ax hence x = u. Thus
fa [L \ {u}] ⊂ L \ {u}.
(7)
(7) is equivalent to saying that L \ {u} is a Caylet set. This is a consequence of the following: Claim 3.2. A subset S of L is Cayley if and only if u ∈ / S and fa [S] ⊂ S for any a ∈ L. Proof. Because of left cancellativity the condition a ∈ / aS is clearly equivalent to u ∈ / S. Now let a ∈ L, s ∈ S. Then (as)fa (s) = a. Hence if fa (s) ∈ S, then the second condition holds for a. Conversely, if (as)s′ = a for some s′ ∈ S, then (as)fa (s) = (as)s′ , so that by left cancellativity, fa (s) = s′ . As for quasi-associativity, we note that Remark 3.3. In any left quasi-group, the quasi-associative subsets form a complete boolean algebra of sets. It is evidently clear that {u}, u being the unit of a left loop L is quasi-associative, hence by Remark 3.3, L \ {u} is quasi-associative. In view of the above observations, we get the following not so surprising result. Proposition 3.4. For any left loop L with unit u, the groupoid graph GG(L, L \ {u}) is a complete graph. Now, Sn acts transitively on V (GG(L, L \ {u})). Let us turn to the machinery developed in section 2. That is, we fix a vertex a ∈ L, and realise the stabiliser (Sn )a of a from which left cosets in Sn are described.
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We have that the map f : (L, (Sn )a ) −→ Sn defined by f (a, α) = aα is a bijection and hence L is a factor of Sn . We therefore have the following result. Theorem 3.5. Every left loop is a factor of a group. This concludes what we had set out to describe: that left loops are intimately related to groups.
References [1] A.A. Albert: “Quasigroups I”, Trans. Amer. Math. Soc., Vol. 54, (1943), pp. 507–520. [2] W. D¨orfler: “Every regular graph is a quasi-regular graph”, Discrete Math., Vol. 10, (1974), pp. 181–183. [3] E. Mwambene: Representing graphs on Groupoids: symmetry and form, Thesis (PhD), University of Vienna, 2001. [4] G. Gauyacq: “On quasi-Cayley graphs”, Discrete Appl. Math., Vol. 77, (1997), pp. 43–58. [5] C. Praeger: “Finite Transitive permutation groups and finite vertex-transitive graphs”, In: G. Sabidussi and G. Hahn (Eds.): Graph Symmetry: Algebraic Methods and Applications, NATO ASI Series, Vol. 497, Kluwer Academic Publishers, The Netherlands, Dordrecht, 1997. [6] G. Sabidussi: “Vertex-transitive graphs”, Monatsh. Math., Vol. 68, (1964), pp. 426– 438.
CEJM 3(2) 2005 251–259
Tensor products of symmetric functions over Z2 Karl Heinz Dovermann1∗, Jason Hanson2† 1
2
Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
visiting: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA DigiPen Institute of Technology, Redmond, WA
Received 16 December 2004; accepted 4 February 2005 Abstract: We calculate the homology and the cycles in tensor products of algebras of symmetric function over Z2 . c Central European Science Journals. All rights reserved.
Keywords: Symmetric Functions, differential graded algebras MSC (2000): 13D07, 13N10
1
Statement of Results
We calculate the homology and the cycles in tensor products of the differential graded algebra of symmetric functions over Z2 , the integers modulo 2. The need for this calculation arises in a project in which we show that closed smooth manifolds with cyclic group actions have equivariant real algebraic models [3] and [4]. There we need to calculate the ordinary equivariant cohomology of some classifying spaces with cyclic group action, and the symmetric functions and the differential ∇ arise in a way explained in some detail in Section 5. Some special cases of our results can be extracted from [6] and [5]. Let Fa := Z2 [z1 , . . . , za ] be the polynomial ring in a variables of dimension 1. The natural differential ∇ on Fa is the sum of the partial derivatives. This derivative is ∗ †
E-mail:
[email protected] E-mail:
[email protected]
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obtained from the standard rules of differentiation, linearlity and Leibnitz’ rule (the product rule), under the assumptions that the derivative of each zj is the constant function 1. Then ∇2 = 0, and (Fa , ∇) is a differential graded algebra. In Fa we consider the subalgebra Sa of symmetric functions. The r-th elementary symmetric function is denoted by σr . It is elementary to compute its derivative: ∇σr = (a − r + 1)σr−1 for a ≥ r ≥ 1 and ∇σ0 = 0.
(1)
The formula depends only on the parity of a, the number of variables, and not on its specific value. Consider a sequence A = (a(0), . . . , a(k)) of nonnegative integers and set SA = Sa(0) ⊗ · · · ⊗ Sa(k) .
(2)
As a tensor product, SA inherits a natural differential operator, which we still denote by ∇. We use an additional subscript to distinguish the factor to which an elementary symmetric function belongs. Specifically, σj,r is the r-th elementary symmetric function in the j-th factor of the tensor product, where 0 ≤ j ≤ k and 1 ≤ r ≤ a(j). The letter Z denotes the cycles of the indicated DGA. Theorem 1.1. Suppose A = (a(0), . . . , a(k)) and a(t) is odd for some fixed value of t between 0 and k. Then the differential graded algebra SA is acyclic. Set D(A) = {σt,1 + σj,1 | j 6= t, a(j) odd} D o (A) = {σj,2s , σt,1 σj,2s + σj,2s+1 | a(j) = 2nj + 1 and 1 ≤ s ≤ nj } D e (A) = {σj,2s−1 , σt,1 σj,2s−1 + σj,2s | a(j) = 2nj and 1 ≤ s ≤ nj }. 2 The cycles in SA are Z(SA ) = Z2 [{σt,1 } ∪ D(A) ∪ D o (A) ∪ D e (A)].
In our next two results, A = (2n0 , . . . , 2nk ) is a sequence of even nonnegative integers. The first result describes the homology of the DGA (SA , ∇), and is an immediate consequence of Corollary 4.2. 2 Corollary 1.2. Set T (A) = {σm,2j | 0 ≤ m ≤ k, 1 ≤ j ≤ nm }. Then
H∗ (SA , Z2 ) = Z2 [T (A)]. Next, we describe the cycles in (SA , ∇) as a module over the ring 2 ΛA = Z2 [{σm,2j−1 , σm,2j | 0 ≤ m ≤ k, 1 ≤ j ≤ nm }] ⊆ SA
Define the sets BA = B′A ∪ {1}, where B′A
=
∇b | b =
k Y
σm,2am (1) · · · σm,2am (tm ) , b has at least two factors,
m=0
and 1 ≤ am (1) < · · · < am (tm ) ≤ nm for all 0 ≤ m ≤ k .
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We define AA to be the module over ΛA with generating set BA . Proposition 1.3. If A = (2n0 , . . . , 2nk ), then Z(SA ) = AA . We prove this proposition in Section 4. In Remark 4.3 we discuss the structural difference between Theorem 1.1 and Proposition 1.3.
2
Some preliminary remarks
The formula for the derivatives of the elementary function in (1) is equivalent to: ∇σ2i =
(
σ2i−1
if a is even
0
if a is odd
and ∇σ2i+1 =
( 0
σ2i
if a is even if a is odd.
Remark 2.1. The homology of Sa(0) ⊗ · · · ⊗ Sa(k) is trivial if any one of the a(t) is odd. Suppose a(t) is odd, and f is any cycle, then f = ∇(σt,1 f ) is a boundary. Remark 2.2. In precise terms, σk ∈ Sa is the k-th symmetric function in a variables. Throughout we will work only with the symmetric functions and use the formulas for the derivatives. In this sense, a statement which we prove for Sa will also hold for the algebra S′a generated by the elementary symmetric functions of degree ≤ a in a + 2t variables. To avoid the introduction of further notation, we identify Sa and S′a .
3
Proof of Theorem 1.1
We prove the theorem by induction. The starting point of the induction is the following special case Theorem 1.1. Proposition 3.1. Suppose A = (1, . . . , 1) is a sequence of (k + 1) ones. Then the DGA (SA , ∇) is acyclic and its cycles are the subalgebra Z(SA , ∇) = Z2 [{z02 } ∪ {z0 + zj | 1 ≤ j ≤ k}] ⊆ Z2 [z0 , . . . , zk ]. We should start out with a sequence A consisting of zeros and ones, but S0 is trivial, and so we ignore the zeros and suppress the corresponding trivial factors in the tensor product. In Theorem 1.1, t denotes a distinguished position in which A has an odd entry. Without loss of generality, this position is t = 0 in the proposition. Set Sa(j) = S1 = Z2 [zj ], then σj,1 = zj , so that we may write SA = S1 ⊗ · · · ⊗ S1 = Z2 [z0 , . . . , zk ]. We note that D o (A) = D e (A) = ∅. This establishes the proposition as a special case of Theorem 1.1.
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We expressed Z(SA , ∇) as a polynomial ring, and this means that the variables (or generators) need to be algebraically independent. This is easy to check for the given set of generators, because each new generator involves a new variable. Proof. Remark 2.1 tells us that (SA , ∇) is acyclic. Let B denote the algebra generated by z02 and z0 + zj for 1 ≤ j ≤ k. Apparently ∇(z02 ) = 0 and ∇(z0 + zj ) = 0 for 1 ≤ j ≤ k, so that B ⊆ Z(SA , ∇). It remains to show that Z(SA , ∇) ⊆ B. More specifically, we will show that the boundary of any monomial q = z0m1 · · · zkmk in (SA , ∇) belongs to B, i.e., that ∇(q) ∈ B. Consider the quotient Z2 [z0 , . . . , zk ]/B. In this quotient we identified each zj with z0 because z0 + zj belongs to the generators of B. We may also reduce the exponent of z0 by 2 because z02 is a generator of B. In conclusion, Z2 [z0 , . . . , zk ]/B ∼ = Z2 , and the nonzero class is [z0 ]. Here and below we indicate equivalence classes in the quotient by square brackets. To conclude our argument, we show that [∇(q)] = 0 ∈ Z2 [z0 , . . . , zk ]/B. After identifying variables, we may suppose that q = z0m . If m is even, then ∇(q) = 0. If q is of odd degree, then ∇(q) is of even degree and belongs to B. In either case [∇(q)] vanishes and our argument is complete. In preparation of our inductive proof of Theorem 1.1, we study what happens to the cycles of the DGA when we increase one entry in the sequence A = (a(0), . . . , a(k)) by two. Suppose A is a sequence of nonnegative integers and its t-th term is 2n + 1. Let A′ be the sequence whose t-th entry is 2n + 3, and which agrees with A in all other places. To avoid double indexing, we just write σj for σt,j . Proposition 3.2. For A and A′ as above Z(SA′ ) = Z(SA ) ⊗ Z2 [σ2n+2 , σ1 σ2n+2 + σ2n+3 ] Proof. We use the abbreviation A = Z(SA ) ⊗ Z2 [σ2n+2 , σ1 σ2n+2 + σ2n+3 ]. Apparently, ∇σ2n+2 = 0 and ∇(σ1 σ2n+2 + σ2n+3 ) = 0, so that A ⊆ Z(SA′ ). We will show Z(SA′ ) ⊆ A. Consider an element f ∈ SA′ and express it in the form X 2j 2j+1 i i f= gi,2j σ2n+2 σ2n+3 + gi,2j+1σ2n+2 σ2n+3 , i,j≥0
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where ga,b ∈ SA . If f is a cycle, then X 2j 2j+1 2j i i i+1 0= ∇(gi,2j )σ2n+2 σ2n+3 + ∇(gi,2j+1)σ2n+2 σ2n+3 + gi,2j+1σ2n+2 σ2n+3 , i,j≥0
and we need to show that f belongs to A. A comparison of coefficients gives us the equations: ∇g0,2j = 0 and ∇gi,2j+1 = 0 and ∇gi,2j + gi−1,2j+1 = 0 for i > 0 and j ≥ 0. In the following it will be useful to observe that 2 2 (σ1 σ2n+2 + σ2n+3 )2 + σ12 σ2n+2 = σ2n+3 ∈ A. 2j Cycles of the first kind: Since g0,2j ∈ Z(SA ), the summands g0,2j σ2n+3 of f are in A. Cycles of the second kind: For a pair (i, j) with i > 0 we look at a pair of summands of f : 2j 2j+1 i i−1 b = gi,2j σ2n+2 σ2n+3 + gi−1,2j+1σ2n+2 σ2n+3 2j i−1 = σ2n+2 σ2n+3 (gi,2j σ2n+2 + gi−1,2j+1 σ2n+3 ).
As we have seen, each of the three factors of b, and hence b itself, is a cycle. We like to show that b ∈ A. The first two factors of b are in A. It remains to be shown that the third factor b′ = gi,2j σ2n+2 + gi−1,2j+1 σ2n+3 is in A. Observe that gi−1,2j+1 ∈ Z(SA ), and gi−1,2j+1(σ1 σ2n+2 + σ2n+3 ) ∈ A. Showing that b′ ∈ A is equivalent to showing that the cycle b′′ = b′ + gi−1,2j+1 (σ1 σ2n+2 + σ2n+3 ) = (gi,2j + gi−1,2j+1 σ1 )σ2n+2 is in A. Observe that (gi,2j + gi−1,2j+1σ1 ) ∈ Z(SA ) and σ2n+2 ∈ A. It follows that b′′ ∈ A, and so b ∈ A. Taken together, our cycles of the first and second kind make up all the summands of f , thus f ∈ A. This completes the proof. As an immediate consequence of Theorem 3.1 and Proposition 3.2 we obtain the computation of the cycles in SA if all entries in A are odd. If the length of A is one, and this one nonzero entry is 2n + 1, then we have Corollary 3.3. The differential graded algebra (S2n+1 , ∇) is acyclic and Z(S2n+1 ) = Z2 [σ12 , σ2 , . . . , σ2n , σ1 σ2 + σ3 , . . . , σ1 σ2n + σ2n+1 ]. Suppose that A = (a(1), . . . , a(k)) is a sequence of nonnegative integers, a(t) is odd, and a(s) = 2n is even. Let A′ be identical to A, with the one exception that the entry in position s is 2n + 2. To avoid double indexing, we write σj for σs,j .
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Proposition 3.4. For A and A′ as above Z(SA′ ) = Z(SA ) ⊗ Z2 [σ2n+1 , σt,1 σ2n+1 + σ2n+2 ]. Proof. The proof is the same as the one of Proposition 3.2, except for a shift in grading by 1. Proof of Theorem 1.1. We compute Z(SA ) inductively, starting out with a sequence Ao which has a 1 in those places where A has an odd entry and a 0 in those places where A has an even entry. For A = Ao , Theorem 1.1 specializes to Proposition 3.1 (as explained after the statement of the proposition), and this proposition we proved already. Propositions 3.2 and 3.4 describe the effect on the cycles of the differential graded algebra when an odd, resp. even, entry is increased by two. Either of these increases adds two algebra generates to a basis for the cycles, exactly as it is described in the assertion of Theorem 1.1.
4
Symmetric functions of an even number of variables
Throughout this section A = (a(0), . . . , a(k)) = (2n0 , . . . , 2nk ) is a sequence of even nonnegative integers. We specify a position t in this sequence. To simplify notation we set a(t) = 2nt = 2n. Let A′ be the sequence whose t-th entry is 2n + 2, and which agrees with A in all other places. To avoid double indexing, we just write σj for σt,j . Proposition 4.1. Let A and A′ be as above. Then any cycle f in SA′ can be expressed in the form 2 2N f = b + c0 + c2 σ2n+2 + · · · + c2N σ2n+2 , where b is a boundary in SA′ , and c0 , . . . , c2N are cycles in SA . Corollary 4.2. Let A and A′ be as above. Then 2 H∗ (SA′ , Z2 ) = H∗ (SA , Z2 ) ⊗ Z2 [σ2n+2 ].
Proof of Proposition 4.1. Express f ∈ SA′ in the form X 2j 2j+1 i i f= gi,2j σ2n+1 σ2n+2 + gi,2j+1σ2n+1 σ2n+2 , i,j≥0
where the ga,b are in SA . Assuming that f is a cycle we find that X 2j 2j+1 2j i i i+1 0= ∇(gi,2j )σ2n+1 σ2n+2 + ∇(gi,2j+1)σ2n+1 σ2n+2 + gi,2j+1σ2n+1 σ2n+2 . i,j≥0
Comparison of coefficients provides us with the equations
∇g0,2j = 0 and ∇ (gi,2j+1) = 0 and ∇ (gi,2j ) + gi−1,2j+1 = 0
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257
for i > 0 and j ≥ 0. Cycles: As ∇g0,2j = 0, we may set c2j = g0,2j . Boundaries: For i ≥ 1 we have 2j+1 2j+1 2j i−1 i−1 i ∇ gi,2j σ2n+1 σ2n+2 = ∇(gi,2j )σ2n+1 σ2n+2 + gi,2j σ2n+1 σ2n+2
2j+1 2j i−1 i = gi−1,2j+1 σ2n+1 σ2n+2 + gi,2j σ2n+1 σ2n+2 .
Together, these terms give us the summand b called for in the proposition. As one may easily verify, the exhibited boundaries and cycles together make up all summands of f , so that our proposition is proved. Proof of Proposition 1.3. The assertion is that Z(SA ) = AA . By definition, AA is a module over the ring ΛA , generated by the set BA . It is trivial to verify that all elements in ΛA and of BA belong to Z(SA ), so that AA ⊆ Z(SA ). Next we show that Z(SA ) ⊆ AA . The given basis elements for H∗ (SA , Z2 ) given in Corollary 1.2 belong to AA , and it remains to be shown that the boundaries B(SA ) in SA belong to AA . Let us describe the boundaries in SA . Let h be a monomial in SA . We express it in the form k 2n k 2n k 2n m m m Y Y Y Y Y Y ǫ(m,j) s(m,j) r(m,j) σm,j = λ · β, σm,j · σm,j = h= m=0 j=1
m=0 j=1
m=0 j=1
and break it up as a product λ · β. In the first double product we collected all the factors of h that belong to ΛA , and we abbreviated it as λ. In particular, s(m, j) = r(m, j) if j is odd, and s(m, j) is the largest even summand of r(m, j) if j is even. Consequently, ǫ(m, j) is 0 or 1 if j is even according to whether r(m, j) is even or odd. Because λ is a cycle, we have ! k Y ǫ(m,1) ǫ(m,2n ) (3) ∇h = λ · ∇ σm,1 · · · σm,2nm m . m=0
Each boundary is the sum of boundaries of monomials, and (3) tells us what they look like. We distinguish cases based on the number of factors of β. If β has no factor, then ∇h = 0, and there is nothing to be proved. If β has exactly one factor, say σm,2j , then ∇h = λσm,2j−1 ∈ ΛA ⊂ AA . If β has at least two factors, then ∇h = λ · ∇β, and β ∈ B′A , so that ∇h ∈ AA . In conclusion, all Z2 -homology classes of (SA , ∇) are represented by elements in AA . The boundaries of all monomials in (SA , ∇), and hence the boundaries of all elements in (SA , ∇) are in AA . Hence Z(SA ) ⊂ AA , and our proof is complete. Remark 4.3. If the sequence A has at least one odd entry, then we are able to write down a basis (algebraically independent generating set) of the algebra Z(SA ). If all entries of A are even, then we are able to describe Z(SA ) only as a module AA over a ring ΛA with a generating set BA . Still, even this generating set is not a basis, and we do not know whether the algebra Z(SA ) has a basis, or if the module AA over ΛA has a basis. Even
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if A = (6) consists of only one even entry, then BA is not linearly independent. Here is a relation in between the module generators over ΛA : 0 = σ5 ∇(σ2 σ4 ) + σ3 ∇(σ2 σ6 ) + σ1 ∇(σ4 σ6 ) = σ5 (σ1 σ4 + σ2 σ3 ) + σ3 (σ1 σ6 + σ2 σ5 ) + σ1 (σ3 σ6 + σ4 σ5 ).
5
Topological motivation
In [4] we need to calculate certain equivariant singular bordism groups. Well known techniques, as established in [2] and [8], reduce the bordism calculation to the computation of the homology group H ∗ (E(C) ×C F, Z2 ) (4) where C is a cyclic group, E(C) is the universal C-space (contractible with free action of C), and F is a product of equivariant Grassmannians. In the calculation one uses the Leray-Serre spectral sequence of the fibration F → E(C) ×C F → B(C). The local coefficient system H(F, Z2 ) turns out to be simple, so that E2p,q = H p (B(C), Z2 ) ⊗ H q (F, Z2 ). The cohomology of each factor of F is an algebra of symmetric functions [1]. The cohomology of B(C) can be found in [7]. The spectral sequence collapses at the E2 -level if the transgression is trivial, and this happens when the order of C is an odd number. So, in this case the calculation of (4) is easily completed. If the order of C is twice an odd number, then C may act nontrivially on some of the factors of F. Denote the product of these factors by Fb . Consider the spectral sequence of the fibration Fb → E(C) ×C Fb → B(C). The cohomology of Fb is a tensor product of algebras of symmetric function SA , as discussed earlier in the paper. The sequence A of indices depends on the factors of Fb . Let x(n) denote the nonzero element in H n (B(C), Z2 ). An element u ∈ SA is a polynomial in variables of degree 1, and one shows that the transgression maps each of them to x(2) . The algebra structure of the spectral sequence allows us to write down the formula for the differential at the E2 -level of the spectral sequence for a typical generator: d2 (x(p) ⊗ u) = x(p+2) ⊗ ∇(u). Here ∇ is the differential of the DGA discussed throughout the paper. This allows us to compute the E3 -term of the spectral sequence: ( {x(p) ⊗ u | u ∈ Zq (SA )} if 0 ≤ p ≤ 1 E3p,q ∼ (5) = {x(p) ⊗ u | u ∈ Hq (SA )} if 2 ≤ p.
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The spectral sequence collapses at the E3 -level and thus we obtain the calculation of H ∗ (E(C) ×C Fb , Z2 ). p,q In conclusion, our paper provides the calculation of E3p,q = E∞ . The general form of the generators is needed in [3], and the precise calculation for the cycles is needed in [4], if at least one entry in the sequence A is odd.
References [1] A. Borel: Topics in the Homology Theory of Fibre Bundles, Lecture Notes in Mathematics, Vol. 36, Springer Verlag, Berlin, Heidelberg, New York, 1967. [2] P.E. Conner and E.E. Floyd: Differentiable Periodic Maps, Ergebnisse der Mathematik, Vol. 33, Springer Verlag, Berlin–Heidelberg–New York, 1964. [3] K.H. Dovermann and A.G. Wasserman: Algebraic Realization for Cyclic Group Actions with one Isotropy Type, preprint. [4] K.H. Dovermann et al: Algebraic Realization for cyclic group actions, in preparation. [5] J.S. Hanson: Bordism and Algebraic Realization, Thesis (PhD.), University of Hawaii at Manoa, 1998. [6] C. Kosniowski: Actions of Finite Abelian Groups, Research Notes in Mathematics, Vol. 18, Pitman, London–San Francisco–Melbourne, 1978. [7] J.-P. Serre: “Cohomologie modulo 2 des complexes d’Eilenberg-MacLane”, Comm. Math. Helv., Vol. 27, (1953), pp. 198–232. [8] R.E. Stong: “Unoriented Bordism and Actions of Finite Groups”, Memoirs of the Amer. Math. Soc., Vol. 103, (1970).
CEJM 3(2) 2005 260–272
Multiple Prime covers of the Riemann sphere Aaron Wootton∗ Department of Mathematics, University of Arizona, 617 North Santa Rita, Tucson AZ85721, USA
Received 14 June 2004; accepted 8 March 2005 Abstract: A compact Riemann surface X of genus g > 2 which admits a cyclic group of automorphisms Cq of prime order q such that X/Cq has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p 6= q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p. c Central European Science Journals. All rights reserved.
Keywords: Automorphism group, compact Riemann surface, hyperelliptic curve MSC (2000): 14H30, 14H37, 30F10, 30F60, 20H10
1
Introduction
A compact Riemann surface X of genus g > 2 which admits a cyclic group of automorphisms Cq of prime order q such that X/Cq has genus 0 is called a cyclic q-gonal surface or a q-gonal surface for brevity. The group Cq is called a q-gonal group for X. If in addition Cq is normal in the full automorphism group of X, then we call X a normal cyclic q-gonal surface or a normal q-gonal surface. As they are the central focus of this paper, we call a surface X which is both cyclic q-gonal and cyclic p-gonal for primes p 6= q a multiple prime surface. The primary aim of this paper is to classify all multiple prime surfaces. By classify, we mean find the full automorphism group and the signature for the normalizer of a surface group for each such surface. There are a number of interesting consequences of this classification, two of the most interesting are the following. ∗
E-mail:
[email protected]
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Theorem. A cyclic q-gonal surface can be p-gonal for at most one other prime p. Theorem. If X is a multiple prime surface which is cyclic q-gonal and cyclic p-gonal, then any element from a q-gonal group commutes with any element from a p-gonal group. We start in Section 2 by developing a number of general results regarding automorphism groups of compact Riemann surfaces, uniformization and Fuchsian groups - discrete subgroups of PSL (2, R). Following this in Section 3, we shall examine a number of preliminary results more specific to the situation we are considering. Section 4 will present one of the main steps needed for this classification. Specifically, we shall show that the group generated by a q-gonal and p-gonal group in the full automorphism group of a multiple prime surface X is cyclic of order pq. With these results, in Section 5 we shall complete the classification and prove explicitly that a cyclic q-gonal surface can be p-gonal for at most one other prime p. The author would like to express his gratitude to the referees for their useful comments and in particular for their input toward a complete proof of Theorem 4.5.
2
Fuchsian Groups and Uniformization
In this section, we develop the necessary theory regarding automorphism groups of compact Riemann surfaces, uniformization and Fuchsian groups. Let X denote a compact Riemann surface of genus g > 2 and G a group of automorphisms of X. Uniformization implies that X is conformally equivalent to a quotient of the upper half plane H by a torsion free Fuchsian group Λ called a surface group or a surface kernel for X. Under such a realization, a group G is a group of automorphisms of X if and only if G = Γ/Λ for some Fuchsian group Γ containing Λ as a normal subgroup, see [5]. It follows that the full automorphism group Aut(X) of X is the quotient group N(Λ)/Λ where N(Λ) denotes the normalizer of Λ in PSL(2, R). Since G is a group of automorphisms acting on X, we can form the quotient space X/G which can be endowed with a unique structure making the map πG : X → X/G a holomorphic map between compact Riemann surfaces. The group Γ is a group of biholomorphic maps acting on H, so we can form the quotient space H/Γ with a complex structure so that the map πΓ : H → H/Γ is holomorphic. Let πΛ : H → H/Λ = X denote the smooth unramified cover of X by H. Then X/G can be identified with H/Γ and after identification, we have πΓ = πG ◦ πΛ : πΓ
H
πΛ
H/Λ = X
πG
H/Γ = X/G
Fig. 1 Holomorphic quotient maps and surface identifications.
For a Fuchsian group Γ with compact orbit space H/Γ of genus g, a presentation for
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Γ is: Γ=
mr 1 ha1 , b1 , . . . , ag , bg , c1 , . . . , cr |cm 1 , . . . , cr ,
r Y
ci
i=1
g Y
[aj , bj ]i
j=1
where the quotient map πΓ branches over r points with ramification indices mi for 1 6 i 6 r. The presentation of such a group is described by the tuple (g; m1 , . . . , mr ) called the signature of Γ, g the orbit genus of Γ and m1 , . . . , mr the periods of Γ. Notice that if Γ is a surface group for a surface of genus g, since it is torsion free, it must have signature (g; −). Given a tuple (g; m1 , . . . , mr ), it is natural to ask when this tuple is the signature for some Fuchsian group Γ. The statement of the answer to this question dates back to Poincar´e and a complete rigorous proof was published by Maskit in [6]. It was shown that a tuple (g; m1, . . . , mr ) is the signature for a Fuchsian group Γ if and only if it satisfies the P inequality 2g − 2 + ri=1 (1 − m1i ) > 0. Among other interesting consequences, this result can be used to prove that the automorphism group G of a compact Riemann surface of genus g > 2 satisfies |G| 6 84(g − 1), a bound often referred to as the Hurwitz bound. We shall now interpret this information into results we shall be using. If X is a cyclic q-gonal surface, let Cq be a q-gonal group for X and if Λ is a surface group for X, let Γq denote the Fuchsian group with Γq /Λ = Cq . Let G denote a subgroup of the normalizer in Aut (X) of Cq and let Γ be the Fuchsian group with Γ/Λ = G. Since Cq is normal in G, it follows that the group K = G/Cq acts by automorphism on the quotient space X/Cq . Let πK denote the quotient map of the surface X/Cq by K. After appropriate identifications, we get the tower of Galois covers illustrated in Figure 2. Since πΛ is unramified, our remarks imply that Γq has signature (0; q, . . . , q ) where r is the number | {z } r times
of branch points of the map πCq . With a little more work, we can find the the signature of Γ. πΓ
πΓq
H
πΛ
H/Λ
πCq
H/Γq
πK
H/Γ
πG
Fig. 2 Holomorphic quotient maps and surface identifications.
The map πK is a Galois map from the Riemann sphere to itself and the branching properties of such maps are well known. We summarize them in Table 1. The branching data is a vector whose length is the number of branch points of the quotient map πK and whose entries are the ramification indices of ramification points above these branch points. To find the possible signatures for Γ, we use the fact that we know complete branching
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Group Branching Data Cn
(n, n)
Dn
(2, 2, n)
A4
(2, 3, 3)
S4
(2, 3, 4)
A5
(2, 3, 5)
Table 1 Groups of automorphisms of the Riemann sphere and branching data.
data of the maps πK and πCq . It is then a simple matter of determining whether or not any branch points of πCq coincide with any ramification points of πK . We summarize below. Proposition 2.1. Let K be the group Γ/Γq . (i ) If K 6= Cn and (m1 , m2 , m3 ) is the branching data of the quotient map πK , the signature of Γ is (0; am1 , bm2 , cm3 , q, . . . , q) where a, b, and c are either 1 or q | {z } s times
depending upon whether any branch points of πCq coincide with ramification points of πK . For such a Γ, the signature of Γq is (0; q . . . , q) where | {z } r times
r = s|K| +
(a − 1)|K| (b − 1)|K| (c − 1)|K| + + . (q − 1)m1 (q − 1)m2 (q − 1)m3
(ii ) If K = Cn , the signature of Γ is (0; an, bn, q, . . . , q ) where a, and b are either 1 or q | {z } s times
depending upon whether any branch points of πCq coincide with ramification points of πK . For such a Γ, the signature of Γq is (0; q, . . . , q ) where | {z } r times
r = sn +
Proof. See [8], Proposition 3.1.
3
(a − 1)|K| (b − 1)|K| + . (q − 1)n (q − 1)n
Preliminary Results
Automorphism groups of compact Riemann surfaces and in particular, cyclic q-gonal surfaces, have been the focus of much research in the last century. In this section, we shall present such results which are specific to the problem we are considering. The first result we examine restricts the different genera we need to consider when studying such surfaces.
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Lemma 3.1. If X is cyclic q-gonal with q-gonal group Cq and the quotient map πCq : X → X/Cq is branched over r points, then the genus of X is ( 2r − 1)(q − 1). In particular, the smallest genus (greater or equal to 2) for which such a group can exist is g = 21 (q − 1) for q > 5 and 2 for q = 2 or 3. Proof. The group Γq will have signature (0; q, . . . , q). Since the surface group Λ is torsion | {z } r times
free, all elliptic generators will have non-trivial image in the quotient group Cq = Γq /Λ. Using the Riemann-Hurwitz formula, we get r
and thus
1 q 1 r qX (1 − ) = −q + r(1 − ) = −q + (q − 1), g − 1 = −q + 2 i=1 q 2 q 2 r r r g = −q + 1 + (q − 1) = −(q − 1) + (q − 1) = ( − 1)(q − 1). 2 2 2
The main results we produce rely heavily on the fact that if the size of the automorphism group of a surface is sufficiently large in relation to its genus, then there are a very small number of possibilities for the signature of the normalizer for a surface group for such a surface. Lemma 3.2. Suppose that G is an automorphism group of a compact Riemann surface X of genus g and |G| > 13(g − 1). If Λ is a surface group for X and Γ the Fuchsian group with Γ/Λ = G, then Γ is a triangle group - a Fuchsian group whose signature has three periods and orbit genus 0 - with one of the signatures tabulated in Table 3.
Case Signature Additional Conditions 1
(0; 3, 3, n)
46n65
2
(0; 2, 6, 6)
3
(0; 2, 5, 5)
4
(0; 2, 4, n)
5 6 n 6 10
5
(0; 2, 3, n)
7 6 n 6 78
Table 2 Signatures for large automorphism groups.
Proof. We extend the proof of Lemma 3.18 in [3]. (gΓ ; m1 , . . . , mr ). Since |G| > 13(g − 1) we have r
Suppose that Γ has signature r
|G| X 1 13(g − 1) X 1 g − 1 = |G|(gΓ − 1) + (1 − ) > 13(g − 1)(gΓ − 1) + (1 − ). 2 i=1 mi 2 m i i=1
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Simplifying, we get
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r
X 1 2 > 2(gΓ − 1) + (1 − ). 13 m i i=1
From this last inequality, it is clear that gΓ = 0. After simplifying, this gives the inequality r
r X 1 28 > >r− 2 mi 13 i=1 and thus 5>
56 > r. 13
If r = 4, then 4
311 28 3 1 28 X 1 4> = + + > + >4 78 13 2 3 13 i=1 mi
which is clearly not the case. Hence we must have r = 3. We now need to consider the different possible signatures. Order the periods of Γ so that m1 6 m2 6 m3 . Since r = 3, we know that 1 1 11 1 + + > . m1 m2 m3 13 By simple calculation, this implies that m1 6 3. Assume that m1 = 3. It follows that 1 1 20 + > . m2 m3 39 Further simple calculations show that m2 = 3, and under these circumstances, we have m3 < 6. If we assume m1 = 2 it follows that 1 1 9 + > m2 m3 26 and consequently m2 6 6. If m2 = 6, then m3 must also be 6, if m2 = 5, then m3 is also 5, if m2 = 4, then 5 6 m3 6 10 and if m2 = 3 then 7 6 m3 6 78. The following result specifies conditions on the periods of the normalizer of a cyclic q-gonal surface. Lemma 3.3. Suppose that X is a cyclic q-gonal surface and Γ is the normalizer of a surface group for X. Then the signature for Γ must have periods divisible by q and have orbit genus 0. Proof. Since X is cyclic q-gonal, there will exist an intermediate subgroup Γq of Γ and a surface group Λ for X with signature (0; q, . . . , q). Since Λ is torsion free, the periods of Γq must be induced by (conjugates to powers of) elliptic generators of Γ. Hence Γ must have elements whose order is divisible by q and in particular must have periods divisible
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by q. It must have orbit genus 0 because it contains a Fuchsian group, Γq , whose orbit genus is 0. The next two results are important because they relate information about the genus of a compact Riemann surface X and the structure of groups which can act on X. Both results are due to Accola, see [1] and [2]. Theorem 3.4. Let X be a compact Riemann surface of genus g. Suppose that X admits a finite group of automorphisms G with subgroups G1 . . . . , Gn such that G = ∪ni=1 Gi and Gi ∩ Gj is trivial for i 6= j. Let gi be the genus of the surface X/Gi for 1 6 i 6 n and let g0 be the genus of the surface X/G. Then (n − 1)g + |G|g0 =
n X
|Gi |gi.
i=1
Theorem 3.5. If X of genus g is cyclic q-gonal and g > (q − 1)2 , then X is normal cyclic q-gonal.
4
The Group Generated by a p-gonal and q-gonal Group
Suppose X is a multiple prime surface which is cyclic p-gonal and q-gonal for primes p 6= q and let Cp and Cq denote p-gonal and q-gonal groups for X respectively. We shall show that the group generated by Cp and Cq is cyclic of order pq through a series of Lemmas. We shall first show that the group G generated by Cp and Cq has order pq. To prove this, we shall make essential use of the fact that if G has order greater than pq, then neither Cp nor Cq can be normal in G, and in fact Cp ∩ NG (Cq ) = 1 and Cq ∩ NG (Cp ) = 1. Following this, we shall show that any group of order pq generated by Cp and Cq is necessarily cyclic. Without loss of generality, we shall henceforth assume that p > q. Lemma 4.1. Suppose X is a multiple prime surface which is q-gonal for q ∈ {2, 3, 5, 7}. If X is p-gonal for p 6= q, then the group G generated by a p-gonal group and a q-gonal group has order pq. Proof. By Theorem 3.5, if g > (q − 1)2 , then X is normal q-gonal and so Cq will be normal in the full automorphism group Aut(X) of X. In particular, it will be normal in G and consequently the order of G must be pq. Therefore given q, by Lemma 3.1, we just need to consider surfaces of genus g=
n (q − 1) 2
where 1 6 n 6 2(q − 1). Since we are assuming q ∈ {2, 3, 5, 7}, this means we only need consider surfaces of genus g 6 36. For all such genera, Breuer developed lists of all automorphism groups and corresponding signatures for Fuchsian groups in [3]. Therefore, we can proceed through these lists to explicitly show that no surfaces exist admitting
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automorphism groups with the specified properties. To illustrate, we shall examine the case q = 7 in more detail. For q = 7, we need to consider surfaces of genus 3k where 1 6 k 6 12. Assuming the result for q ∈ {2, 3, 5}, we can loop through all these possible genera and use Lemma 3.1 to find all primes in addition to q = 7 which occur for that genus. For genus g = 6, there are cyclic 7-gonal and cyclic 13-gonal surfaces. By Lemma 3.2, if a surface were cyclic 7-gonal and cyclic 13-gonal, the normalizer of a surface group for X would have orbit genus 0 and periods divisible by 7 and 13. By observation of Breuer’s list for genus g = 6, we see that no such signature exists and hence there exists no surface of genus 6 which is 7-gonal and 13-gonal. Similar arguments holds for all genera g 6= 36 for which there exist 7-gonal and 13-gonal surfaces. For genus g = 36, there does exist a surface which is 7-gonal and 13-gonal, but in this case G = Aut(X) is cyclic of order pq. Identical arguments hold for all other possible choices of p and each corresponding choice for g. Lemma 4.2. There does not exist G with |G| > 13pq for any choice of p and q. Proof. If |G| > 13pq, X cannot be normal cyclic q-gonal, so it follows that |G| > 13pq > 13q 2 > 13(q − 1)2 > 13g > 13(g − 1). Therefore if Λ is a surface group for X and Γ is the Fuchsian group with Γ/Λ = G, then Γ must have one of the signatures given in Table 3. For each of these signatures, since we are assuming that p > q, the only possible choices for q are 2, 3, 5, and 7. However, by Lemma 4.1, if q ∈ {2, 3, 5, 7}, then |G| = pq. Thus there exists no surface X with |G| > 13pq. Lemma 4.3. There does not exist G with q = 11 and |G| = 121p or |G| = 132p for any choice of p > 11. Proof. If |G| = 6 pq, then |G| = apq for some integer a > 1. Assuming |G| = apq for some a > 1, it follows that Cp ∩ NG (Cq ) = 1 and Cq ∩ NG (Cp ) = 1. Using the Sylow Theorems, this implies there exist integers a1 and a2 , both divisors of a, and b1 and b2 such that a1 q = b1 p + 1
(1)
a2 p = b2 q + 1.
(2)
a1 > b1
(3)
a2 < b2 .
(4)
and Since p > q, it also follows that and If a = 11, then a1 = 1 or 11, so (1) implies that either 121 = b1 p + 1 or 11 = b1 p + 1. In the latter case, b1 p = 10 so the only possible choices for p are 2 or 5. Both choices
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contradict our assumption that p > q. In the former case, b1 p = 120. This implies the only possible choices for p are 2, 3 and 5 which also contradicts our assumption that p > q. Now suppose a = 12. In this case, we can have a1 = 1, 2, 3, 4, 6, or 12. For a1 = 1, 2 or 3, the only possibilities for p are less than 11 contradicting our assumption that p > q. For a1 = 4, we get p = 43, but there are no possible values of g 6 100 = (11 − 1)2 for which both 11 and 43 are admissible. For a1 = 6, we get p = 13 and the possible genera are g = 30, 60 and 90. However, for each of these choices of g, |G| = 1716 > 13(g − 1) so they each reduce to the cases considered in Lemma 4.1. Finally, if a1 = 12, we get p = 131 and g = 65. In this case, |G| = 17292 > 84(g − 1) which contradicts the Hurwitz bound. Thus we cannot have q = 11 and |G| = 121p or |G| = 132p for any choice of p > 11. Lemma 4.4. There does not exist G with |G| = apq for 2 6 a 6 12 and p > q > a. Proof. Let a1 , a2 , b1 and b2 be as defined in (1) and (2) of the proof of Lemma 4.3. By Lemma 4.1, we may assume that q > 11. (1) and (2) imply that q(a1 a2 − b1 b2 ) = a2 + b1 .
(5)
Also, we get
|G| = apq >
number of elements of order p and q
= a1 q(p − 1) + a2 p(q − 1) = (a1 + a2 )pq − (a1 q + a2 p)
(6)
> (a1 + a2 )pq − 2pq = (a1 + a2 + 2)pq which implies that a1 + a2 < 14. Therefore, since b1 < a1 , we get q(a1 a2 − b1 b2 ) = a2 + b1 < a1 + a2 < 14.
(7)
It follows that since q > 11, we only need consider the two cases q = 11 and q = 13. To finish the problem, since b1 < a1 and a1 6 12, we can loop over all possibilities with q = 11 or 13. For each such pair (p, q), we can calculate the genus for p = a1bq−1 1 of each surface with g 6 (q − 1)2 which is admissible for both p and q. The only possibility we obtain is q = 13, p = 103, a = 8 and g = 102. In this case however, |G| = 10712 > 84 ∗ 101 = 84(g − 1) which contradicts the Hurwitz bound. Therefore there does not exist G with |G| = apq for 2 6 a 6 12 and p > q > a. We now have the necessary tools to prove the main result of this section.
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Theorem 4.5. Suppose X is a multiple prime surface which is cyclic p-gonal and cyclic q-gonal for primes p > q. If Cp and Cq are a cyclic p-gonal and a cyclic q-gonal group for X respectively, then the group of automorphisms G generated by Cp and Cq is cyclic of order pq. Proof. By Lemma’s 4.1-4.4, we know that |G| = pq. Therefore we just need to show that it is cyclic. Since p and q are distinct primes, any group of order pq will either be cyclic of order pq or a semi-direct product Cp ⋊ Cq provided p ≡ 1 mod (q). Assuming the latter case, such a group admits a partition into p groups of order q and 1 group of order p. Since all groups of order q are conjugate in G, the quotient space X/H will have the same genus for any such group H. Applying Theorem 3.4, it follows that X/Cq must have genus strictly greater than 0 contrary to our assumption that Cq is a q-gonal group for X. Therefore G must be cyclic of order pq.
5
The Classification of Multiple Prime Surfaces
We now have the necessary results to find all multiple prime surfaces. We start by fixing some notation. Let X denote a multiple prime surface which is p-gonal and q-gonal for primes q < p. Let Cp and Cq denote a p-gonal group and a q-gonal group respectively for X and let G 6 Aut(X) denote the group generated by Cp and Cq . Let Λ denote some fixed surface group for X and Γp , Γq and ΓG the Fuchsian groups with Γp /Λ = Cp , Γq /Λ = Cq and ΓG /Λ = G respectively. Before we calculate the possible full automorphism groups for X, we need the following result. Lemma 5.1. If p > 3 then the signature of ΓG is either (0; p, p, q, q) or (0; p, q, pq). If p = 3 and q = 2, the only possible signature for ΓG is (0; 2, 2, 3, 3). Proof. Since X/Cp has genus 0 and G/Cp = Cq , Proposition 2.1 implies that ΓG has signature (0; aq, bq, p, . . . , p) for a and b either 1 or p. Likewise, since X/Cq has genus 0 and G/Cq = Cp , Proposition 2.1 implies that ΓG has signature (0; cp, dp, q, . . . , q) for c and d either 1 or q. This implies that there are exactly two periods divisible by p and two periods divisible by q. Since the number of periods of a Fuchsian group with orbit genus 0 has to be at least 3, the only possible signatures are (0; p, p, q, q) and (0; p, q, pq). If p = 3 and q = 2, there is no Fuchsian group with signature (0; 2, 3, 6), so in this case the only possibility is (0; 2, 2, 3, 3). Theorem 5.2. Either X has full automorphism group Cp × Cq and the normalizer of Λ has signature (0; pq, p, q), or the dihedral group Dpq is an automorphism group of X and either: (i ) Dpq is the full automorphism group of X and the normalizer of Λ has signature (0; 2, 2, p, q), (ii ) Cq ⋊ D2p is the full automorphism group of X and the normalizer of Λ has signature
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(0; 2, 2p, 2q) where D2q ⋉ Cp has presentation hx, y, z|x2 , y 2q , z p , xyxy, xzxz, yzy −1 zi, (iii ) GL(2, 3) is the full automorphism group of X, which has genus 2, and the normalizer of Λ has signature (0; 2, 3, 8). Proof. We examine the two cases individually. (i ) ΓG has signature (0; p, q, pq). In this case, ΓG is a Fuchsian triangle group. For such groups, Singerman’s list, [7], gives us complete knowledge regarding the signatures for each Fuchsian group Γ with Γ > ΓG and [Γ : ΓG ] < ∞. By inspection of this list, the only instances in which there exists a Fuchsian group Γ > ΓG is when either q = 2 or q = 3. Therefore, if q > 5, it follows that ΓG must be the normalizer of Λ and hence the full automorphism group of X is Cp × Cq . When q = 2, ΓG is contained in a Fuchsian group Γ with signature (0; 2, 3, 2p). Since q = 2 and the genus g of X satisfies g > 2, Theorem 3.5 implies that X will be normal cyclic 2-gonal. If Λ is normal in Γ, then Γ2 must also be normal in Γ and so the group Γ/Γ2 = K will be a group of automorphisms of the Riemann sphere. However, for all possibles choices of K and a, b, and c (or a and b when K = Cn ), the signature (0; 2, 3, 2p) does not satisfy Proposition 2.1 (since p > 3). Consequently, Λ cannot be normal in Γ and so ΓG must be the normalizer of Λ and hence Cp × C2 must be the full automorphism group of X. When q = 3, ΓG is contained in a Fuchsian group Γ with signature (0; 2, 3, 3p). By inspection of Breuer’s lists, [3], for genera g = 2, 3, and 4, there is no choice of p or g for which the signature (0; 2, 3, 3p) occurs. Therefore, if there exists Λ with Λ ⊳ ΓG and Λ ⊳ Γ where Γ has signature (0; 2, 3, 3p), the surface X = H/Λ must have genus g > 5. In particular, Theorem 3.5 implies that X is normal cyclic 3-gonal. As with the case q = 2, if Λ is normal in Γ, then Γ3 must also be normal in Γ and so the group Γ/Γ3 = K will be a group of automorphisms of the Riemann sphere. For all possibles choices of K and a, b, and c (or a and b when K = Cn ), the signature (0; 2, 3, 3p) does not satisfy Proposition 2.1 so ΓG must be the normalizer of Λ and hence Cp × C3 is the full automorphism group of X. (ii ) ΓG has signature (0; p, p, q, q). If q = 2, then X is normal cyclic q-gonal. If q > 2, applying the Riemann-Hurwitz formula to the map πG : X → X/G and using the fact that p > q, we get pq (p − 1) (q − 1) g − 1 = −pq + 2 +2 2 p q = pq − p − q = (p − 1)(q − 1) > (q − 1)2 + 1. Consequently Theorem 3.5 implies that X is a normal cyclic q-gonal surface for any choice of q. In particular, if N is the normalizer of Λ, then Γq is also normal in N and N/Γq = K is a group of automorphisms of the Riemann sphere. We shall first show that K necessarily contains a dihedral subgroup.
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The signature of ΓG is (0; p, p, q, q), and this signature appears in Singerman’s list, [7]. Specifically, there exists a Fuchsian group Γ with signature (0; 2, 2, p, q) in which ΓG is normal of index 2. It is easy to show that the only epimorphism from ΓG onto Cp × Cq with torsion free kernel maps the first two periods to elements of order p which are inverse and the second two periods to similar elements of order q. Applying Theorem 5.1 of [4], it follows that the kernel Λ will also be normal in Γ with signature (0; 2, 2, p, q) and quotient group Γ/Λ = Dpq . Therefore, since Λ 6 Γq 6 Γ 6 N and Dp = Γ/Γq 6 K, it follows that K contains a dihedral subgroup. This immediately implies that K cannot be cyclic. Moreover, since p > q, we cannot have p = 2, and unless p = 3 or 5, the only possibility for K is a dihedral group. Therefore, we shall first consider the cases where p = 3 or p = 5. When p = 5, we must have q = 2 or q = 3. If q = 2, then ΓG has signature (0; 2, 2, 5, 5) and X has genus 4. If we assume that K is not dihedral, the only possibility for K is A5 , so the order of Γ/Λ would be divisible by 120. Checking Breuer’s list for genus 4, there is no signature for K = A5 and q = 2 satisfying Proposition 2.1 and so no surface exists whose automorphism group has these properties. If q = 3 then ΓG has signature (0; 3, 3, 5, 5) and X has genus 8. An identical argument works in this case. When p = 3, we must have q = 2, the signature of ΓG is (0; 2, 2, 3, 3) and the genus of X is 2. If we assume that K is not dihedral, the possibilities for K are S4 , A4 and A5 . If K = A5 , the order of Γ/Λ would be divisible by 120 and no such group exists for genus 2. If K is either A4 or S4 , then the order of Γ/Λ would be divisible by 24. Checking Breuer’s list for genus 2, the signature (0; 3, 3, 4) with K = A4 and automorphism group SL(2, 3) occurs and the signature (0; 2, 3, 8) with K = S4 and automorphism group GL(2, 3) occurs. Using Theorem 5.1 of [4], it can be shown that any surface kernel of orbit genus 2 normal in a Fuchsian group with signature (0; 3, 3, 4) is also normal in a Fuchsian group with signature (0; 2, 3, 8). In particular, since we are trying to find the full automorphism of all multiple prime surfaces, we only need consider the signature (0; 2, 3, 8). Therefore, suppose X is a surface of genus 2 with automorphism group GL(2, 3) and a surface group of X is normal in a Fuchsian group with signature (0; 2, 3, 8). Such a surface is necessarily cyclic 2-gonal as all surfaces of genus 2 are cyclic 2-gonal. It is cyclic 3-gonal since the only elements of order 3 in the automorphism group of a genus 2 surface are generators of cyclic 3-gonal groups. As there are no larger automorphism groups for genus g = 2, GL(2, 3) must be the full automorphism group of X. Hence there exists a genus 2 multiple prime surface with full automorphism group GL(2, 3) and the normalizer for such a surface has signature (0; 2, 3, 8). As we remarked previously, if p > 7, then K = Dn for some n divisible by p. Through simple calculation, we see that the only possible choice satisfying Proposition 2.1 is K = D2p with corresponding signature (0; 2, 2p, 2q). For any choice of p and q, this signature never appears in Singerman’s list, [7], and so there is no Fuchsian group containing a group with this signature of finite index. Consequently,
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if Λ is normal in a Fuchsian group with signature (0; 2, 2p, 2q), then it must be the normalizer N of Λ. Using Theorem 5.3.1 of [9], the only possible quotient group of N by a surface group Λ of a normal cyclic q-gonal surface is Cq ⋊ D2p . It is easy to see that such a surface is also p-gonal (and in fact normal p-gonal), hence the result. We finish by proving one of the most interesting consequences of our work - the fact that a multiple prime surface can admit cyclic prime covers of the Riemann sphere for at most two different primes. The result is a simple consequence of our analysis. Theorem 5.3. Suppose X is a cyclic q-gonal surface. Then X is cyclic p-gonal for at most one other prime p. Proof. Suppose X is a multiple prime surface of a fixed genus g that is cyclic q-gonal and cyclic p-gonal. By results in the proof of Theorem 5.2, if ΓG has signature (0; p, q, pq), then Cpq is the full automorphism group of X, so the result follows. Therefore, we shall assume that the signature of ΓG is (0; p, p, q, q). In this case, by the proof of Theorem 5.2, we may asumme that X is normal cyclic q-gonal. Then the genus g satisfies g = (p − 1)(q − 1), so p = g/(q − 1) + 1. Likewise, if we assume that X is cyclic r-gonal, we get r = g/(q − 1) + 1 and thus r = p. The result follows.
References [1] R.D.M. Accola: “Strongly Branched Covers of Closed Riemann Surfaces”, Proc. of the AMS, Vol. 26(2), (1970), pp. 315–322. [2] R.D.M. Accola: “Riemann Surfaces with Automorphism Groups Admitting Partitions”, Proc. Amer. Math. Soc., Vol. 21, (1969), pp. 477–482. [3] T. Breuer: Characters and Automorphism Groups of Compact Riemann Surfaces, Cambridge University Press, 2001. [4] E. Bujalance, F.J. Cirre and M.D.E. Conder: “On Extendability of Group Actions on Compact Riemann Surfaces”, Trans. Amer. Math. Soc., Vol. 355, (2003), pp. 1537–1557. [5] A.M. Macbeath: “On a Theorem of Hurwitz”, Proceedings of the Glasgow Mathematical Association, Vol. 5, (1961), pp. 90–96. [6] B. Maskit: “On Poincar´e’s Theorem for Fundamental Polygons”, Advances in Mathematics, (1971), Vol. 7, pp. 219–230. [7] D. Singerman: “Finitely Maximal Fuchsian Groups”, J. London Math. Soc., Vol. 2(6), (1972), pp. 29–38. [8] A. Wootton: “Non-Normal Bely˘ı p-gonal Surfaces”, In: Computational Aspects of Algebraic Curves, Lect. Notes in Comp., (2005), to appear. [9] A. Wootton: “Defining Algebraic Polynomials for Cyclic Prime Covers of the Riemann Sphere”, Dissertation, (2004).
CEJM 3(2) 2005 273–281
Generalizations of coatomic modules M. Tamer Ko¸san1∗, Abdullah Harmanci2† 1
Department of Mathematics, Faculty of Sciences and Arts, Kocatepe University, Afyon, Turkey 2 Department of Mathematics, Faculty of Science, Hacettepe University, 06532-Beytepe, Ankara, Turkey
Received 18 November 2004; accepted 8 March 2005 Abstract: For a ring R and a right R−module M , a submodule N of M is said to be δsmall in M if, whenever N + X = M with M/X singular, we have X = M . Let ℘ be the P class of all singular simple modules. Then δ(M ) = {L ≤ M | L is a δ-small submodule of M } = RejM (℘) = ∩{N ⊂ M : M/N ∈ ℘}. We call M δ−coatomic module whenever N ≤ M and M/N = δ(M/N ) then M/N = 0. And R is called right (left) δ−coatomic ring if the right (left) R−module RR (R R) is δ−coatomic. In this note, we study δ−coatomic modules and ring. We prove M = ⊕ni=1 Mi is δ-coatomic if and only if each Mi (i = 1, ..., n) is δ-coatomic. c Central European Science Journals. All rights reserved.
Keywords: δ- small module, coatomic module MSC (2000): 16D60, 16D99, 16S90
Throughout this paper, our ring R is associative with identity, and modules M are unitary right R−modules. N ≤ M will mean N is a submodule of M. Let N ≤ M. N is said to be small submodule of M, denoted by N ≪ M, in M whenever L ≤ M and M = N + L then M = L. For any R-module M, we write Rad(M), Soc(M), E(M) and Z(M) for the radical, socle, injective hull and singular submodule of M, respectively. M is said to be singular(or non-singular) if M = Z(M)(or Z(M) = 0). M is called coatomic if every submodule N of M, Rad(M/N) = M/N implies M/N = 0, equivalently every proper submodule of M is contained in a maximal submodule of M (see, namely [1,3,4]). A submodule N of a module M is called δ−small in M, N ≪δ M, if N + K 6= M for any proper submodule K of M with M/K singular. Further, for a module M the submodule ∗ †
E-mail:
[email protected],
[email protected] E-mail:
[email protected]
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δ(M) is generated by all δ− small submodules of M (see, [8]). The paper deals with δ− coatomic modules as a generalization of coatomic modules, i.e. the modules M such that for every submodule N of M with δ(M/N) = M/N it is M/N = 0. Several basic properties and characterizations of δ− coatomic modules and rings are given in the main part 2. We will refer to [1, 2, 6, 7] for all undefined notions used in the text, and also for basic facts concerning coatomic and singular modules.
1
δ-small submodule and the functor δ(M)
Following [8], N is said to be δ−small, denoted by N ≪δ M, in M if N + K 6= M for any proper submodule K of M with M/K singular. If N is any small submodule of M, then N is δ−small submodule of M. Clearly, any singular δ-small submodule of M is small submodule. For the reader’s convenience, we record here some of the known results which will be used repeatedly in the sequel. Lemma 1.1. Let M be a module. Then 1. Let N ≪δ M and M = X + N. Then M = X ⊕ Y , for a projective semisimple submodule Y with Y ⊂ N. 2. For submodules N, K, L of M with K ⊆ N, we have i. N ≪δ M if and only if K ≪δ M and N/K ≪δ M/K ii. N + L ≪δ M if and only if N ≪δ M and L ≪δ M. 3. If K ≪δ M and f : M → N is a homomorphism, then f (K) ≪δ N. In particular, if K ≪δ M ⊆ N, then K ≪δ N 4. Let K1 ⊆ M1 ⊆ M, K2 ⊆ M2 ⊆ M and M = M1 ⊕ M2 . Then K1 ⊕ K2 ≪δ M1 ⊕ M2 if and only if K1 ≪δ M1 and K2 ≪δ M2 . Proof.
See [8] Lemma 1.2 and 1.3. P For a module M, let δ(M) = {L ≤ M | L is a δ-small submodule of M}.
2
Proof.
2
Lemma 1.2. Let ℘ be the class of all singular simple modules. Then 1. δ(M) = RejM (℘) = ∩{N ⊂ M : M/N ∈ ℘} 2. If f : M → N is an R−homomorphism then f (δ(M)) ⊂ δ(N). 3. If M = ⊕i∈I Mi then δ(M) = ⊕i∈I δ(Mi ). 4. If every proper submodule of M contained in a maximal submodule of M, then δ(M) is the unique largest δ-small submodule of M. See [8] Lemma 1.5.
Remark : It is clear that, in general, δ(M) need not be δ-small in M. But if M is a coatomic module, i.e. every proper submodule of M is contained in a maximal submodule
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of M, then δ(M) is δ-small in M by Lemma 1.2(4). Lemma 1.3 Let M be a module. Then the following hold: (1) If δ(M) is δ−small in M and K/δ(M) is also δ−small in M/δ(M) where K ≤ M, then K is δ−small in M. (2) If δ(M) is δ−small in M, then δ(M/δ(M)) = 0. Proof. 1. Let K/δ(M) be a δ−small submodule of M/δ(M) and M = K + L with M/L singular. M/(L + δ(M)) is singular as an homomorphic image of the singular module M/L, and since M/δ(M) = K/δ(M)+(L+δ(M))/δ(M) and K/δ(M) is δ−small submodule of M/δ(M), M = L + δ(M). Being δ(M) is δ−small in M and M/L singular, we then have M = L and so K is δ−small in M. 2. is clear from the first part. 2 Now we give a characterization of M/δ(M). Proposition 1.4. Let M be an R-module. 1. If, for any submodule A of M, there exists a decomposition M = M1 ⊕ M2 such that M1 ≤ A and A ∩ M2 ≪δ M2 , then M/δ(M) is semisimple. 2. If, for every submodule A of M, there exists a submodule B of M such that M = A + B and A ∩ B ≪δ M, then M/δ(M) is semisimple. Proof. 1. Let δ(M) ≤ N ≤ M. Then N/δ(M) ≤ M/δ(M). By assumption, there exists a submodule A of N such that M = A ⊕ B and N ∩ B ≪δ B for some submodules B of M. Hence M/δ(M) = N/δ(M) ⊕ ((B + δ(M)/δ(M)). 2. Let δ(M) ≤ N ≤ M. By hypothesis, there exists a submodule K of M such that M = N + K and N ∩ K ≪δ M. Then N ∩ K ≤ δ(M). Hence M/δ(M) is semisimple by [5, Proposition 2.1]. 2
2
δ-coatomic Modules and Rings
Let M be an R module. We call M δ-coatomic if every submodule N of M, δ(M/N) = M/N implies M/N = 0. The ring R is called right(or left) δ-coatomic if the right(or left) R−module RR (or R R) is δ-coatomic. We can give another definition of δ-coatomic module. Lemma 2.1. Let M be a module. The following are equivalent. 1. M is δ-coatomic. 2. Every proper submodule K of M is contained in a maximal submodule N with M/N singular.
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Proof. 1 ⇒ 2: Let K be any proper submodule of M. By 1, δ(M/K) 6= M/K. Hence there esists a singular simple module S and homomorphism f from M/K to S. Let Ker(f ) = N/K. Then N is maximal in M and M/N is singular. 2 ⇒ 1: Let K be a proper submodule of M. Assume that δ(M/K) = M/K. We prove M/K = 0. By (ii) there exists a submodule N of M such that K ≤ N and M/N is singular simple. Let p denote the canonical epimprphism from M/K onto M/N. Since Ker(p) = N/K, δ(M/K) ≤ N/K. By assumption M/K = N/K ,and so M = N. This contradaction completes the proof. 2 Theorem 2.2. Let M be an R module with δ(M) ≪δ M. Then M is δ−coatomic if it satisfies one of the following conditions. (1) M/δ(M) is semisimple. (2) For every submodule A of M, there exists a submodule B of M such that M = A+ B and A ∩ B ≪δ M. Proof. 1. Suppose that M/δ(M) is semisimple and δ(M) ≪δ M. For any submodule N of M, let δ(M/N) = M/N. Since M/δ(M) is semisimple, there exists a submodule K of M with δ(M) ≤ K and M/δ(M) = ((N + δ(M))/δ(M)) ⊕ K/δ(M). Then M = N + K and N ∩ K ≤ δ(M). Hence M/N = (N + K)/N ∼ = K/(N ∩ K). Let p denote the canonical epimorphism K/(N ∩ K) → K/δ(M). By Lemma 1.1, K/δ(M)) = p(K/(N ∩ K)) = p(δ(K/(N ∩ K))) ≤ δ(K/δ(M))), and by Lemma 1.3, δ(M/δ(M)) = 0, and so δ(K/δ(M))) = 0. Hence K/(N ∩ K) = 0. Thus M/N = 0. 2. Assume that, for every submodule A of M, there exists a submodule B of M such that M = A + B and A ∩ B ≪δ M. By Proposition 1.4, M/δ(M) is semisimple. Hence M is δ-coatomic by preceeding paragraph. 2 Lemma 2.3. Let M be a module. Then the following holds. (1) If X ≤ δ(M) and X is δ-coatomic, then X ≪ M. (2) If M is δ-coatomic, then δ(M) ≪ M. In either case δ(M) ≪δ M. Proof. (1) Assume that X ≤ δ(M) and X is δ-coatomic module. Let M = X + Y for some submodule Y of M. We show that M = Y . Assume that M 6= Y . Then X 6= X ∩ Y . By hypothesis and Lemma 2.1 there exists a maximal submodule X ′ of X such that X ∩ Y ≤ X ′ ≤ X and X/X ′ is singular simple. Hence M/(X ′ + Y ) is singular simple since X/X ′ ∼ = (X + Y )/(X ′ + Y ) = M/(X ′ + Y ). It follows that X ′ ≤ δ(M) ≤ X ′ + Y and X ′ + Y ≤ δ(M) + Y ≤ X ′ + Y , and therefore M = X ′ + Y . Hence X = X ′ . This contradicts the fact that X ′ is maximal submodule of X. Thus X is small in M and so δ−small in M. (2) Suppose that M is δ-coatomic module. Let M = δ(M) + Y for some Y ≤ M. Assume that M 6= Y . By Lemma 2.1, there exists Y ≤ Y ′ ≤ M with M/Y ′ singular simple. By Lemma 1.1, δ(M) ≤ Y ′ . Hence M = Y ′ . This contradicts the fact that Y ′ is maximal submodule of M. Hence δ(M) is small in M and so δ−small in M. 2 Theorem 2.4. For an R-module M, the following are equivalent.
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1. M/δ(M) is semisimple and every submodule of δ(M) is δ-coatomic. 2. For every submodule A of M, there exists a submodule B of M such that M = A+B and A ∩ B ≪δ M, and every submodule of M is δ-coatomic. Proof. Note that under the assumptions 1 and 2, δ(M) ≪δ M by Lemma 2.3 and Proposition 1.4. 1 ⇒ 2: For any submodule A of M, let M/δ(M) = ((A + δ(M))/δ(M) ⊕ B/δ(M) for some submodule B of M. Then M = A + B and A ∩ B ≤ δ(M). Since δ(M) ≪δ M, by Lemma 1.1, A ∩ B ≪δ M. Let X be a submodule of M. We show that X is δ−coatomic. Assume that δ(X/A) = X/A for some submodule A of X. Then M/δ(M) = ((A + δ(M))/δ(M)) ⊕ B/δ(M) for some submodule B of M since M/δ(M) is semisimple. Then M = A + B and A ∩ B ≤ δ(M). It is easy to check that (X + δ(M))/(A + δ(M)) = δ( (X + δ(M))/(A + δ(M)) ) ≤ δ(M/(A + δ(M))) δ(M/(A + δ(M))) ∼ = δ(B/δ(M)) ≤ δ(M/δ(M)). By Lemma 1.3, δ(M/δ(M)) = 0. Hence A + δ(M) = X + δ(M), and so X = A + (X ∩ δ(M)). Then X/A ∼ = (X ∩ δ(M))/(A ∩ δ(M)). Since every submodule of δ(M) is δ-coatomic by hypothesis, X ∩ δ(M) is a δ−coatomic submodule of δ(M). Since δ((X ∩δ(M))/(A∩δ(M))) = (X ∩δ(M))/(A∩δ(M)), we have that X ∩δ(M) = A∩δ(M). Hence A = X. 2 ⇒ 1 is clear by Proposition 1.4. 2 Proposition 2.5. Let 0 → K → M → N → 0 be an exact sequence of modules. 1. If M is δ-coatomic module, then N is δ-coatomic. 2. If K and N are δ-coatomic modules, then M is δ-coatomic. In particular, any direct summand of a δ-coatomic module is δ-coatomic. Proof. 1. We may assume that K ≤ M and N = M/K. Let U be a submodule of N. Suppose that δ(N/U) = N/U. Then we find submodule L of M with L/K = U. Then δ(M/L) = M/L. Since M is a δ-coatomic module, M/L = 0. This implies that N/U = 0. It follows that N is δ-coatomic. 2. Suppose that K and N are δ-coatomic modules. Let L be any proper submodule of M. Case a. M/K = (L + K)/K. Then M = L + K. Since K is δ−coatomic, there exists a maximal submodule K ′ of K such that K ∩ L ≤ K ′ ≤ K and K/K ′ singular simple. Since K/K ′ ∼ = (K + L)/(K ′ + L) = M/(K ′ + L), M/(K ′ + L) is singular simple. Hence M is δ-coatomic by Lemma 2.1. Case b. M/K 6= (L + K)/K. Then M 6= L + K. Since N is δ-coatomic and N ∼ = M/K, ′ ′ ′ ∼ there exists a submodule K /K of M/K such that (M/K)/(K /K) = M/K is singular simple and (L + K)/K ≤ K ′ /K. Then M is δ-coatomic by Lemma 2.1. 2
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Proposition 2.6. Let M = ⊕ni=1 Mi be a finite direct sum of modules Mi (i = 1, ..., n). Then M is δ-coatomic if and only if each Mi (i = 1, ..., n) is δ-coatomic. Proof. It is sufficent by induction on n to prove this is the case when n = 2. Let M1 and M2 be δ-coatomic modules and M = M1 ⊕ M2 . We consider the following exact sequence; 0 → M1 → M = M1 ⊕ M2 → M2 → 0 Hence M = M1 ⊕ M2 is δ-coatomic module if and only if M1 and M2 are δ-coatomic modules by Proposition 2.5. 2 Theorem 2.7. Let M be an R-module. Then the following are equivalent : (1) M is semisimple and singular module (2) M is δ-coatomic and every submodule N of M with M/N singular simple is a direct summmand of M. Proof. (1) ⇒ (2) : Let K be any maximal submodule of M. Then M = N ⊕ K where N is a simple submodule of M. By (1), M/K is simple singular. Then δ(M) ≤ K for every maximal submodule K of M. Hence δ(M) = 0. Assume that δ(M/N) = M/N for some N ≤ M. Let M = N ⊕ N ′ for some submodule N ′ of M. Then 0 = δ(N ′ ) = N ′ ∼ = M/N. Hence M is δ−coatomic. The rest is clear. (2) ⇒ (1): Let A be a submodule of M. By Zorn’s lemma, we may find K ≤ M such that K is maximal with respect to the property A ∩ K = 0. Then A ⊕ K is essential in M. Now assume that A ⊕ K is proper submodule of M. Since M is δ-coatomic, there exists a submodue N of M such that A ⊕ K ≤ N and M/N is singular simple by Lemma 2.1. By assumption, N is direct summand of M. Let M = N ⊕ N ′ for some submodule N ′ of M. Then (A ⊕ K) ∩ N ′ = 0. It follows that N ′ = 0 and M = N. This contradicts to being N maximal submodule. Hence M = A ⊕ K and M is semisimple module. Since M/K ∼ = A is singular and A is an arbitrary submodule of M, M is singular module. 2 Proposition 2.8. Let M be a semisimple module. Then M has a decomposition M = M1 ⊕ M2 , where M1 is δ−coatomic singular submodule and M2 is nonsingular. Proof. Let M be a δ−coatomic module. Let L ≤ M be a submodule maximal with respect to the property L ∩ Z(M) = 0. Assume that M 6= L ⊕ Z(M). Then there exists a maximal submodule K such that L⊕Z(M) ≤ K and M/K is singular. Let M = K ⊕K ′ . Then K ′ ≤ Z(M). Hence K ′ = 0. This is a contradiction. Thus M = L ⊕ Z(M). By Theorem 2.7, Z(M) is singular δ−coatomic and L is nonsingular submodule. 2 Corollary 2.9. Every singular semisimple module is δ−coatomic. Proof.
Clear from Proposition 2.8.
2
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A pair (P, f ) is called a projective δ-cover of the module M if P is projective right R-module and f is an epimorphism of P onto M with Ker(f ) ≪δ P . Lemma 2.10 Let M = A + B. If M/A has a projective δ-cover, then B contains a submodule A′ of A such that M = A + A′ and A ∩ A′ ≪δ A′ . Proof. Let π : B → M/A the natural homomorphism and f : P → M/A be a projective δ-cover. Since P is projective, there exists g : P → B such that πg = f and Ker(f ) is δ-small in P . Then (πg)(P ) = f (P ) and A ∩ g(P ) = g(Ker(f )). Hence M = A + g(P ) and A ∩ g(P ) = g(Ker(f )). Since Ker(f ) is δ-small in P , g(Ker(f )) is δ-small in g(P ) and so A ∩ g(P ) is δ-small in g(P ). 2 Lemma 2.11 Let A be any submodule of M. Assume that M/A has a projective δ-cover. Then there exists a submodule A′ such that M = A + A′ and A ∩ A′ ≪δ A′ . Proof.
Let B = M in Lemma 2.10.
2
We call a projective module M δ-semiperfect if every homomorphic image of M has a projective δ-cover. Lemma 2.12 For any projective R-module M, the following are equivalent: 1. M is δ-semiperfect. 2. For any N ≤ M, M has a decomposition M = M1 ⊕ M2 for some submodules M1 , M2 with M1 ≤ N and M2 ∩ N ≪δ M2 . Proof.
See [8, Lemma 2.4].
2
Theorem 2.13 Let M be a δ−semiperfect module and δ(M) <<δ M. Then M is δcoatomic. Proof. Let M be a δ−semiperfect module. Let A be any submodule. By Lemma 2.11, there exists a submodule A′ such that M = A + A′ such that A ∩ A′ <<δ A′ . Then by Theorem 2.2, M is δ−coatomic. 2 Proposition 2.14 is mentioned in [8], we give a brief and short proof for the sake of completeness Proposition 2.14 For any ring R, δ(R) is δ−small in R. Proof. Let I be a right ideal in R. Assume that R = δ(R) + I with R/I singular. Suppose that I is proper and let K be a maximal right ideal containing I. Then R/K is singular simple right R−module. Hence δ(R) ≤ K. This is a contradiction. Thus for any right ideal I such that R = δ(R) + I with R/I singular we have R = I. By definition δ(R) ≪δ R. 2
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Proposition 2.15 Let R be a δ-semiperfect ring. Then R is left and right δ−coatomic ring. Proof. R is right δ-coatomic ring from Theorem 2.13 and Proposition 2.14. By symmetry, R is also left δ-coatomic ring. 2 Theorem 2.16 let R be a ring. Then each right ideal of R with δ(R/I) = R/I is direct summand. Proof. Let I be a right ideal of R. Suppose that δ(R/I) = R/I. Then all maps from R/I to singular simple right R−modules is zero. Assume that I is an essential right ideal. Let K be a maximal right ideal containing I. Then R/K is singular simple right R−module. Since R/K is an image of R/I and δ(R/I) = R/I, R = K. This is a contradiction. Hence I is not essential. Let L be a maximal right ideal with respect to the property I ∩ L = 0. Then I ⊕ L is essential in R. Assume that I ⊕ L is proper. Let T be a maximal right ideal containing I ⊕ L. Then R/T is singular simple image of R/I. This is a contradiction again. Hence R = I ⊕ L. 2 The following result is well known and also easy to prove. Theorem 2.17 The following are equivalent for a ring R. (1) R is semisimple artinian. (2) Every maximal right ideal of R is a direct summand of RR . Proof.
[9, Lemma 2.1 ].
2
Remark : If I is an essential right ideal in the ring R, then R/I is singular right R−module. The converse is also true. In module case it takes the form: for a nonsingular module B and A ≤ B, B/A is singular if and only if A is A is essential in B (See [2, Proposition 1.21]). Any maximal right ideal in a ring is essential right ideal or direct summand. For δ−coatomic rings, this is not the case in general for maximal right ideals. Theorem 2.18 Let R be a right δ−coatomic ring. Then (1) Every simple right R−module is singular. (2) Every maximal right ideal in R is essential right ideal. Proof. (1). Let I be a maximal right ideal in R. If δ(R/I) = R/I, by hypothesis R = I. It is not possible. Hence δ(R/I) = 0. Then there exists a nonzero homomorphism f : R/I → S where S is a singular simple right R−module. Hence f is an isomorphism and so R/I is singular right R−module. (2). Let I be a maximal right ideal in R. We claim that I is an essential right ideal. Assume that I is not essential right ideal and let R = I ⊕ K for some right ideal K. If δ(R/I) = R/I, by hypothesis R = I. It is not possible. Hence δ(R/I) 6= R/I. By (1), R/I is nonzero singular simple right R−module. By the preceding remark, I is essential right ideal in R. This contradicts the assumption. Thus I is direct sumand. 2
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Examples 2.19 1. Consider the integers Z as Z-module. Then δ(Z) = 0 and for any prime integer p, δ(Z/pZ) = 0 since Z/pZ is singular simple Z-module. Hence Z is δcoatomic Z-module. But the rational numbers Q as Z-module is not δ-coatomic since every cyclic submodule of Q is small and so δ(Q) = Q. 2. Let M be a local module with unique maximal submodule Rad(M) = δ(M). Then M is δ-coatomic. 3. Let M denote the Z-module Z. By Lemma 2.12, M is not δ-semiperfect module. Since every proper submodule is essential and contained in a maximal submodule, by Lemma 2.1, M is δ-coatomic.
References [1] F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Springer-Verlag, New York, 1974. [2] K.R. Goodearl: Ring Theory : Nonsingular Rings and Modules, Dekker, New York, 1976. [3] G. Gungoroglu: “Coatomic Modules”, Far East J. Math. Sci., Special Volume, Part II, (1998), pp. 153–162. [4] F. Kasch: Modules and Rings, Academic Press, 1982. [5] C. Lomp: “On Semilocal Modules and Rings”, Comm. Alg., 27(4), (1999), pp. 1921– 1935. [6] S.H. Mohamed and B.J. M¨ uller: Continuous and discrete modules, London Math. Soc. LNS 147, Cambridge Univ. Press, Cambridge, 1990. [7] R. Wisbauer: Foundations of Module and Ring Theory, Gordon and Breach, Reading, 1991. [8] Y. Zhou: “Generalizations of Perfect, Semiperfect and Semiregular Rings”, Algebra Colloquium, Vol. 7(3), (2000), pp. 305–318. [9] M.Y. Yousif and Y. Zhou: “Semiregular, Semiperfect and Perfect Rings relative to an ideal”, Rocky Mountain J. Math., Vol. 32(4), (2002), pp. 1651–1671.
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A Newton - Kantorovich - SOR type theorem B´ela Finta∗ Mathematics and Computer Sciences Department, Petru Maior University of Tg. Mures, 4300, Tg. Mure¸s, Romania
Received 29 November 2003; accepted 8 October 2004 Abstract: In this paper we propose a new method for solving nonlinear systems of equations in finite dimensional spaces, combining the Newton-Raphson’s method with the SOR idea. For the proof we adapt Kantorovich’s demonstration given for the Newton-Raphson’s method. As applications we reobtain the classical Newton-Raphson’s method and the author’s NewtonKantorovich-Seidel type result. c Central European Science Journals. All rights reserved.
Keywords: Iterative method, Newton method, SOR method MSC (2000): 46-00, 65-00
1
Introduction
We consider the functions fi : Di ⊂ Rn → R, where Di 6= ∅ for i = 1, n, f : D ⊂ Rn → Rn , D ⊂ ∩ni=1 Di , D 6= ∅, f = [f1 , f2 , . . . , fn ] and the corresponding equation f (x) = θRn where θRn means the null vector of the space Rn . We assign to this equation another equation of the form ϕ(x) = x, where ϕ : D ⊂ n R → Rn is called the iterative function. This correspondence is such that the solution of the equation f (x) = θRn will be a solution of the equation ϕ(x) = x, i.e. a fixed point for ϕ, and conversely. A known local numerical method to solve the equation f (x) = θRn is the NewtonRaphson-Kantorovich method with the iterative function given by ϕ : D ⊂ Rn → Rn , ϕ(x) = x−[f ′ (x)]−1 f (x), where x = [x1 x2 . . . xn ]T and f (x) = [f1 (x) f2 (x) . . . fn (x)]T are column vectors and [f ′ (x)]−1 means the inverse matrix of the Jacobian matrix [f ′ (x)] of the function f at point x. The existence and convergence of the iterative sequence {xk }k∈N generated by xk+1 = ϕ(xk ) is assured by the well known Kantorovich theorem [6]: ∗
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Theorem 1.1. Let f : D ⊂ Rn → Rn , be a Fr´echet differentiable function on the open subset D 6= ∅. Let D0 6= ∅, D0 ⊂ D be a convex subset and x0 ∈ D0 . Assume the following conditions are fulfilled: i) there exist [f ′ (x0 )]−1 and k[f ′ (x0 )]−1 k ≤ α0 ; ii) k[f ′ (x0 )]−1 f (x0 )k ≤ β0 ; iii) k[f ′ (x)] − [f ′ (y)]k ≤ γ · kx − yk for every x, y ∈ D0 ; iv) h0 = α0 · β0 · γ ≤ 21 ; v) the sphere B0 = B(x0 , r0 ) = {x ∈ Rn | kx − x0 k ≤ r0 } ⊂ D0 , with r0 = √ 1− 1−2h0 · β0 . h0 Then the equation f (x) = θRn has a solution x∗ ∈ D0 and the iterative sequence {xk }k∈N given by the Newton’s iterative method xk+1 = ϕ(xk ), k ∈ N is well defined and it converges to x∗ . The error at the k-th iteration is estimated by kxk − x∗ k ≤ k 21−k (2h0 )2 −1 · β0 . This method is often combined with the SOR method. In [7] a Newton-KantorovichSOR method is given. The purpose of this work is to show a new theorem of this kind. We assign to the iterative function ϕ : D ⊂ Rn → Rn , D 6= ∅, ϕ = [ϕ1 , ϕ2 , . . . , ϕn ] the following iterative function Φ : D ′ ⊂ Rn → Rn , D ′ 6= ∅, Φ = [Φ1 , Φ2 , . . . , Φn ], where Φi : D ′ ⊂ Rn → R for i = 1, n are given by the following formulae: Φ1 (x) = ϕ1 (x), Φ2 (x) = ϕ2 (ω21 Φ1 (x) + (1 − ω21 )x1 , x2 , . . . , xn ), .. . Φi (x) = ϕi (ωi1 Φ1 (x) + (1 − ωi1 )x1 , ωi2 Φ2 (x) + (1 − ωi2 )x2 , . . . , ωi,i−1Φi−1 (x) + (1 − ωi,i−1 )xi−1 , xi , . . . , xn ), .. . Φn (x) = ϕn (ωn1Φ1 (x) + (1 − ωn1 )x1 , ωn2 Φ2 (x) + (1 − ωn2 )x2 , . . . , ωn,n−1Φn−1 (x) + (1 − ωn,n−1 )xn−1 , xn ), where Ω = (ωij )i,j=1,n is a strictly lower triangular matrix of real numbers in [0, 1] such that ωij = 0 for every 1 ≤ i ≤ j ≤ n. This function is based on the SOR idea and it is used in [3] and [4] to accelerate the convergence of the iterative sequence obtained by the nonlinear Jacobi method. We mention that x∗ ∈ D ′ ∩ D is a fixed point for Φ if and only if it is a fixed point for ϕ.
2
Main part
First let us introduce some definitions. Let us consider for i = 1, n the matrices Ai of order n and the column vectors Bi of size n × 1.
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Definition 2.1. By the symbol [A1 A2 . . . An ]T we understand a matrix of order n whose row i, where i = 1, n is formed by the row i of the matrix Ai and the symbol [B1 B2 . . . Bn ] means a matrix of order n, whose column i is formed by the column vector Bi . We define the product of two such matrices [A1 A2 . . . An ]T · [B1 B2 . . . Bn ] = C, where C is a column vector, the elements of which we obtain by using the usual matrix multiplication of the row i from the first matrix by the column i of the second matrix for i = 1, n. Then we arrange these results in the column vector C. We can observe that in this way we take exactly the diagonal elements of the product of two matrices of order n, arranged in the form of a column vector. Now we are ready to build a new iterative function ϕ∗ in a similar way to the function Φ by using the form of the Newton’s iterative function ϕ. Suppose that f is infinitely differentiable so that we can build the function ϕ∗ : D ∗ ⊂ Rn → Rn , ϕ∗ = [ϕ∗1 , ϕ∗2 , . . . , ϕ∗n ] , D ∗ 6= ∅ in the following way, by using definition 2.1:
ϕ∗ (x) = [ϕ∗1 (x) ϕ∗2 (x) . . . ϕ∗i (x) . . . ϕ∗n (x)]T = [x1 x2 . . . xi . . . xn ]T − −[[f ′ (x)]−1 [(f (ω21 ϕ∗1 (x) + (1 − ω21 )x1 , x2 , . . . , xn ))′ ]−1 . . . [(f (ωi1 ϕ∗1 (x) + (1 − ωi1 )x1 , ωi2 ϕ∗2 (x) + (1 − ωi2 )x2 , . . . , ωi,i−1 ϕ∗i−1 (x) + +(1 − ωi,i−1 )xi−1 , xi , . . . , xn ))′ ]−1 . . . [(f (ωn1 ϕ∗1 (x) + +(1 − ωn1 )x1 , ωn2 ϕ∗2 (x) + (1 − ωn2 )x2 , . . . , ωn,n−1ϕ∗n−1 (x) + +(1 − ωn,n−1 )xn−1 , xn ))′ ]−1 ]T · ·[f (x)f (ω21 ϕ∗1 (x) + (1 − ω21 )x1 , x2 , . . . , xn ) , . . . , f (ωi1 ϕ∗1 (x) + (1 − ωi1 )x1 , ωi2 ϕ∗2 (x) + (1 − ωi2 )x2 , . . . , ωi,i−1 ϕ∗i−1 (x) + (1 − ωi,i−1 )xi−1 , xi , . . . , xn ), . . . , f (ωn1 ϕ∗1 (x) + (1 − ωn1 )x1 , ωn2 ϕ∗2 (x) + (1 − ωn2 )x2 , . . . , ωn,n−1 ϕ∗n−1 (x) + (1 − ωn,n−1 )xn−1 , xn )].
We introduce the following notations: for i = 1, n, fi∗ : Di∗ ⊂ Rn → Rn , Di∗ 6= ∅,
f1∗ (x) = f (x), f2∗ (x)f (ω21 ϕ∗1 (x) + (1 − ω21 )x1 , x2 , . . . , xn ) , . . . , ∗ fi (x) = f ωi1 ϕ∗1 (x) + (1 − ωi1 )x1 , ωi2 ϕ∗2 (x) + (1 − ωi2 )x2 , . . . , ∗ ωi,i−1 ϕi−1 (x) + (1 − ωi,i−1 )xi−1 , xi , . . . , xn , . . . , ∗ fn (x) = f ωn1 ϕ∗1 (x) + (1 − ωn1 )x1 , ωn2ϕ∗2 (x) + (1 − ωn2 )x2 , . . . , ∗ ωn,n−1ϕn−1 (x) + (1 − ωn,n−1 )xn−1 , xn ,
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where we suppose that D ∗ ⊂ ∩ni=1 Di∗ . So we have: ϕ∗ (x) = [ϕ∗1 (x) ϕ∗2 (x) . . . ϕ∗i (x) . . . ϕ∗n (x)]T [x1 x2 . . . xi . . . xn ]T − ′
′
′
′
−[[f1∗ (x)]−1 [f2∗ (x)]−1 . . . [fi∗ (x)]−1 . . . [fn∗ (x)]−1 ]T · · [f1∗ (x)f2∗ (x) . . . fi∗ (x) . . . fn∗ (x)] , ′ −1 ′ where for i = 1, n fi∗ (x) means the inverse matrix of the Jacobian matrix fi∗ (x) of the function fi∗ at point x and fi∗ (x) are column vectors of size n × 1. If we use the symbolical notations: ′
′
′
′
′
[f ∗ (x)]−1 = [[f1∗ (x)]−1 [f2∗ (x)]−1 . . . [fi∗ (x)]−1 . . . [fn∗ (x)]−1 ]T and [f ∗ (x)] = [f1∗ (x)f2∗ (x) . . . fi∗ (x) . . . fn∗ (x)] then we obtain the new Newton-SOR type ′ −1 ∗ iterative function: ϕ∗ : D ∗ ⊂ Rn → Rn , ϕ∗ (x) = x − f ∗ (x) [f (x)] . We mention ∗′ ∗ that we did not define the function f , f (x) does not mean the Jacobian matrix, and ∗′ −1 ′ f (x) does not mean the inverse matrix of the matrix f ∗ (x) . Now we are able to announce and to demonstrate the main theorem of this work which assures us the existence and the convergence of the iterative sequence {xk }k∈N given by the iterative method xk+1 = ϕ∗ (xk ). In this theorem and in the proof of this theorem we will consider the vector norm given by k x k=k (x1 , x2 , . . . , xn ) k= max {|xi |} = kxk∞ 1≤i≤n
on the linear space Rn and the corresponding matrix norm (row norm) ( n ) X k A k=k (aij )i,j=1,n k max |aij | 1≤i≤n
j=1
on the linear space of the matrices of order n, denoted by L(Rn ) [1]. In this way L(Rn ) becomes a Banach algebra [2]. Theorem 2.2. Let f : D ⊂ Rn → Rn be a function smooth enough so that there exist ′ the Jacobian matrices fi∗ (x) of the functions fi∗ : D ∗ ⊂ Rn → Rn at the points x ∈ D ∗ for i = 1, n, where D 6= ∅, D ∗ 6= ∅ are open sets. Let D0 6= ∅, D0 ⊂ D ∗ be a convex subset and x0 ∈ D0 . Assume the following conditions are fulfilled: ′ −1 ∗′ 0 i) there exist fi∗ (x0 ) (the inverse matrix of the Jacobian matrix fi (x ) of the
∗′ 0 −1 ∗ 0 function fi at point x ) and fi (x ) ≤ α0 for every i = 1, n; ′ −1 ii) if f ∗ (x) exists (see the symbolical notation) at a point x ∈ D ∗ then −1 −1 ′ −1 ′ f ∗ (x) (the inverse matrix of the matrix f ∗ (x) ) exists;
−1 ∗ 0
′
iii) f ∗ (x0 ) [f (x )] ≤ β0 ;
′ ′ iv) fi∗ (x) − fi∗ (y) ≤ γ · kx − yk for every x, y ∈ D0 and i = 1, n;
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∗′ −1 −1 ∗′ −1 −1
≤ γ · kx − yk for x, y ∈ D0 for which f ∗′ (x) −1 v) f (x) − f (y)
∗′ −1 and f (y) exist; vi) h0 = α0 β0 γ ≤ 12 ;
vii) the sphere B0 = B(x0 , r0 ) = {x ∈ Rn /kx − x0 k ≤ r0 } ⊂ D0 with r0 =
√ 1− 1−2h0 h0
·
β0 . Then the equation f (x) = θRn has a solution x∗ ∈ D0 and the iterative sequence {xk }k∈N given by the iterative method xk+1 = ϕ∗ (xk ), k ∈ N is well defined and it converges to x∗ . The error at the k-th iteration is estimated by kxk − x∗ k ≤ 21−k · k (2h0 )2 −1 β0 . To prove this theorem we need the following lemmas: Lemma 2.3. (Proposition 2.5 in [2]). If A ∈ L(Rn ) and k I − A k< 1 then there exists 1 the inverse matrix A−1 and k A−1 k≤ 1−kI−Ak , where I denotes the unit matrix. Lemma 2.4. If A, B ∈ L(Rn ) such that there exists A−1 and k A−1 k≤ α, k A − B k≤ β α with α · β < 1 then there exists B −1 and kB −1 k ≤ 1−αβ . Proof. Let us consider M = A−1 · (A − B) = I − A−1 · B, so kMk ≤ kA−1 k · kA − Bk ≤ α · β < 1. 1 1 By virtue of lemma 2.3 there exists (I − M)−1 and k(I − M)−1 k ≤ 1−kM ≤ 1−αβ . k −1 −1 −1 Consequently the matrix B = A · (I − M) has the inverse B = (I − M) · A and α kB −1 k ≤ k(I − M)−1 k · kA−1 k ≤ 1−αβ .
Lemma 2.5. ([9]) Let h : D ⊂ Rn → Rn , where D is an open set, be a Fr´echet differentiable function on the convex subset D0 ⊂ D, D0 6= ∅. If the Fr´echet differential h′ has the Lipschitz property on D0 , i.e. k h′ (u) − h′ (v) k≤ L· k u − v k for every u, v ∈ D0 , then k h(u) − h(v) − h′ (v)(u − v) k≤
L · k u − v k2 2
for every u, v ∈ D0 . Proof of Theorem 2. We begin with two observations: a) The proof given below is valid when h0 6= 0. Now, we study the case h0 = α0 β0 γ = ′ 0. By virtue of the condition i) α0 6= 0, because the inverse matrix [fi∗ (x0 )]−1 cannot be the null matrix. If β0 = 0, then xk = x0 for every k ∈ N. We have the following two subcases: if γ = 0, then we can choose the new values β0′ > 0 and γ ′ > 0 such that
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β0′ γ ′ ≤ 12 · α10 , so 0 < h′0 = α0 β0′ γ ′ ≤ 12 . If γ 6= 0, then we can choose the new value 0 < β0′ ≤ 21 · α10 γ such that 0 < h′0 = α0 β0′ · γ ≤ 12 . In both cases, by virtue of the proof given below for h′0 instead of h0 = 0, we obtain that x0 will be the solution x∗ = x0 of the equation f (x) = θRn . Consequently, we can suppose that β0 6= 0. If γ = 0, then by virtue ′ of the condition iv) we have that [fi∗ (x)] are constant matrices for every x ∈ D0 and ′ i = 1, n. Consequently, ([f ∗ (x)]−1 )−1 are also constant matrices, so the condition v) is fulfilled for γ = 0. In this case we obtain the special Newton-Kantorovich-SOR iterative ′ function ϕ∗ : D ∗ ⊂ Rn → Rn , ϕ∗ (x) = x − [f ∗ (x0 )]−1 [f ∗ (x)] for every x ∈ D ∗ . We can choose a new value 0 < γ ′ ≤ 12 · α01β0 such that 0 < h′0 = α0 β0 γ ′ ≤ 21 and for this value h′0 6= 0 instead of h0 we apply the proof given below, so we obtain a similar result for the convergence of the special Newton - Kantorovich - SOR method as well. √ b) the function T : 0, 21 → R, defined by T (0) = 1 and T (h) = 1− h1−2h , when 0 < h ≤ 12 , is continuous at point h = 0 and 1 ≤ T (h) ≤ 2 for every h ∈ 0, 12 . ′ −1 From the condition i) it follows that f ∗ (x0 ) exists (see the symbolical notation), ′ −1 because row i of this matrix is exactly row i of the matrix fi∗ (x0 ) for every
i = 1, n. −1 ′
From the definition of the matrix norm and from the conditions fi∗ (x0 ) ≤ α0 for
−1
′ every i = 1, n we can obtain that f ∗ (x0 ) ≤ α0 as well. We observe that
h i−1
1
∗ 0 0
x − x0 = ϕ∗ (x0 ) − x0 x0 − f ∗′ (x0 ) f (x ) − x =
h
∗′ 0 i−1 ∗ 0
= f (x ) f (x )
≤ β0
using condition iii). An elementary calculation shows us that β0 ≤ r0 (or see b)). Consequently, kx1 − x0 k ≤ r0 , so x1 ∈ B0 = B(x0 , r0 ). Using condition iv), from the above and from conditions vi) and i), we obtain successively:
h i h ′ i
1 1
∗′ 1
< ≤
fi (x ) − fi∗ (x0 ) ≤ γ · x1 − x0 ≤ γ · β0 ≤ 2α0 α0
h
∗′ 0 i−1 −1
≤ fi (x )
,
−1
∗′ 0 −1 −1
∗′
where fi (x ) means the reciprocal value of the number fi (x0 ) . Now we
′ ′ −1
′
apply Lemma 2.4, where we choose A = fi∗ (x0 ) , B = fi∗ (x1 ) , α = fi∗ (x0 ) , β =
∗′ 1 ∗′ 0
f (x ) − f (x ) . We can observe that the inverse matrix of the matrix A is A−1 = i ∗′i 0 −1 ′ fi (x ) , i.e. the inverse matrix of the Jacobian matrix fi∗ (x0 ) , which exists by virtue of condition i) and
h
h i h ′ i
∗′ 0 i−1
∗′ 1 ∗ 0
α · β = fi (x ) · f (x ) − f (x )
< 1. i i
′ ′ −1 So the matrix fi∗ (x1 ) is invertible, and therefore fi∗ (x1 ) exists for every i = 1, n.
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′ −1 Consequently, f ∗ (x1 ) exists. Lemma 2.4 implies that
∗′ 0 −1
h
i
fi (x )
∗′ 1 −1
f (x ) ≤
′ ≤ i
∗′ 0 −1 ′ 1 − fi (x ) · fi∗ (x1 ) − fi∗ (x0 ) α0 α0 ≤ = 1 − α0 γβ0 1 − h0
∗′ 1 −1 α0 , so f (x ) for every i = 1, n. We define α1 = 1−h
i
≤ α1 for every i = 1, n. This 0 means that condition i) is true if we choose x1 in place of x0 , and α1 in place of α0 . ′ −1 ∗′ 1 −1 −1 Next we apply Lemma 2.3 for A = f ∗ (x0 ) f (x ) which exists by virtue of condition ii).
∗′ 0 −1 ∗′ 1 −1 −1
≤ We observe that kI − Ak = · f (x )
I − f (x )
h
h i−1 −1 h ′ i−1 −1
∗′ 0 i−1 ′
f ∗ (x0 ) ≤ − f ∗ (x1 )
≤
f (x ) ·
1 ≤ α0 · γ · x0 − x1 ≤ α0 · γ · β0 = h0 ≤ < 1, 2 where we used condition v) and vi). So A is invertible and ! h ′ i−1 h ′ i−1 −1 −1 = A−1 = f ∗ (x0 ) f ∗ (x1 ) h ′ i−1 h ′ i−1 −1 ∗ 1 ∗ 0 = f (x ) f (x ) ,
where the first superscript -1 is included in the symbol, but the second and third super index -1 is used for the inverse matrix. Consequently: h ′ i−1 h ′ i−1 −1 ∗ ∗ 1 0 f (x ) f (x ) = A−1 , ′ −1 ′ −1 so f ∗ (x1 ) = A−1 f ∗ (x0 ) . Therefore h ′ i−1 h ′ i−1 f ∗ (x1 ) f ∗ (x1 ) A−1 f ∗ (x0 ) f ∗ (x1 ) .
If we take the vector and matrix norm we obtain:
h
∗′ 1 i−1 ∗ 1 −1 h ∗′ 0 i−1 ∗ 1
f (x )
f (x ) f (x ) f (x ) = A
≤
h
∗′ 0 i−1 ∗ 1 −1
≤ kA k · f (x ) f (x )
.
1 1 By using Lemma 2.3 again we obtain that kA−1 k ≤ 1−kI−Ak ≤ 1−h , so 0
h
h
∗′ 1 i−1 ∗ 1
∗′ 0 i−1 ∗ 1 1
f (x )
f (x ) ≤ · f (x ) f (x )
. 1 − h0
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−1 ∗ 1
′
In order to majorize f ∗ (x0 ) [f (x )] , we consider the function g : D ∗ ⊂ Rn → Rn , ′ −1 g(x) = x − f ∗ (x0 ) · [f ∗ (x)]. We can observe that g is Fr´echet differentiable on D0 and we want to calculate [g ′(x)], the Jacobian matrix of the function g at point x. Let g = [g1 , g2 , . . . , gn ], so we obtain the matrix form: g(x) = [g1 (x) g2 (x) . . . gi(x) . . . gn (x)]T [x1 x2 . . . xi . . . xn ]T − ′
′
′
′
−[[f1∗ (x0 )]−1 [f2∗ (x0 )]−1 . . . [fi∗ (x0 )]−1 . . . [fn∗ (x0 )]−1 ]T · [f1∗ (x)f2∗ (x) . . . fi∗ (x) . . . fn∗ (x)]. ′ −1 ′ −1 Let fi∗ (x0 ) i denote row i of the matrix fi∗ (x0 ) of order n (which means the ∗′ 0 inverse matrix of the Jacobian matrix fi (x ) of the function fi∗ at point x0 ) for ′ −1 ′ −1 i = 1, n, then gi (x) = xi − fi∗ (x0 ) i fi∗ (x). We mention that fi∗ (x0 ) i is a row vector with n elements of the form [ai1 ai2 . . . ain ], and fi∗ (x) is a column vector, also ′ with n elements. h So the function gi is well i defined. In the h Jacobian matrix [g (x)], i ∂gi ∂gi ∂gi ∂gi ∂gi ∂gi (x), (x), . . . , (x) . Consequently, (x) (x) . . . (x) = row i is given by ∂x ∂x2 ∂xn ∂x1 ∂x2 ∂xn 1 ∗′ 0 −1 ∗′ ∗′ [0 . . . 1 . . . 0]− fi (x ) i . fi (x) , where the number 1 is in position i and fi (x) is the ∗ Jacobian matrix of the function fi∗ at point x, with components fi∗ = (fi1∗ , fi2∗ , . . . , fin ). Consequently, ( n ) X ∂gi ∂g i ′ ′ k[g (x)] − [g (y)]k = max (x) − (y) ∂xj 1≤i≤n ∂x j j=1 from the definition of the matrix norm. But we have: X ∗ n n X n ∗ X ∂gi ∂gi ∂fik ∂f aik · (x) − ik (y) ≤ ∂xj (x) − ∂xj (y) = ∂xj ∂xj j=1 j=1 k=1
≤ = = ≤ = =
n X n X
∗ ∂fik ∂fik∗ |aik | · (x) − (y) = ∂xj ∂xj j=1 k=1 n X n ∗ X ∂fik∗ ∂f ik |aik | · (x) − (y) = ∂xj ∂xj k=1 j=1 ! n n ∗ X X ∂fik∗ ∂f ik ≤ |aik | · (x) − (y) ∂xj ∂x j j=1 k=1 ( n )! n ∗ X X ∂f ∗ ∂f ik (x) − ik (y) |aik | · max = ∂xj 1≤k≤n ∂xj j=1 k=1 ! ( n ) n ∗ X X ∂f ∗ ∂f ik (x) − ik (y) = |aik | · max ∂xj 1≤k≤n ∂xj j=1 k=1 ! ! n n
h i h ′ i X X ′
|aik | · fi∗ (x) − fi∗ (y) ≤ |aik | · γ · kx − yk, k=1
k=1
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by virtue of condition iv. Consequently, k[g ′ (x)] − [g ′ (y)]k ≤ max
1≤i≤n
= max
1≤i≤n
( n X k=1
|aik |
)
(
n X k=1
|aik |
!
· γ · kx − yk
)
=
h
∗′ 0 i−1
· γ · kx − yk
f (x ) · γ · kx − yk ≤ α0 · γ · kx − yk
∗′ 0 −1 according to the definition of the matrix f (x ) and the above demonstrated fact
∗′ 0 −1 that f (x ) ≤ α0 . But [g ′ (x0 )] = On , where On is the null matrix of order n, because: h ′ i−1 h ′ i ∂gi 0 ∂gi 0 ∂gi 0 (x ) (x ) . . . (x ) = [0 . . . 1 . . . 0] − fi∗ (x0 ) fi∗ (x0 ) = ∂x1 ∂x2 ∂xn i = [0 . . . 1 . . . 0] − [0 . . . 1 . . . 0] = [0 . . . 0 . . . 0] so
1
1 h ∗′ 0 i−1 ∗ 1 0 ′ 0 1 0
g(x ) − g(x ) − [g (x )](x − x ) = x − f (x ) f (x ) −
h
h ′ i−1
∗′ 0 i−1 ∗ 1 0 ∗ 0 ∗ 0
−x + f (x ) f (x ) − On f (x )
= f (x )
.
Now we use Lemma 2.5, the mean inequality or the generalized Lagrange’s formula given by Wertheim, choosing h = g, because the following inequality is true: k[g ′ (x)] − [g ′(y)]k ≤ α0 · γ · kx − yk for every x, y ∈ B0 ⊂ D0 with constant L = α0 γ. According to Lemma 2.5, we obtain that
1
g(x ) − g(x0 ) − g ′(x0 ) (x1 − x0 ) ≤ 1 · α0 γ · x1 − x0 2 ≤ 2 1 1 ≤ α0 γβ02 = β0 · h0 . 2 2 Consequently, we have the following estimation:
h
h
∗′ 1 i−1 ∗ 1
∗′ 0 i−1 ∗ 1 1
f (x )
f (x ) f (x )
≤ 1 − h0 · f (x )
≤ 1 1 ≤ · β0 · h0 . 1 − h0 2
∗′ 1 −1 ∗ 1 h0 Now we use the notation β1 = 21 · 1−h · β , so f (x ) [f (x )]
≤ β1 . Therefore we 0 0 obtain a condition similar to iii) for β1 instead of β0 . We verify the condition vi) by using the new values α1 and β1 . Indeed we have: α0 1 h0 1 h0 · · · β0 γ = α0 β0 · γ · · = 1 − h0 2 1 − h0 2 (1 − h0 )2 1 h20 1 = · ≤ , 2 2 (1 − h0 ) 2
h1 = α1 β1 γ =
by elementary calculations.
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Now we choose the sphere B1 = B(x1 , r1 ), where r1 = 1 2
h20 (1−h0 )2
1 2
291
√ 1− 1−2·h1 h1
· β1 . If we substitute
h0 1−h0
and β1 = · · β0 , then we obtain r1 = r0 − β0 after elementary h1 = · calculations. We can show that B1 = B(x1 , r1 ) ⊂ B0 = B(x0 , r0 ). Indeed, let x ∈ B1 be an element. So kx − x1 k ≤ r1 and kx − x0 k ≤ kx − x1 k + kx1 − x0 k ≤ r1 + β0 = r0 − β0 + β0 = r0 , i.e. x ∈ B0 . By using mathematical induction we can show that for every k ∈ N there exist
∗′ k −1
∗′ k −1 fi (x ) and fi (x ) ≤ αk , where
h
i−1 αk−1 1 hk−1 ∗′ k ∗ k
αk = , f (x ) f (x ) ≤ βk , βk = · · βk−1 , 1 − hk−1 2 1 − hk−1 √
k · βk . These hk = αk βk γ ≤ 21 , Bk = B(xk , rk ) ⊂ Bk−1 B(xk−1 , rk−1), and rk = 1− h1−2h k k relations imply that the sequence {x }k∈N given by the Newton-SOR method is well defined and included in B0 . In order to show convergence of the iterative sequence we h2 1 2 2 22 observe that hk = 12 · (1−hk−1 ≤ ··· ≤ 2 ≤ 2 · (2 · hk−1 ) , i.e. 2hk ≤ (2hk−1 ) ≤ (2hk−2 ) k−1 ) k
(2h0 )2 . At the same time we have βk =
1 hk−1 · · βk−1 ≤ hk−1 · βk−1 ≤ hk−1 · hk−2 · βk−2 ≤ · · · ≤ 2 1 − hk−1 k −1
≤ hk−1 · hk−2 . . . h1 · h0 · β0 ≤ 2−k · (2h0 )2
· β0 ,
2k −1
i.e. βk ≤ 2−k · (2h0 ) · β0 , and we obtain that βk → 0 when k → ∞. From the relation √ 1− 1−2hk rk = ·βk , by performing elementary calculations we get rk ≤ 2βk (or see b)). The hk sequence of spheres {Bk }k∈N is a decreasing sequence of spheres whose diameters tend to zero. By using Cantor’s theorem there exists a unique point x∗ included in every Bk such that limk→∞ xk = x∗ . In the following we will show that x∗ is a solution to the equation
−1
−1 −1 −1 ′ ′ ∗ ∗
is a f (x) = θRn . Since f (x) has the Lipschitz property,
f (x)
continuous functional on the compact set B0 and, according to the Weierstrass theorem,it
∗′ −1 −1
≤M is bounded on B0 , i.e. there exists a constant M > 0 such that
f (x)
for every x ∈ B0 . Then
h ′ i−1 −1 h ′ i−1
∗ k
f (x ) ≤ f ∗ (xk ) f ∗ (xk ) ≤
f ∗ (xk )
h h ′ i−1 i−1 −1
∗′ ∗ k ∗ k k
f (x ) ≤ M · βk , ≤ f (x )
· f (x )
so f ∗ (xk ) → On when k → ∞ and from the continuity of f1∗ = f it follows that f (x∗ ) = θRn . To evaluate the error at the k-th iteration, we have: k −1
kxk − x∗ k ≤ rk ≤ 2βk ≤ 21−k · (2h0 )2
· β0 ,
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B. Finta / Central European Journal of Mathematics 3(2) 2005 282–293
because x∗ , xk ∈ Bk . Corollary 2.6. If we choose the relaxation matrix, Ω = On , i.e. the null matrix of order n, then Φ = ϕ and ϕ∗ (x) = [ϕ∗1 (x) ϕ∗2 (x) . . . ϕ∗i (x) . . . ϕ∗n (x)]T [x1 x2 . . . xi . . . xn ]T − −[[f ′ (x)]−1 [f ′ (x)]−1 . . . [f ′ (x)]−1 . . . [f ′ (x)]−1 ]T · · [f (x)f (x) . . . f (x) . . . f (x)] = x − [f ′ (x)]−1 f (x) = ϕ(x), So we reobtain the classical Newton-Raphson-Kantorovich method and Theorem 1.1 follows from Theorem 2.2. Corollary 2.7. Let us choose ωij = 1 for every n ≥ i > j ≥ 1. Then we obtain the nonlinear Seidel method, so the iterative function Φ is given by the following formulae [8]: Φ1 (x) = ϕ1 (x), Φ2 (x) = ϕ2 (Φ1 (x), x2 , . . . , xn ), .. . Φi (x) = ϕi (Φ1 (x), Φ2 (x), . . . , Φi−1 (x), xi , . . . , xn ) .. . Φn (x) = ϕn (Φ1 (x), Φ2 (x), . . . , Φn−1 (x), xn ) In this way we reobtain the author’s method where the Newton-Raphson’s iteration is combined by the Seidel idea [5]. So we can build the function ϕ∗ : D ∗ ⊂ Rn → Rn , ϕ∗ = [ϕ∗1 , ϕ∗2 , . . . , ϕ∗n ] , D ∗ 6= ∅ in the following way, by using definition 1:
ϕ∗ (x) = [ϕ∗1 (x) ϕ∗2 (x) . . . ϕ∗i (x) . . . ϕ∗n (x)]T [x1 x2 . . . xi . . . xn ]T − − [[f ′ (x)]−1 [(f (ϕ∗1 (x), x2 , . . . , xn ))′ ]−1 . . . [(f (ϕ∗1 (x), ϕ∗2 (x), . . . , ϕ∗i−1 (x), xi , . . . , xn ))′ ]−1 . . . [(f (ϕ∗1 (x), ϕ∗2 (x), . . . , ϕ∗n−1 (x), xn ))′ ]−1 ]T · · [f (x)f (ϕ∗1 (x), x2 , . . . , xn ) . . . f (ϕ∗1 (x), ϕ∗2 (x), . . . , ϕ∗i−1 (x), xi , . . . , xn ) . . . f (ϕ∗1 (x), ϕ∗2 (x), . . . , ϕ∗n−1 (x), xn )]. We introduce the following notations: for i = 1, n, fi∗ : Di∗ ⊂ Rn → Rn , Di∗ 6= ∅, f1∗ (x) = f (x), f2∗ (x) = f (ϕ∗1 (x), x2 , . . . , xn ) , . . . , fi∗ (x) = f ϕ∗1 (x), ϕ∗2 (x) . . . , ϕ∗i−1 (x), xi , . . . , xn , . . . , fn∗ (x) = f ϕ∗1 (x), ϕ∗2 (x), . . . , ϕ∗n−1 (x), xn ,
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where we suppose that D ∗ ⊂ ∩ni=1 Di∗ . So we have: ϕ∗ (x) = [ϕ∗1 (x) ϕ∗2 (x) . . . ϕ∗i (x) . . . ϕ∗n (x)]T = [x1 x2 . . . xi . . . xn ]T − h i−1 h ′ i−1 h ′ i−1 h ′ i−1 T ∗′ ∗ ∗ − f1 (x) f2 (x) . . . fi (x) . . . fn∗ (x) · · [f1∗ (x)f2∗ (x) . . . fi∗ (x) . . . fn∗ (x)] ,
′ −1 ′ where for i = 1, n fi∗ (x) means the inverse matrix of the Jacobian matrix fi∗ (x) of the function fi∗ at point x and fi∗ (x) are column vectors of size n × 1. If we use the symbolical notations: ∗′
−1
[f (x)]
h i−1 h ′ i−1 h ′ i−1 h ′ i−1 T ∗′ ∗ ∗ = f1 (x) f2 (x) . . . fi (x) . . . fn∗ (x)
and [f ∗ (x)] = [f1∗ (x)f2∗ (x) . . . fi∗ (x) . . . fn∗ (x)] then we obtain the new Newton-Raphson ′ −1 ∗ Seidel type iterative function: ϕ∗ : D ∗ ⊂ Rn → Rn , ϕ∗ (x) = x − f ∗ (x) [f (x)] . We ∗′ ∗ mention that we did not define the function f , f (x) does not mean the Jacobian ′ −1 ′ matrix, and f ∗ (x) does not mean the invers matrix of the matrix f ∗ (x) . Consequently, from Theorem 2.2 with the same formal formulation we obtain the author’s result [5].
References [1] N.S. Bahvalov: Numerical Methods, Technical Press, Budapest, 1977 (in Hungarian). [2] R.G. Douglas: Banach Algebra Techniques in Operator Theory, Academic Press, New York and London, 1972. [3] B. Finta: “Note about the iterative solutions of the nonlinear operator equations in finite dimensional spaces”, Research Seminars, Department of Mathematics, Technical University of Tg.Mures, Romania, Vol. 3, (1994), pp. 49–79. [4] B. Finta: “Note about a method for solving nonlinear system of equations in finite dimensional spaces”, Studia Univ. Babes-Bolyai, Romania, Mathematica, XL, Vol. 1, (1995), pp. 59–64. [5] B. Finta: “A Newton-Kantorovich-Seidel Type Theorem”, Publ. Univ. of Miskolc, Series D. natural Sciences, Hungary, Vol. 38, (1998), pp. 31–40, . [6] L.V. Kantorovich and G.P. Akilov: Functional Analysis in Normed Spaces, Academic Press, New York, 1978. [7] J. Ortega and W. Rheinboldt: Local and global convergence of generalized linear iterations, Numerical solution of nonlinear problems, Soc. Ind. Appl. Math., Philadelphia, 1970. [8] F. Szidarovszky and S. Yakowitz: Principles and Procedures of Numerical Analysis, Plenum Press, New York and London, 1978. [9] V.A. Wertheim: “On the conditions for the application of Newton’s method”, D.A.N., Vol. 110, (1956), pp. 719–722.
CEJM 3(2) 2005 294–308
Higher order valued reduction theorems for classical connections Josef Janyˇska∗† Department of Mathematics, Masaryk University, Jan´aˇckovo n´am. 2a, 602 00 Brno, Czech Republic
Received 21 October 2004; accepted 21 December 2004 Abstract: We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the 2nd order valued reduction theorems we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection. c Central European Science Journals. All rights reserved.
Keywords: Natural bundle, natural operator, classical connection, reduction theorem MSC (2000): 53C05, 58A32, 58A20
1
Introduction
It is well known that natural operators of linear symmetric connections on manifolds and of tensor fields which have values in bundles of geometrical objects of order one can be factored through curvature tensors, tensor fields and their covariant differentials. These results are known as the first (the operators of connections only) and second reduction theorems. The history of the first and second reduction theorems goes back to the paper by Christoffel, [1], and the paper by Ricci and Levi Civita, [11], respectively. For further references see [5, 9, 12, 16]. In [12] the proof for algebraic operators (concomitants) is given. In [5] the first and the second reduction theorems are proved for all natural differential operators by using the modern approach of natural bundles and natural differential operators, [5, 8, 10, 15]. The reduction theorems play a very important role in ∗
E-mail:
[email protected] This paper has been supported by the Grant Agency of the Czech Republic under the Project number GA 201/02/0225. †
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theoretical physics. Namely, if we represent linear connections on manifolds as principal connections on the principal bundles of first order frames, then the reduction theorems are in fact higher order versions of Utiyama’s theorem (the first reduction theorem) and Utiyama’s invariant interaction (the second reduction theorem), [17]. In this paper we generalize the reduction theorems for natural operators which have values in higher order natural bundles. For these theorems we shall use the name higher order valued reduction theorems for classical connections. As an example we discuss natural (0,2)-tensor fields on the cotangent bundle of a manifold. In this paper we use the terms ”natural bundle” and “natural operator” in the sense of [5, 8, 10, 15]. Namely, a natural operator is defined to be a system of local operators AM : C ∞ (F M ) → C ∞ (GM ), such that AN (fF∗ s) = fG∗ AM (s) for any section (s : M → F M ) ∈ C ∞ (F M ) and any (local) diffeomorphism f : M → N , where F, G are two natural bundle functors and fF∗ s = F f ◦s◦f −1 . A natural operator is said to be of order r if, for all sections s, q ∈ C ∞ (F M ) and every point x ∈ M , the condition jxr s = jxr q implies AM s(x) = AM q(x). Then we have the induced natural transformation AM : J r F M → GM such that AM (s) = AM (j r s), for all s ∈ C ∞ (F M ). The correspondence between natural operators of order r and the induced natural transformations is bijective. In this paper we shall identify natural operators with the corresponding natural transformations. Any natural bundle functor F of order r is given by its standard fibre SF which is a left Grm -manifold, where Grm = inv J0r (IRm , IRm )0 is the r-th order differential group. A classification of natural operators between natural bundles is equivalent to the classification of equivariant maps between standard fibers. A very important tool in classifications of equivariant maps is the orbit reduction theorem, [5, 7, 8]. Let p : G → H be a Lie group epimorphism with kernel K, M be a left G-space, N and Q be left H-spaces, and π : M → Q be a p-equivariant surjective submersion, i.e. π(gx) = p(g)π(x) for all x ∈ M, g ∈ G. Given p, we can consider every left H-space N as a left G-space by gy = p(g)y, g ∈ G, y ∈ N. Theorem 1.1. If each π −1 (q), q ∈ Q is a K-orbit in M, then there is a bijection between the G-maps f : M → N and the H-maps ϕ : Q → N given by f = ϕ ◦ π. All manifolds and maps are assumed to be smooth. The sheaf of (local) sections of a fibered manifold p : Y → X is denoted by C ∞ (Y ), C ∞ (Y , IR) denotes the sheaf of (local) functions.
2
Preliminaries
Let M be an m-dimensional manifold. If (xλ ), λ = 1, . . . , m, is a local coordinate chart, then the induced coordinate charts on T M and T ∗ M will be denoted by (xλ , x˙ λ ) and (xλ , x˙ λ ) and the induced local bases of sections of T M and T ∗ M will be denoted by (∂λ ) and (dλ ), respectively.
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Definition 2.1. We define a classical connection to be a connection Λ : T M → T ∗M ⊗ T T M TM
of the vector bundle pM : T M → M which is linear and torsion free. The coordinate expression of a classical connection Λ is of the type Λ = dλ ⊗ ∂λ + Λλ µ ν x˙ ν ∂˙µ ,
with
Λµ λ ν = Λν λ µ ∈ C ∞ (M , IR) .
Classical connections can be regarded as sections of a 2nd order natural bundle Cla M → M , [5]. The standard fibre of the functor Cla will be denoted by Q = IRm ⊗ ⊙2 IRm∗ , elements of Q are said to be formal classical connections, the induced coordinates on Q are said to be formal Christoffel symbols and will be denoted by (Λµ λ ν ). The action α : G2m × Q → Q of the group G2m on Q is given in coordinates by (Λµ λ ν ) ◦ α = (aλρ (Λσ ρ τ a ˜σµ a ˜τν − a ˜ρµν )) , where (aλµ , aλµν ) are the coordinates on G2m and ˜ denotes the inverse element. Remark 2.2. Let us note that the action α gives in a natural way the action r r αr : Gr+2 m × Tm Q → Tm Q
given by the jet prolongation of the action α. r+2 Remark 2.3. Let us consider the group epimorphism πr+1 : Gr+2 → Gr+1 and its m m r+2 r+2 r+2 λ kernel Br+1 = Ker πr+1 . We have the induced coordinates (aµ1 ...µr+2 ) on Br+1 . Then the r+2 restriction α ¯ r of the action αr to Br+1 has the following coordinate expression
(Λµ1 λ µ2 , . . . , Λµ1 λ µ2 ,µ3 ...µr+2 ) ◦ α ¯r = (Λµ1 λ µ2 , . . . , Λµ1 λ µ2 ,µ3 ...µr+1 , Λµ1 λ µ2 ,µ3 ...µr+2 − a˜λµ1 ...µr+2 ) , where (Λµ1 λ µ2 , Λµ1 λ µ2 ,µ3 , . . . , Λµ1 λ µ2 ,µ3 ...µr+2 ) are the induced jet coordinates on Tmr Q. The curvature tensor of a classical connection is a section R[Λ] : M → W M := T ∗ M ⊗ V T M ⊗ 2 T ∗ M with coordinate expression R[Λ] = Rν ρ λµ dν ⊗ ∂ρ ⊗ dλ ∧ dµ ,
where the coefficients are Rν ρ λµ = ∂µ Λλ ρ ν − ∂λ Λµ ρ ν + Λµ σ ν Λλ ρ σ − Λλ σ ν Λµ ρ σ . Let us note that the curvature tensor is a natural operator R[Λ] : C ∞ (Cla M ) → C ∞ (W M ) which is of order one, i.e., we have the associated G3m -equivariant mapping, called the formal curvature map of classical connections, R : Tm1 Q → W := ST ∗ ⊗T ⊗V2 T ∗
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with coordinate expression (wν ρ λµ ) ◦ R = (Λλ ρ ν,µ − Λµ ρ ν,λ + Λµ σ ν Λλ ρ σ − Λλ σ ν Λµ ρ σ ) ,
(1)
where (wν ρ λµ ) are the induced coordinates on the standard fibre W = IRm∗ ⊗IRm ⊗
V2
IRm∗ .
Let V M be a first order natural vector bundle over M . Let us set Vr M = V M ⊗ ⊗ T ∗ M , V (k,r) M = Vk M × . . . × Vr M , k ≤ r, V (r) M := V (0,r) M . Let us denote by r
M
M
V = IRn or Vr or V (k,r) the standard fibres of V M or Vr M or V (k,r) M , respectively. The r-th order covariant differential of sections of V M with respect to classical connections is a natural operator ∇r : J r−1 Cla M × J r V M → Vr M . M
We shall denote by the same symbol its corresponding Gr+1 m -equivariant mapping ∇r : Tmr−1 Q × Tmr V → Vr . We shall set ∇(k,r) := (∇k , . . . , ∇r ) : J r−1 Cla M × J r V M → V (k,r) M M
(r) := and the same for the corresponding Gr+1 ∇(0,r) . m -equivariant mapping. Especially ∇
Remark 2.4. For any section σ : M → V M we have Alt(∇2 σ) = pol(R[Λ], σ) ,
(2)
where Alt is the antisymmetrization and pol(R[Λ], σ) is a bilinear polynomial. Namely, Alt(∇2 R[Λ]) is a quadratic polynomial of R[Λ]. If (v A ) are coordinates on V , then (v A , v A λ , . . . , v A λ1 ...λr ) are the induced jet coordinates on Tmr V (symmetric in all subscripts) and (V A λ1 ...λr ) are the canonical coordinates on Vr . Then ∇r is of the form (V A λ1 ...λr ) ◦ ∇r = v A λ1 ...λr + pol(Tmr−1Q × Tmr−1 V ) ,
(3)
where pol is a quadratic homogeneous polynomial on Tmr−1 Q × Tmr−1 V . Remark 2.5. Let us recall the 1st and 2nd Bianchi identities of classical connections given in coordinates by R(ν ρ λµ) = 0 ,
Rν ρ (λµ;σ) = 0 ,
respectively, where ; denotes the covariant differential with respect to Λ and (. . . ) denotes the cyclic permutation.
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The first k-th order valued reduction theorem
Let us introduce the following notations. Let W0 M := W M , Wi M = W M ⊗⊗i T ∗ M , i ≥ 0. Let us set W (k,r)M = Wk M ×M . . . ×M Wr M , k ≤ r. We put W (r) M = W (0,r) M . Then Wi M and W (k,r) M are natural bundles of order one and the corresponding standard fibers will be denoted by Wi and W (k,r) , respectively, where W0 := W , Wi = W ⊗⊗i IRm∗ , i ≥ 0, and W (k,r) = Wk ×. . .×Wr . We denote by Ri : Tmi+1 Q → Wi the Gi+3 m -equivariant map associated with the i-th covariant differential of curvature tensors of classical connections ∇i R[Λ] : C ∞ (Cla M ) → C ∞ (Wi M ) . The map Ri is said to be the formal curvature map of order i of classical connections. Let Ci ⊂ Wi be a subset given by identities of the i-th covariant differentials of the curvature tensors of classical connections, i.e., by covariant differentials of the Bianchi identities and the antisymmetrization of second order covariant differentials, see Remark 2.4 and Remark 2.5. So Ci is given by the following system of equations w(ν ρ λµ)σ1 ...σi = 0 , wν
ρ
(λµσ1 )σ2 ...σi
(4)
= 0,
(5)
wν ρ λµσ1 ...[σj−1 σj ]...σi + pol(W (i−2) ) = 0 ,
(6)
where j = 2, . . . , i and [..] denotes the antisymmetrization. (k,r) Let us set C (r) = C0 ×. . .×Cr and denote by Cr(k−1) , k ≤ r, the fiber in r (k−1) ∈ C (k−1) (k,r) of the canonical projection prrk−1 : C (r) → C (k−1) . For r < k we set Cr(k−1) = ∅. Let us note that there is an affine structure on the projection prrr−1 : C (r) → C (r−1) , [5]. Then we set R(k,r) := (Rk , . . . , Rr ) : Tmr+1 Q → W (k,r) ,
R(r) := R(0,r) ,
(7)
(k,r)
which has values in CR(k−1) (j k λ) , , for any j0r+1 λ ∈ Tmr+1 Q. In [5] it was proved that C (r) is 0
a submanifold in W (r) and the restriction of R(r) to C (r) is a surjective submersion. Then we can consider the fiber product Tmk Q ×C (k−1) C (r) and denote it by Tmk Q × C (k,r) . First we shall prove the technical Lemma 3.1. If r + 1 ≥ k ≥ 1, then the restricted map (πkr+1 , R(k,r) ) : Tmr+1 Q → Tmk Q × C (k,r) is a surjective submersion.
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Proof. To prove surjectivity of (πkr+1 , R(k,r)) it is sufficient to consider the commutative diagram R(r)
Tmr+1 Q −−−→ r+1 πk y
R(k−1)
C (r) r prk−1 y
Tmk Q −−−−→ C (k−1)
All morphisms in the above diagram are surjective submersions which implies that for any element j0k λ ∈ Tmk Q the restriction of R(r) to the fibre (πkr+1 )−1 (j0k λ) is a surjective (k,r) submersion of the fibre (πkr+1 )−1 (j0k λ) on the fibre (prrk−1 )−1 (R(k−1) (j0k λ)) ≡ CR(k−1) (j k λ) 0
which proves that the mapping (πkr+1 , R(k,r) ) is surjective. To prove that (πkr+1 , R(k,r) ) is a submersion we shall consider the above diagram for k = r. From the formal covariant differentials of (1) it follows that R(r,r) = Rr is an affine morphism over R(r−1) (with respect to the affine structures on πrr+1 : Tmr+1 Q → Tmr Q and prrr−1 : C (r) → C (r−1) ) which has constant rank. So the surjective morphism (πrr+1 , Rr ) : Tmr+1 Q → Tmr Q × C (r,r) has constant rank and hence is a submersion. (πkr+1 , R(k,r) ) is then a composition of surjective submersions. Let F be a natural bundle functor of order k ≥ 1, i.e., SF is a left Gkm -manifold. Theorem 3.2. Let r + 2 ≥ k. For every Gr+2 m -equivariant map f : Tmr Q → SF there exists a unique Gkm -equivariant map g : Tmk−2 Q × C (k−2,r−1) → SF satisfying r f = g ◦ (πk−2 , R(k−2,r−1) ) .
Proof. Let us consider the space Sr := IRm ⊗ ⊙r IRm∗ with coordinates (sλ µ1 µ2 ...µr ). Let us consider the action of Grm on Sr given by s¯λ µ1 µ2 ...µr = sλ µ1 µ2 ...µr − a ˜λµ1 ...µr .
(8)
From Remark 2.3 and (8) it is easy to see that the symmetrization map σr : Tmr Q → Sr+2 given by (sλ µ1 µ2 ...µr+2 ) ◦ σr = Λ(µ1 λ µ2 ,µ3 ...µr+2 ) , is equivariant. We have the Gr+2 m -equivariant map r ϕr := (σr , πr−1 , Rr−1) : Tmr Q → Sr+2 × Tmr−1 Q × Wr−1 .
On the other hand we define a Gr+2 m -equivariant map ψr : Sr+2 × Tmr−1 Q × Wr−1 → Tmr Q
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over the identity of Tmr−1 Q by the following coordinate expression Λµ λ ν,ρ1 ...ρr = sλ µνρ1 ...ρr + lin(wµ λ νρ1 ...ρr − pol(Tmr−1 Q)) ,
(9)
where lin denotes a linear combination with real coefficients which arises in the following way. We recall that Rr−1 gives the coordinate expression, given by formal covariant differentials of (1), Λµ λ ν,ρ1 ...ρr − Λµ λ ρ1 ,νρ2 ...ρr = wµ λ νρ1 ...ρr − pol(Tmr−1 Q) .
(10)
We can write Λµ λ ν,ρ1 ...ρr = sλ µνρ1 ...ρr + (Λµ λ ν,ρ1 ...ρr − Λ(µ λ ν,ρ1 ...ρr ) ) . Then the term in brackets can be written as a linear combination of terms of the type Λµ λ ν,ρi ρ1 ...ρi−1 ρi+1 ...ρr − Λµ λ ρi ,νρ1 ...ρi−1 ρi+1 ...ρr , i = 1, . . . , r, and from (10) we get (9). Moreover, ψr ◦ ϕr = idTmr Q . Then the map f ◦ ψr : Sr+2 × Tmr−1 Q × Wr−1 → SF satisfies the conditions of the orbit r+2 r+1 reduction Theorem 1.1 for the group epimorphism πr+1 : Gr+2 m → Gm and the surjective submersion pr2,3 : Sr+2 × Tmr−1 Q × Wr−1 → Tmr−1 Q × Wr−1 . Indeed, the space Sr+2 is a r+2 r+2 Br+1 -orbit. Moreover, (8) implies that the action of Br+1 on Sr+2 is simply transitive. r+1 r−1 Hence there exists a unique Gm -equivariant map gr−1 : Tm Q × Wr−1 → SF such that the following diagram ψr
Sr+2 × Tmr−1 Q × Wr−1 pr2,3 y
Tmr−1 Q × Wr−1
f
Tmr Q
−−−→
−−−→ SF
r (πr−1 ,Rr−1 ) y
id
y
idSF
r−1 Q×W Tm r−1
gr−1
−−−−−−−−−→ Tmr−1 Q × Wr−1 −−−→ SF
commutes. So f ◦ ψr = gr−1 ◦ pr2,3 and if we compose both sides with ϕr , by considering r r pr2,3 ◦ϕr = (πr−1 , Rr−1 ), we obtain f = gr−1 ◦ (πr−1 , Rr−1 ) . In the second step we consider the same construction for the map gr−1 and obtain the commutative diagram ψr−1 ×idWr−1
(Sr+1 × Tmr−2 Q × Wr−2 ) × Wr−1 y
pr2,3 × idWr−1
Tmr−2 Q × W (r−2,r−1)
−−−−−−−−→
id
Tmr−1 Q × Wr−1 y
r−1 (πr−2 ,Rr−2 )×idWr−1
T r−2 Q×W (r−2,r−1)
gr−1
−−−→ SF y
idSF
gr−2
−−−m−−−−−−−−−→ Tmr−2 Q × W (r−2,r−1) −−−→ SF
So that there exists a unique Grm -equivariant map gr−2 : Tmr−2Q × W (r−2,r−1) → SF such r−1 r that gr−1 = gr−2 ◦ ((πr−2 , Rr−2 ) × idWr−1 ), i.e., f = gr−2 ◦ (πr−2 , R(r−2,r−1) ) . Proceeding in this way we get in the last step a unique Gkm -equivariant map gk−2 : Tmk−2 Q × W (k−2,r−1) → SF such that r f = gk−2 ◦ (πk−2 , R(k−2,r−1) ) .
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Putting g the restriction of gk−2 to Tmk−2Q × C (k−2,r−1) we prove Theorem 3.2. In the above Theorem 3.2 we have found a map g which factors f , but we did not r prove that (πk−2 , R(k−2,r−1) ) : Tmr Q → Tmk−2 Q × C (k−2,r−1) satisfy the orbit conditions, r namely we did not prove that (πk−2 , R(k−2,r−1) )−1 (j0k−2λ, r (k−2,r−1) ) is a Bkr+2 -orbit for any (j0k−2 λ, r (k−2,r−1)) ∈ Tmk−2 Q × C (k−2,r−1) . Now we shall prove it. ´ ∈ T r Q satisfy Lemma 3.3. If (j0r λ), (j0r λ) m r r ´ , (πk−2 , R(k−2,r−1) )(j0r λ) = (πk−2 , R(k−2,r−1) )(j0r λ)
´ = (j r λ). then there is an element h ∈ Bkr+2 such that h . (j0r λ) 0 Proof. Consider the orbit set Tmr Q/Bkr+2 . This is a Gkm -set. Clearly the factor projection k p : Tmr Q → Tmr Q/Bkr+2 is a Gr+2 m -map. By Theorem 3.2 there is a Gm -equivariant map g : Tmk−2 Q × C (k−2,r−1) → Tms Q/Bkr+2 r r r ´ = satisfying p = g ◦ (πk−2 , R(k−2,r−1) ). If (πk−2 , R(k−2,r−1) )(j0r λ) = (πk−2 , R(k−2,r−1) )(j0s λ) k−2 k−2 ´ so j r λ and j r λ ´ are in the (j0 λ, r (k−2,r−1)), then p(j0r λ) = g(j0 λ, r (k−2,r−1) ) = p(j0r λ), 0 0 same Bkr+2 -orbit.
It is easy to see that Tmk−2Q×C (k−2,r−2) is closed with respect to the action of the group Gkm . The corresponding natural bundle of order k is J k−2 Cla M × C (k,r) M . Then, as a M
direct consequence of Theorem 3.2, we obtain the first k-order valued reduction theorem for classical connections. Theorem 3.4. Let F be a natural bundle functor of order k ≥ 1 and let r + 2 ≥ k. All natural differential operators f : C ∞ (Cla M ) → C ∞ (F M ) which are of order r are of the form f (j r Λ) = g(j k−2Λ, ∇(k−2,r−1) R[Λ]) where g is a unique natural operator g : J k−2 Cla M × C (k−2,r−1) M → F M . M
Remark 3.5. From the proof of Theorem 3.2 it follows that the operator g is the restriction of a natural operator defined on the natural bundle J k−2 Cla M × W (k−2,r−1) M . M
4
The second k-th order valued reduction theorem
Remark 2.4 defines for r = 2 the equation V A [λµ] − pol(C0 , V ) = 0
(E2 )
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on C0 × V2 and for r > 2 the system of equations V A µ1 ...[µs−1 µs ]...µr − pol(C (r−2) , V (r−2) ) = 0
(Er )
on C (r−2) × V (r) . The r-th Ricci subspace Z (r) ⊂ C (r−2) × V (r) is defined by solutions of (E2 ), . . . , (Er ), r ≥ 2. For r = 0 we set Z (0) = V and for r = 1 we set Z (1) = V (1) . In [5] it was proved that Z (r) is a submanifold in C (r−2) × V (r) and (R(r−2) , ∇(r) ) : Tmr−1 Q × Tmr V → Z (r) is a surjective submersion. For r > k − 1 we can consider the projection prrk−1 : Z (r) → Z (k−1) (k,r) and denote by Zz (k−1) its fiber in z (k−1) ∈ Z (k−1) . Then we shall denote by Tmk−2 Q × Tmk−1 V × Z (k,r) the fiber product (Tmk−2Q × Tmk−1 V ) ×Z (k−1) Z (r) . Lemma 4.1. If r + 1 ≥ k ≥ 1, then the restricted map r−1 r (πk−2 × πk−1 ) × (R(k−2,r−2) , ∇(k,r) ) : Tmr−1 Q × Tmr V → Tmk−2Q × Tmk−1 V × Z (k,r)
is a surjective submersion. Proof. The proof of Lemma 4.1 follows from the commutative diagram Tmr−1 Q × Tmr V r−1 r πk−2 ×πk−1 y
(R(r−2) ,∇(r) )
−−−−−−−→
(R(k−3) ,∇(k−1) )
Z (r) r prk−1 y
Tmk−2Q × Tmk−1 V −−−−−−−−−→ Z (k−1)
r−1 r where all morphisms are surjective submersions. Hence (πk−2 × πk−1 ) × (R(k−2,r−2) , ∇(k,r) ) is surjective. For k = r the map (R(r−2,r−2) = Rr−2 , ∇(r,r) = ∇r ) is an affine morphism r−1 r over (R(r−3) , ∇(r−1) ) with constant rank, i.e., (πr−2 × πr−1 ) × (Rr−2 , ∇r ) is a submersion. r−1 r (πk−2 × πk−1 ) × (R(k−2,r−2) , ∇(k,r) ) is then a composition of surjective submersions.
Theorem 4.2. Let SF be a left Gkm -manifold. If r + 1 ≥ k ≥ 1, then for every Gr+1 m r−1 r k equivariant map f : Tm Q × Tm V → SF there exists a unique Gm -equivariant map g : Tmk−2Q × Tmk−1 V × Z (k,r) → SF such that r−1 r f = g ◦ ((πk−2 × πk−1 ) × (R(k−2,r−2) , ∇(k,r) )) .
Proof. Consider the map r (idTmr−1Q ×πk−1 , ∇(k,r) ) : Tmr−1 Q × Tmr V → Tmr−1Q × Tmk−1V × V (k,r)
and denote by Ve (k,r) ⊂ Tmr−1 Q×Tmk−1 V ×V (k,r) its image. By (3), the restricted morphism f ∇(k,r) : Tmr−1 Q × Tmr V → Ve (k,r) is bijective for every j0r−1 λ ∈ Tmr−1 Q, so that f ∇(k,r) is an equivariant diffeomorphism. Define e (k−2,r−2) : Ve (k,r) → T k−2 Q × T k−1 V × Z (k,r) R m m
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by e (k−2,r−2) (j r−1 λ, j k−1 µ, v) = (j k−2 λ, j k−1 µ, R(k−2,r−2) (j r−1 λ), v) , R 0 0 0 0 0
e (k−2,r−2) is a surjective submersion. (j0r−1 λ, j0k−1µ, v) ∈ Ve (k,r) . By Lemma 3.1 R e (k−2,r−2) satisfies the orbit conditions Thus, Lemma 3.1 and Lemma 3.3 imply that R k k for the group epimorphism πkr+1 : Gr+1 m → Gm and there exists a unique Gm -equivariant map g : Tmk−2 Q × Tmk−1V × Z (k,r) → SF such that the diagram
Ve (k,r)
e(k−2,r−2) R y
e (k,r) )−1 (∇
−−−−−−→
f
Tmr−1 Q × Tmr V
−−−→ SF
r−1 r (k−2,r−2) (k,r) ((πk−2 ×πk−1 )×(R ,∇ )) y
Tmk−2 Q × Tmk−1V × Z (k,r)
id
−−−→
g
idSF y
Tmk−2Q × Tmk−1 V × Z (k,r) −−−→ SF
e (k−2,r−2) . Composing both sides with f commutes. Hence f ◦ (f ∇(k,r) )−1 = g ◦ R ∇(k,r) , by r−1 r e (k−2,r−2) ◦ f considering R ∇(k,r) = ((πk−2 × πk−1 ) × (R(k−2,r−2) , ∇(k,r))), we get r−1 r f = g ◦ ((πk−2 × πk−1 ) × (R(k−2,r−2) , ∇(k,r))) .
Tmk−2 Q × Tmk−1 V × Z (k,r) is closed with respect to the action of the group Gkm . The corresponding natural bundle of order k is J k−2 Cla M × J k−1 V M × Z (k,r) M . M
M
Then the second k-order valued reduction theorem can be formulated as follows. Theorem 4.3. Let F be a natural bundle of order k ≥ 1 and let r + 1 ≥ k. All natural differential operators f : C ∞ (Cla M × V M ) → C ∞ (F M ) of order r with respective M
sections of V M are of the form
f (j r−1Λ, j r Φ) = g(j k−2Λ, j k−1Φ, ∇(k−2,r−2) R[Λ], ∇(k,r) Φ) where g is a unique natural operator g : J k−2 Cla M × J k−1 V M × Z (k,r) M → F M . M
M
Remark 4.4. The order (r − 1) of the above operators with respect to classical connections is the minimal order we have to use. The second reduction theorem can be easily generalized for any operator of order s ≥ r − 1 with respect to connections. Then f (j s Λ, j r Φ) = g(j k−2Λ, j k−1Φ, ∇(k−2,s−1) R[Λ], ∇(k,r) Φ) . Remark 4.5. If Λ is a linear non-symmetric connection on M , then there exists its e + T , where Λ e is the classical connection obtained by the symmetrization splitting Λ = Λ of Λ and T is the torsion tensor of Λ. Then all natural operators of order r defined on Λ are of the form e j r T ) = g(j k−2 Λ, e j k−1 T, f e ∇ f(k,r) T ) . ∇(k−2,r−1) R[Λ], f (j r Λ) = f (j r Λ,
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Remark 4.6. If g is a metric field on M , then there exists the unique classical Levi Civita connection Λ given by the metric field g. Then, applying the second reduction theorem, we get that all natural operators of order r ≥ 1 defined on g are of the form f (j r g) = f (j r−1 Λ, j r g) = h(j k−2 Λ, j k−1 g, ∇(k−2,r−2)R[Λ]) = h(j k−1 g, ∇(k−2,r−2)R[Λ]) .
5
Natural (0,2)-tensor fields on the cotangent bundle
Typical applications of higher order valued reduction theorems are classifications of natural tensor fields on the tangent (or cotangent) bundle of a manifold endowed with a classical connection, or lifts of tensor fields to the tangent (or cotangent) bundle by means of a classical connection, see [2, 3, 6, 13, 14]. As a direct consequence of Theorem 3.2, Theorem 4.3 and Remark 4.5 we get Corollary 5.1. Let (M , Λ) be a manifold endowed with a linear (non-symmetric) connection Λ. Then any natural tensor field Φ on T M or T ∗ M of order r is of the type e j 1 T, f e f Φ(u, j r Λ) = Φ(u, Λ, ∇(r−1) R[Λ], ∇(2,r) T ) ,
e is the classical connection given by the where u ∈ T M or u ∈ T ∗ M , respectively, Λ symmetrization of Λ and T is the torsion tensor of Λ.
Corollary 5.2. Let (M , Λ, Ψ) be a manifold endowed with a linear (non-symmetric) connection Λ and a tensor field Ψ. Then any natural tensor field Φ on T M or T ∗ M of order s with respect to Λ and of order r, s ≥ r − 1, with respect to Ψ is of the type e j 1 T, j 1 Ψ, ∇ f(s−1) R[Λ], e f Φ(u, j s Λ, j r Ψ) = Φ(u, Λ, ∇(2,s) T, f ∇(2,r) Ψ) ,
where u ∈ T M or u ∈ T ∗ M , respectively.
As a concrete example let us classify all (0,2)-tensor fields on T ∗ M given by a linear (non-symmetric) connection Λ. Theorem 5.3. Let (M , Λ) be a manifold endowed with a linear (non-symmetric) connection Λ. Then all finite order natural (0,2)-tensor fields on T ∗ M are of the maximal order one and they form a 14-parameter family of operators with coordinate expression
Φ = A x˙ λ x˙ µ + C1 x˙ λ Tρ ρ µ + C2 x˙ µ Tρ ρ λ + C3 x˙ ρ Tλ ρ µ
(11)
+ F1 Tρ ρ λ Tσ σ µ + F2 Tσ ρ λ Tρ σ µ + F3 Tρ ρ σ Tλ σ µ
+ G1 Tρ ρ λ;µ + G2 Tρ ρ µ;λ + G3 Tλ ρ µ;ρ + H1 Rρ ρ λµ + H2 Rλ ρ ρµ dλ ⊗ dµ + B dλ ⊗ (d˙λ + Λλ ρ µ x˙ ρ dµ ) + C (d˙λ + Λλ ρ µ x˙ ρ dµ ) ⊗ dλ , where A, B, C, Ci, Fi , Gi , Hj , i = 1, 2, 3, j = 1, 2, are real constants.
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Proof. Let us denote by S = IRm∗ × ⊗2 IRm∗ × IRm∗ ⊗ IRm × IRm ⊗ IRm∗ × ⊗2 IRm the ¯ standard fibre of ⊗2 T ∗ (T ∗ M ). The coordinates on S will be denoted by (x˙ λ , φλµ , φλ µ¯ , φλµ , ¯ φλ¯µ ). Then we have the following action of the group G2m on S ¯ x¯˙λ = a ˜µλ x˙ µ , φ¯λ¯µ = aλρ aµσ φρ¯σ¯ , φ¯λµ = a ˜ρλ a ˜σµ φρσ + aρλ aασβ a ˜βµ a ˜κα x˙ κ φρ σ¯ + aσµ aαρβ a ˜βλ a ˜κα x˙ κ φρ¯σ
+ aαρβ a ˜βλ a ˜κα x˙ κ aγσδ a ˜δµ a ˜νγ x˙ ν φρ¯σ¯ , ¯ φ¯λ µ = aλρ a ˜σµ φρ¯σ + aλρ aασβ a˜βµ a˜κα x˙ κ φρ¯σ¯ .
φ¯λ µ¯ = a ˜ρλ aµσ φρ σ¯ + aµσ aαρβ a ˜βλ a ˜κα x˙ κ φρ¯σ¯ , ¯
First let us discuss φλ¯µ . We have, by Corollary 5.1, ¯
¯
λ e λ ,T λ ,T λ ,R e ρ φλ¯µ = φλ¯µ (x˙ λ , Λ µ ν µ ν µ ν,σ ν λµ;σ1 ;...;σi , Tµ ν;σ1 ;...;σj ) ,
i = 0, . . . , r − 1, j = 2, . . . , r. The equivariance with respect to homotheties (c δµλ ) implies ¯
¯
−(j+1) e λ , c−1 T λ , c−2 T λ , c−(i+2) R e ρ c2 φλ¯µ = φλ¯µ (c−1 x˙ λ , c−1 Λ Tµ λ ν;σ1 ;...;σj ) µ ν µ ν µ ν,σ ν λµ;σ1 ;...;σi , c ¯
which implies, by the homogeneous function theorem, [5], that φλ¯µ is a polynomial of λ e λ , c in T λ , c in T λ , d in R e ρ orders a in x˙ λ , b in Λ µ ν 0 µ ν 1 µ ν,σ i ν λµ;σ1 ;...;σi and ej in Tµ ν;σ1 ;...;σj such that 2 = −a − b − c0 − 2 c1 −
r−1 X
(i + 2) di −
i=0
r X
(j + 1) ej .
(12)
j=2
The equation (12) has no solution in natural numbers, so we get by the homogeneous ¯ function theorem that φλ¯µ is independent of all variables and so it has to be absolute invariant, hence ¯
φλ¯µ = 0 .
(13)
¯
For φλ µ¯ and φλ µ we get from the equivariance with respect to the homotheties (c δµλ ) that they are polynomials of orders satisfying the equation (12) with 0 on the left hand ¯ side. So also φλ µ¯ and φλ µ are independent of all variables and they have to be absolute invariant, hence φλ µ¯ = B δλµ ,
¯
φλ µ = C δµλ .
(14)
Finally φλµ has to be a polynomial of orders satisfying the equation (12) with −2 on the left hand side. There are 8 possible solutions: a = 2 and the other exponents vanish; a = 1, b = 1 and the other exponents vanish; a = 1, c0 = 1 and the other exponents vanish; b = 2 and the other exponents vanish; b = 1, c0 = 1 and the other exponents vanish; c0 = 2 and the other exponents vanish; c1 = 1 and the other exponents vanish; d0 = 1 and the other exponents vanish. It implies that the maximal order of the operator is one and φλµ is of the form ρωτ e κ + C ρωτ x˙ T κ + D ω1 τ1 ω2 τ2 Λ e κ1 Λ e κ2 φλµ = Aρσ ˙ ρ x˙ σ + Bλµκ x˙ ρ Λ ω τ ω1 τ1 ω2 τ2 λµ x λµκ ρ ω τ λµκ1 κ2
ω1 τ1 ω2 τ2 e κ1 ω1 τ1 ω2 τ2 κ ǫ ωτ ǫ e κ + Eλµκ Λω1 τ1 Tω2 κ2 τ2 + Fλµκ Tω1 κ1 τ1 Tω2 κ2 τ2 + Gωτ λµκ Tω τ,ǫ + Hλµκ Rω τ ǫ , 1 κ2 1 κ2
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ωτ ǫ where Aρσ λµ , . . . , Hλµκ are absolute invariant tensors, i.e.,
e ρ + B x˙ Λ e ρ e ρ φλµ = A x˙ λ x˙ µ + B1 x˙ λ Λ ˙ρ Λ ρ µ 2 µ ρ λ + B3 x λ µ
+ C1 x˙ λ Tρ ρ µ + C2 x˙ µ Tρ ρ λ + C3 x˙ ρ Tλ ρ µ
e ρ Λ e σ e ρ e σ e ρ e σ + D1 Λ ρ λ σ µ + D2 Λ σ λ Λ ρ µ + D3 Λ ρ σ Λ λ µ σ σ e ρ T σ +E Λ e ρ e ρ + E1 Λ ρ λ σ µ 2 σ λ Tρ µ + E3 Λρ σ Tλ µ
σ σ e ρ T σ +E Λ e ρ e ρ + E4 Λ ρ µ σ λ 5 σ µ Tρ λ + E6 Λλ µ Tρ σ
+ F1 Tρ ρ λ Tσ σ µ + F2 Tσ ρ λ Tρ σ µ + F3 Tρ ρ σ Tλ σ µ
e ρ +H R e ρ + G1 Tρ ρ λ,µ + G2 Tρ ρ µ,λ + G3 Tλ ρ µ,ρ + H1 R ρ λµ 2 λ ρµ .
The equivariance with respect to (δµλ , aλµν ) implies B1 = B2 = 0, B3 = B + C, Di = 0, E1 = E4 = 0, E2 = G3 , E3 = −G3 , E5 = −G3 , E6 = −(G1 + G2 ) and the other coefficients are arbitrary. Then φλµ = A x˙ λ x˙ µ + (B + C) Λλρ µ x˙ ρ + C1 x˙ λ Tρ ρ µ + C2 x˙ µ Tρ ρ λ + C3 x˙ ρ Tλ ρ µ + F1 Tρ ρ λ Tσ σ µ + F2 Tσ ρ λ Tρ σ µ + F3 Tρ ρ σ Tλ σ µ σ e ρ T σ ) + G (T ρ e ρ + G1 (Tρ ρ λ,µ − Λ λ µ ρ σ 2 ρ µ,λ − Λµ λ Tρ σ )
e ρ T σ −Λ e ρ T σ −Λ e ρ T σ ) + G3 (Tλ ρ µ,ρ + Λ σ λ ρ µ ρ σ λ µ σ µ ρ λ
e ρ +H R e ρ + H1 R ρ λµ 2 λ ρµ ,
φλ µ¯ = B δλµ ,
¯
φλ µ = C δµλ ,
¯
φλ¯µ = 0 ,
which is the equivariant mapping corresponding to (11). Remark 5.4. Let us note that the canonical symplectic form ω of T ∗ M is a special case of (11). Namely, for C = −B 6= 0 and the other coefficients vanish we get just the scalar multiple of ω = dλ ⊗ d˙λ − d˙λ ⊗ dλ . The invariant description of the tensor fields (11) is the following. We have the canonical Liouville 1-form on T ∗ M given in coordinates by θ = x˙ λ dλ . The operator standing by A is then θ ⊗ θ. Λ gives a 3-parameter family of (1,2) tensor fields on M , [5], given by S(Λ) = C1 IT M ⊗ Tb + C2 Tb ⊗ IT M + C3 T ,
(15)
where Tb is the contraction of the torsion tensor and IT M : M → T M ⊗ T ∗ M is the identity tensor. Then the evaluation hS(Λ), ui gives, by the pullback, three operators standing by C1 , C2 , C3 . The connection Λ defines naturally the following 8 parameter family of (0,2)-tensor fields on M , [5], given by 12 12 12 G(Λ) = F1 C13 (T ⊗ T ) + F2 C31 (T ⊗ T ) + F3 C12 (T ⊗ T )
e + H C 1 R[Λ] e , + G1 C11 f ∇T + G2 C11 f ∇T + G3 C31 f ∇T + H1 C11 R[Λ] 2 2
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307
ij where Ckl is the contraction with respect to the indicated indices and C11 f ∇T denotes the conjugated tensor obtained by the exchange of subindices. The second 8-parameter subfamily of operators from (11) is then given by the pullback of G(Λ) to T ∗ M . The last two operators are given by the vertical projection
e ∗] : T ∗M → T ∗T ∗M ⊗ V T ∗M , ν[Λ
e ∗ ] = (d˙ + Λ e ρ x˙ dµ ) ⊗ ∂˙ λ ν[Λ λ λ µ ρ
e ∗ dual to Λ. e Then contractions of ν[Λ e ∗ ]⊗ω and ω ⊗ν[Λ e ∗] associated with the connection Λ give the last two operators. So we have the non-coordinate expression of the natural family (11)
Corollary 5.5. All (0,2)-tensor fields on T ∗ M naturally given by a non-symmetric linear connection Λ are of the form Φ(Λ) =A θ ⊗ θ + hS(Λ), ui + G(Λ) +B
C21 (ω
(16)
e ∗ ]) + C C 1 (ν[Λ e ∗ ] ⊗ ω) . ⊗ ν[Λ 3
Remark 5.6. For a symmetric connection Λ the family (16) reduces to the following 5-parameter family Φ(Λ) = A θ ⊗ θ + H1 C11 (R[Λ]) + H2 C21 (R[Λ]) + B C21 (ω ⊗ ν[Λ∗ ]) + C C31 (ν[Λ∗ ] ⊗ ω) .
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CEJM 3(2) 2005 309–317
On the Bochner conformal curvature of K¨ ahler-Norden manifolds Karina Olszak∗ Institute of Mathematics, Wroclaw University of Technology, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, Poland
Received 7 November 2004; accepted 21 February 2005 Abstract: Using the one-to-one correspondence between K¨ ahler-Norden and holomorphic Riemannian metrics, important relations between various Riemannian invariants of manifolds endowed with such metrics were established in my previous paper [19]. In the presented paper, we prove that there is a strict relation between the holomorphic Weyl and Bochner conformal curvature tensors and similarly their covariant derivatives are strictly related. Especially, we find necessary and sufficient conditions for the holomorphic Weyl conformal curvature tensor of a K¨ ahler-Norden manifold to be holomorphically recurrent. c Central European Science Journals. All rights reserved.
Keywords: K¨ ahler-Norden manifold, holomorphic Riemannian manifold, Bochner conformal curvature, Weyl holomorphic conformal curvature MSC (2000): 53C15, 53C50, 53C56
1
Preliminaries
K¨ahler-Norden manifolds. Let M be a real connected n(= 2m)-dimensional differentiable manifold endowed with an almost complex structure J (J 2 = −I, I being the identity transformation) and a pseudo-Riemannian metric g of Norden type (that is, of neutral signature (m, m)) and such that g(JX, JY ) = −g(X, Y ),
(∇X J)Y = 0 for any X, Y ∈ X(M)
(1)
where ∇ is the Levi-Civita connection of g and X(M) is the Lie algebra of smooth vector fields on M. Then the triple (M, J, g) will be said to be a K¨ahler-Norden manifold (it is ∗
E-mail:
[email protected]
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called a K¨ahlerian manifold with Norden metric in [6], and an anti-K¨ahlerian manifold in [1, 2]). Holomorphic Riemannian manifolds. Let M be a complex manifold of complex dimension m. Denote by (M, J) the manifold considered as a real 2m-dimensional manifold with the induced almost complex structure J. The tangent space to (M, J) at p ∈ M and its complexification are denoted by Tp M and TpC M, respectively. The subspaces of TpC M (1,0) (0,1) consisting of complex vectors of type (1, 0) and (0, 1) are denoted by Tp M and Tp M, respectively. The Lie algebras of real smooth vector fields, complex vector fields, complex vector fields of type (1, 0) and complex vector fields of type (0, 1) on M are denoted by X(M), XC (M), X(1,0) (M) and X(0,1) (M), respectively. In the sequel, for any X ∈ X(M), √ b we denote the complex vector field defined by X b = 1 (X − −1 JX) ∈ X(1,0) (M). by X 2 b for a certain X ∈ X(M) ([11], Vol. II). Any Z ∈ X(1,0) (M) is of this form, that is, Z = X A complex Riemannian metric on M is defined to be a symmetric (0, 2)-tensor field G, which is nondegenerate at each point of M and such that G Z 1 , Z 2 = G(Z1 , Z2 ) for any Z1 , Z2 ∈ XC (M), G(Z1 , Z2) = 0 for any Z1 ∈ X(1,0) (M) and Z2 ∈ X(0,1) (M).
The second condition of the above is equivalent to G(JZ1 , JZ2 ) = −G(Z1 , Z2 ) for any Z1 , Z2 ∈ XC (M). Thus, a complex Riemannian metric is completely determined by its values on X(1,0) (M). If M is a complex manifold and G is a complex Riemannian metric on M, then the pair (M, G) is said to be a complex Riemannian manifold ([8] - [10], [21]). For a local holomorphic coordinates system (z α ; 1 6 α 6 m) of a complex Rieman√ nian manifold, let z α = xα + −1y α , with xα = Re z α and y α = Im z α , and next √ √ suppose ∂/∂z α = (1/2)(∂/∂xα − −1 ∂/∂y α ), ∂/∂z α = (1/2)(∂/∂xα + −1 ∂/∂y α ), and GAB = G(∂/∂z A , ∂/∂z B ), A, B = 1, . . . , m, 1, . . . , m. Therefore, we may then express the defining conditions for a complex Riemannian metric G with respect to this system of local coordinates in the form Gαβ = Gαβ , Gαβ = Gαβ = 0. A complex Riemannian manifold (M, G) is said to be holomorphic Riemannian ([8, 9]; also [3, 12, 13, 21]) if additionally the local components Gαβ are holomorphic functions, b = 0, where ∇ b is the Levi-Civita connection of that is, ∂Gαβ /∂z γ = 0, or equivalently ∇J G ([8]). K¨ahler-Norden vs. holomorphic Riemannian. There exists a one-to-one correspondence between K¨ahler-Norden manifolds and holomorphic Riemannian manifolds ([1, 2]; compare also [21] and [19]). Below, we sketch the description of this correspondence. Let (M, J, g) be a K¨ahler-Norden manifold. Since ∇J = 0, the almost complex structure J is integrable. Therefore, the real manifold M inherits the structure of a complex manifold, which for simplicity will also be denoted by M, and J comes from the complex structure in the usual way. To define a complex Riemannian metric G on the complex manifold M it is sufficient to suppose √ b Yb ) = 1 g(X, Y ) − −1 g(X, JY ) , X, Y ∈ X(M), G(X, (2) 2
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and next extend G to be complex Riemannian. Additionally, (M, G) is holomorphic Riemannian. Conversely, a holomorphic Riemannian manifold (M, G) can be considered as a real 2m-dimensional K¨ahler-Norden manifold (M, J, g). Namely, we define J to be the almost complex structure coming from the complex structure of M and suppose b Yb ) , X, Y ∈ X(M). g(X, Y ) = 2 Re G(X, (3) The relations (2) and (3) state the one-to-one correspondence between K¨ahler-Norden structures (J, g) and holomorphic Riemannian metrics G on M.
2
General formulas [19]
In this section, we recall important formulas concerning the main Riemannian invariants obtained for K¨ahler-Norden manifolds in our previous paper [19]. Let (M, J, g) be a K¨ahler-Norden manifold and let (M, gb) be the corresponding holomorphic Riemannian manifold (in the sense explained in the previous section). Here and in the rest of this paper, we write b g instead of G. h Let X (M) denote the Lie algebra of holomorphic vector fields on M. Agreement. Throughout the rest of this paper, without loss of generality, X, Y, . . . will b Yb , . . . ∈ Xh (M). denote arbitrary real smooth vector fields on M such that X,
Under the above agreement, the considered vector fields on M are always infinitesimal automorphisms of the almost complex structure J. Therefore (cf. e.g. [11], Vol. II), [JX, Y ] = [X, JY ] = J[X, Y ],
[JX, JY ] = −[X, Y ],
\ b Yb ] = [X, [X, Y ].
c, we have One notes that for a holomorphic function f and a vector field W c = ((Re f ) W + (Im f ) JW )b. fW
(4)
b b Yb = ∇ [ ∇ XY . X
(5)
By (e1 , e2 , . . . , e2m ) we denote a frame of a tangent space Tp M, which is adapted to the structure (J, g) in the sense that it consists of real vectors such that g(eα , eβ ) = −g(eα′ , eβ ′ ) = δαβ , g(eα , eβ ′ ) = g(eα′ , eβ ) = 0, Jeα = eα′ , Jeα′ = −eα , where the Greek √ indices take on values 1, . . . , m and α′ = α+m. Then assuming b eα = (1/2)(eα − −1 Jeα ), (1,0) we have a frame (b e1 , . . . , b em ) of the space Tp M for which b g (b eα , b eβ ) = (1/2) δαβ . b be the Levi-Civita connections of the K¨ahler-Norden metric g and the Let ∇ and ∇ b is holomorphic, that holomorphic Riemannian metric b g , respectively. The connection ∇ h h b b Yb ∈ X (M) for any X, b Yb ∈ X (M) [3, 12, 13]. For the Levi-Civita connections ∇ is, ∇ X b we have the following basic relation and ∇, b be the Let R be the Riemann curvature tensor field connected with ∇, and let R b holomorphic Riemann curvature tensor field connected with ∇, R(X, Y ) = [∇X , ∇Y ] − ∇[X,Y ] ,
b X, b Yb ) = [∇ b b, ∇ b b] − ∇ b bb. R( X Y [X,Y ]
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b are related by The Riemann curvature tensors R and R
b X, b Yb )Zb = (R(X, Y )Z)b. R(
(6)
Let S, Sb be the Ricci and the holomorphic Ricci curvature tensor fields, respectively, S(X, Y ) = Tr {Z 7→ R(Z, X)Y },
b X, b Yb ) = Tr {Zb 7→ R( b Z, b X) b Yb }. S(
b the Ricci and the holomorphic Ricci operators, respectively, Denote by Q and Q g(QX, Y ) = S(X, Y ),
For S and Q, we have S(JX, JY ) = −S(X, Y ),
bX, b Yb ) = S( b X, b Yb ). g (Q b
S(JX, Y ) = S(X, JY ),
QJ = JQ.
(7)
b are related by The Ricci curvature tensors S, Sb and the Ricci operators Q, Q √ b X, b Yb ) = 1 S(X, Y ) − −1 S(X, JY ) , bX b = QX. d S( Q (8) 2 Let r and r ∗ be the scalar and ∗-scalar curvatures of g, and let rb be the holomorphic b For them, it holds scalar curvature of b g , r = Tr Q, r ∗ = Tr(JQ), rb = Tr Q. √ 1 r − −1 r ∗ . (9) rb = 2
3
Additional operators
Let A be a symmetric (0, 2)-tensor field on M. For X, Y ∈ X(M), define an operator X ∧A Y acting on X(M) by (X ∧A Y )Z = A(Y, Z)X − A(X, Z)Y. Let A satisfy the additional condition (it should be noted that by (1) and (7) the relation of this type is fulfilled by the metric tensor g and the Ricci curvature tensor S) A(JX, JY ) = −A(X, Y ). b to be the complex (0, 2)-tensor field which is completely determined by its Define A values on X(1,0) (M) and for which √ b X, b Yb ) = 1 A(X, Y ) − −1 A(X, JY ) . A( 2 b Yb ∈ X(1,0) (M), define X b ∧ b Yb to be the operator acting on X(1,0) (M) by For X, A
b ∧ b Yb )Zb = A( b Yb , Z) bX b − A( b X, b Z) b Yb . (X A
It is now a straightforward verification that
b ∧ b Yb )Zb = 1 ((X ∧A Y ) Z − (JX ∧A JY ) Z) b. (X (10) A 2 b is additionally a holomorphic tensor field, then (X b ∧ b Yb )Zb ∈ Xh (M) One notes that if A A b Yb , Zb ∈ Xh (M). for any X,
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The holomorphic Weyl conformal curvature
Properties of the real Weyl conformal curvature tensor C in the class of K¨ahler-Norden manifolds were studied by the author in the papers [17, 18]. In the paper [7], the (real) Bochner curvature tensor field B is defined as a conformal invariant on conformally K¨ahler manifolds with B-metric. Following [7], on a K¨ahlerNorden manifold of real dimension > 6, the Bochner tensor appears as 1 (X ∧g QY + QX ∧g Y − JX ∧g QJY − QJX ∧g JY ) n−4 1 + r(X ∧g Y − JX ∧g JY ) − r ∗ (JX ∧g Y + X ∧g JY ) .(11) (n − 2)(n − 4)
B(X, Y ) = R(X, Y ) −
On the other hand, in the presented paper, we will also treat with the holomorphic b which is defined by the standard Weyl (H-Weyl in short) conformal curvature tensor C, formula (see [3, 8, 12]) b X, b Yb ) = R( b X, b Yb ) − C(
1 b b rb b b b b b b QX ∧bg Y + X ∧bg QY − X ∧bg Y . m−2 m−1
(12)
We will show that the Bochner curvature tensor and the holomorphic Weyl conformal curvature tensor are strictly related. At first, we have the following result. Proposition 4.1. For a K¨ahler-Norden manifold, the conformal curvature tensor fields b are related by B and C b X, b Yb )Zb = (B(X, Y )Z)b. C( (13) Proof. By applying (8), (10) and (7), one gets
1 (QX ∧g Y − QJX ∧g JY )Z b, 2 b ∧bg Q bYb )Zb = 1 (X ∧g QY − JX ∧g QJY )Z b. (X 2
(15)
J(X ∧g Y − JX ∧g JY ) = JX ∧g Y + X ∧g JY,
(16)
bX b ∧bg Yb )Zb = (Q
(14)
In the sequel, we will also need the following formula
which can be checked by a direct computation. b ∧bg Yb ), using (10) with A = g, we find at first To transform rb(X
b ∧bg Yb )Zb = 1 rb((X ∧g Y )Z − (JX ∧g JY )Z)b. rb(X 2
By virtue of (4) with f = rb, W = (X ∧g Y )Z − (JX ∧g JY )Z, and (9), (16), the above expression turns into b ∧bg Yb )Zb = 1 rb(X 4
r(X ∧g Y − JX ∧g JY ) − r ∗ (JX ∧g Y + X ∧g JY ) Z b. (17)
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We now apply the formula (12) into the left hand side of (13), and next we use (6) and the obtained equalities (14), (15), (17). Then after certain calculations, regarding the definition of B (11), we obtain the right hand side of (13). Remark 4.2. In view of (13), for a K¨ahler-Norden manifold (M, J, g), we have the following two statements. b = 0 for a (i) In case of dim M = 6, B = 0. In fact by a pure algebraic reason, C holomorphic Riemannian manifold of complex dimension 3. b = 0. (ii) In case of dim M > 6, B = 0 if and only if C Proposition 4.3. For a K¨ahler-Norden manifold of real dimension > 8,
Proof. At first, we write
b b C)( b X, b Yb )Zb = ((∇U B)(X, Y )Z)b. (∇ U
(18)
b b C)( b X, b Yb )Zb = ∇ b b (C( b X, b Yb )Z) b − C( b ∇ b b X, b Yb )Zb − C( b X, b ∇ b b Yb )Zb − C( b X, b Yb )∇ b b Z. b (∇ U U U U U
Next, we apply the formulas (5), (13) and reduce the right hand side of the above equality to the form ((∇U B)(X, Y )Z)b. Lemma 4.4. For the covariant derivative of the Bochner curvature tensor, it holds (∇JU B)(X, Y ) = J(∇U B)(X, Y ). Proof. For an arbitrary vector field U, we have √ b. c = JU b = −1 U JU
(19)
(20)
Using (18) and the first equality of (20), we get
b b C)( b X, b Yb )Zb = ((∇JU B)(X, Y )Z)b. (∇ JU
On the other hand, using (18) and the second equality of (20), we find b b C)( b X, b Yb )Zb = (J(∇U B)(X, Y )Z) b. (∇ JU
Comparing the last two equalities, we obtain (19), as required.
A K¨ahler-Norden manifold is called (i) of parallel Bochner curvature if ∇B = 0; (ii) of recurrent Bochner curvature if B is non-identically zero and for a 1-form ψ, ∇B = ψ ⊗ B.
(21)
At the moment, it is useful to recall the theorem stating that a tensor field fulfilling a recurrence condition (like for instance B realizing (21)) is either everywhere zero or nowhere zero on the manifold (see [20, Th. 1.2]).
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Theorem 4.5. Any K¨ahler-Norden manifold of recurrent Bochner curvature and of real dimension > 8 is necessarily of parallel Bochner curvature. Proof. By applying (21) into (19), we obtain ψ(JU)B(X, Y )Z − ψ(U)JB(X, Y )Z = 0.
(22)
Under our assumption, B does not vanish at any point of M. Therefore, at every point p ∈ M, B(X, Y )Z and JB(X, Y )Z are linearly independent for certain X, Y, Z ∈ Tp (M). Now, from (22), it follows that ψ(U) = 0 for any U ∈ Tp (M). Consequently, ψ = 0 on M, which by (21) gives the desired assertion. A K¨ahler-Norden manifold will be called (i) of parallel H-Weyl conformal curvature b b = 0; (ii) of holomorphically recurrent (H-recurrent in short) H-Weyl conformal if ∇C b is non-identically zero and for a certain holomorphic 1-form ϕ, curvature if C b bC b=ϕ b ∇ b ⊗ C.
(23)
Remark 4.6. In [19], we have investigated the H-recurrence of the holomorphic Riemann curvature. It is clear that the H-recurrence of the holomorphic Riemann curvature implies the H-recurrence of the H-Weyl conformal curvature. But the converse implication does not hold in general, as it is pointed out in the next section. We extend J to act on real 1-forms ϕ by assuming (Jϕ)(X) = ϕ(JX) for any X ∈ X(M). Then J is compatible with the musical isomorphisms acting between tangent and cotangent bundles (cf. [19]). Theorem 4.7. Let (M, J, g) be a K¨ahler-Norden manifold of dimension > 8. (i) (M, J, g) is of H-recurrent H-Weyl conformal curvature if and only if the Bochner tensor B is non-identically zero and (∇U B)(X, Y ) = ϕ(U)B(X, Y ) − ϕ(JU)JB(X, Y ) (24) √ for a certain real 1-form ϕ such that ϕ b = ϕ − −1 Jϕ is a holomorphic 1-form; (ii) (M, J, g) is of parallel H-Weyl conformal curvature if and only if it is of parallel Bochner curvature. Proof. (i) We write down the defining condition (23) in the following explicit way b b C)( b X, b Yb )Zb = ϕ( b )C( b X, b Yb )Z. b (∇ bU U
(25)
To the left hand side of (25) we can apply the formula (18). We should transform now √ the right hand side of (25). To do it write the holomorphic 1-form ϕ b as ϕ b = ϕ − −1Jϕ with ϕ being a certain real 1-form. Then we see that √ b ) = ϕ(U) − −1 ϕ(JU). ϕ( bU (26)
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b W = B(X, Y )Z enable us to find Now, the formulas (13), (26) and (4) with f = ϕ( b U), b )C( b X, b Yb )Zb = ϕ( b )(B(X, Y )Z)b = (ϕ(U)R(X, Y )Z − ϕ(JU)JR(X, Y )Z)b. ϕ( bU bU
Finally, using the above expression and (23), we claim that the condition (25) is equivalent to (24). To have the proof complete, one should also note that at a point of the manifold b = 0 if and only if B = 0. M, C (ii) This is an obvious consequence of (18).
5
Examples
The main idea of obtaining examples of K¨ahler-Norden manifolds of parallel or H-recurrent H-Weyl conformal curvature is to make the complexification of some real (pseudo-)Riemannian metrics. There are many examples of (pseudo-)Riemannian metrics which have parallel or recurrent Weyl conformal curvature; cf.[4, 5, 14, 15, 16], etc. Below, basing on the paper [15], we present only one of the known classes of such manifolds. Fix m ∈ N, m > 3 and assume that the Greek indices run through the range {2, 3, . . . , m − 1}. Let p, q be non-constant functions of an one complex variable only, which are holomorphic on an open connected subset U1 ⊂ C such that q is non-zero on U1 . Let f be a holomorphic function given on the open connected subset U = U1 × Cm−1 ⊂ Cm by X f (z 1 , . . . , z m ) = p(z 1 ) kαβ + q(z 1 ) cαβ z α z β , where cαβ , kαβ are complex constants such that the (m − 2)-by-(m − 2) matrices [cαβ ], P αβ [kαβ ] are symmetric, Rank [cαβ ] > 1, Rank [kαβ ] = m − 2 and k cαβ = 0, k αβ being the entries of the inverse matrix [kαβ ]−1 . Let b g be the holomorphic Riemannian metric defined on U by X gb = f dz 1 ⊗ dz 1 + kαβ dz α ⊗ dz β + dz 1 ⊗ dz m + dz m ⊗ dz 1 .
Then it is a straightforward verification that the metric b g is of H-recurrent H-Weyl con′ 1 formal curvature with ψb = (q /q)dz as its recurrence form. In the case when q is a non-zero constant, this metric is of parallel H-Weyl conformal curvature. And it can be also verified that the holomorphic Riemann curvature is not holomorphically recurrent.
References [1] A. Borowiec, M. Ferraris, M. Francaviglia and I. Volovich: “Almost-complex and almost-product Einstein manifolds from a variational principle”, J. Math. Physics, Vol. 40(7), (1999), pp. 3446–3464. [2] A. Borowiec, M. Francaviglia and I. Volovich: “Anti-K¨ahlerian manifolds”, Diff. Geom. Appl., Vol. 12, (2000), pp. 281–289. [3] E.J. Flaherty, Jr.: “The nonlinear gravitation in interaction with a photon”, General Relativity and Gravitation, Vol. 9(11), (1978), pp. 961–978.
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[4] A. Derdzi´ nski: “On homogeneous conformally symmetric pseudo-Riemannian manifolds”, Colloq. Math., Vol. 40, (1978), pp. 167–185. [5] A. Derdzi´ nski: “The local structure of essentially conformally symmetric manifolds with constant fundamental function, I. The elliptic case, II. The hyperbolic case, III. The parabolic case”, Colloq. Math., Vol. 42, (1979), pp. 59–81; Vol. 44, (1981), pp. 77–95; Vol. 44, (1981), pp. 249–262. [6] G.T. Ganchev and A.V. Borisov: “Note on the almost complex manifolds with Norden metric”, Compt. Rend. Acad. Bulg. Sci., Vol. 39, (1986), pp. 31–34. [7] G.T. Ganchev, K. Gribachev and V. Mihova: “B-connections and their conformal invariants on conformally Kaehler manifolds with B-metric”, Publ. Inst. Math., Vol. 42(56), (1987), pp. 107–121. [8] G. Ganchev and S. Ivanov: “Connections and curvatures on complex Riemannian manifolds”, Internal Report, No. IC/91/41, International Centre for Theoretical Physics, Trieste, Italy, 1991. [9] G.T. Ganchev and S. Ivanov: “Characteristic curvatures on complex Riemannian manifolds”, Riv. Math. Univ. Parma (5), Vol. 1, (1992), pp. 155–162. [10] S. Ivanov: “Holomorphically projective transformations on complex Riemannian manifold”, J. Geom., Vol. 49, (1994), pp. 106–116. [11] S. Kobayashi and K. Nomizu: Foundations of differential geometry, Vol. I, II, Interscience Publishers, New York, 1963, 1969. [12] C.R. LeBrun: “H-space with a cosmological constant”, Proc. Roy. Soc. London, Ser. A, Vol. 380, (1982), pp. 171–185. [13] C. LeBrun: “Spaces of complex null geodesics in complex-Riemannian geometry”, Trans. Amer. Math. Soc., Vol. 278, (1983), pp. 209–231. [14] Z. Olszak: “On conformally recurrent manifolds, II. Riemann extensions”, Tensor N.S., Vol. 49, (1990), pp. 24–31. [15] W. Roter: “On a class of conformally recurrent manifolds”, Tensor N.S., Vol. 39, (1982), pp. 207–217. [16] W. Roter: “On the existence of certain conformally recurrent metrics”, Colloq. Math., Vol. 51, (1987), pp. 315–327. [17] K. Sluka: “On K¨ahler manifolds with Norden metrics”, An. S¸ tiint. Univ. ”Al.I.Cuza” Ia¸si, Ser. Ia Mat., Vol. 47, (2001), pp. 105–122. [18] K. Sluka: “Properties of the Weyl conformal curvature of K¨ahler-Norden manifolds”, In: Steps in Differential Geometry (Proc. Colloq. Diff. Geom. July 25-30, 2000), Debrecen, 2001, pp. 317–328. [19] K. Sluka: “On the curvature of K¨ahler-Norden manifolds”, J. Geom. Physics, (2004), in print. [20] Y.C. Wong: “Linear connexions with zero torsion and recurrent curvature”, Trans. Amer. Math. Soc., Vol. 102, (1962), pp. 471–506. [21] N. Woodhouse: “The real geometry of complex space-times”, Int. J. Theor. Phys., Vol. 16, (1977), pp. 663–670.
CEJM 3(2) 2005 318–330
On almost cosymplectic (−1, µ, 0)-spaces Piotr Dacko∗, Zbigniew Olszak† Institute of Mathematics, Wroclaw University of Technology, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, Poland
Received 30 October 2004; accepted 9 February 2005 Abstract: In our previous paper, almost cosymplectic (κ, µ, ν)-spaces were defined as the almost cosymplectic manifolds whose structure tensor fields satisfy a certain special curvature condition. Amongst other results, it was proved there that any almost cosymplectic (κ, µ, ν)space can be D-homothetically deformed to an almost cosymplectic (−1, µ′ , 0)-space. In the present paper, a complete local description of almost cosymplectic (−1, µ, 0)-spaces is established: ”models” of such spaces are constructed, and it is noted that a given almost cosymplectic (−1, µ, 0)-space is locally isomorphic to a corresponding model. In the case when µ is constant, the models can be constructed on the whole of R2n+1 and it is shown that they are left invariant with respect to Lie group actions. c Central European Science Journals. All rights reserved.
Keywords: Almost cosymplectic manifold, D-homothetic transformation, almost cosymplectic (κ, µ, ν)-space MSC (2000): 53C25, 53D15
1
Preliminaries
By an almost contact metric manifold M is meant a connected, differentiable manifold of dimension 2n + 1 (n > 1), which is endowed with an almost contact metric structure (ϕ, ξ, η, g) ([1]). Such a structure is described by a quadruple: a (1, 1)-tensor field ϕ, a vector field ξ, a 1-form η and a Riemannian metric g such that ϕ2 = −I + η ⊗ ξ,
η(ξ) = 1,
g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ),
where I indicates the identity tensor field on M. In addition, for an almost contact metric structure, the following holds: ϕξ = 0, η ◦ ϕ = 0, η(X) = g(X, ξ) and ξ is a unit vector ∗ †
E-mail:
[email protected] E-mail:
[email protected]
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319
field. Let Φ be the fundamental 2-form associated to an almost contact metric structure by Φ(X, Y ) = g(ϕX, Y ). In the above and in the sequel, X, Y, . . . denote arbitrary smooth vector fields on M unless otherwise stated. An almost contact metric manifold and its almost contact metric structure are called: (a) almost cosymplectic if the forms η and Φ are closed [13]; (b) cosymplectic if they are almost cosymplectic and the almost contact structure (ϕ, ξ, η) is normal, equivalently, ∇ϕ = 0, ∇ being the Levi-Civita connection determined by g [1]. Let M be an almost cosymplectic manifold. Let F be the codimension 1 foliation of M, which is generated by the integrable distribution D = ker η = im ϕ. The leaves (maximal integral submanifolds) N of F inherit the almost K¨ahlerian structures (J, G) induced from M. If the structure (J, G) is K¨ahlerian on any leaf N of F, we say that M is an almost cosymplectic manifold with K¨ahlerian leaves [16]. For M, define (1, 1)-tensor fields A and h by (cf. [14, 16, 8, 9]) AX = −∇X ξ, 1 h = Lξ ϕ, 2
(1) (2)
where L indicates the Lie differentiation operator. The tensors A and h are related by h = Aϕ,
A = ϕh.
(3)
A restricted to a leaf N of F is its shape operator. The main algebraic properties of A and h are the following g(AX, Y ) = g(AY, X), g(hX, Y ) = g(hY, X),
2
Aϕ + ϕA = 0,
hϕ + ϕh = 0,
Aξ = 0,
hA + Ah = 0,
η ◦ A = 0, hξ = 0,
η ◦ h = 0.
D-homothetic deformations
Let M be an almost cosymplectic manifold and (ϕ, ξ, η, g) its almost cosymplectic structure. Let Rη (M) be the subring of the ring of smooth functions f : M → R for which df ∧ η = 0, or equivalently df = df (ξ)η. A D-homothetic deformation of (ϕ, ξ, η, g) into a new almost cosymplectic structure ′ ′ (ϕ , ξ , η ′, g ′ ) is defined as follows [9] ϕ′ = ϕ,
ξ′ =
1 ξ, β
η ′ = βη,
g ′ = αg + (β 2 − α)η ⊗ η,
(4)
where α is a positive constant, β ∈ Rη (M) and β 6= 0 at any point of M. For two almost cosymplectic structures related by (4), we will say that they are D-homothetic. The above notion is a generalization of a D-homothetic deformation of almost contact metric structures, where α, β are constants; see [1, 15, 17], etc.
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Almost cosymplectic (κ, µ, ν)-spaces
If the curvature operator R, R(X, Y ) = [∇X , ∇Y ] − ∇[X,Y ] , of an almost cosymplectic manifold M satisfies the condition R(X, Y )ξ = η(Y )P X − η(X)P Y,
(5)
where P = κI + µh + νA and κ, µ, ν ∈ Rη (M), then M is called an almost cosymplectic (κ, µ, ν)-space, and in this case (ϕ, ξ, η, g) is called an almost cosymplectic (κ, µ, ν)structure on M. Almost cosymplectic manifolds satisfying (5) with P = κI and constant κ were studied in [7]; and with P = κI + µh and constant κ, µ in [10, 11, 12]. Contact metric manifolds satisfying (5) with P = κI + µh and constant κ, µ were extensively studied by many authors; see [1, 2, 3], etc. In the rest of this section, we summarize the main results concerning almost cosymplectic (κ, µ, ν)-spaces, which were obtained in our previous paper [9]. For D-homothetic almost cosymplectic structures (4), if (ϕ, ξ, η, g) is an almost cosymplectic (κ, µ, ν)-structure, then (ϕ′ , ξ ′, η ′ , g ′) is an almost cosymplectic (κ′ , µ′ , ν ′ )-structure with κ′ , µ′ , ν ′ ∈ Rη ′ (M)(= Rη (M)) being related to κ, µ, ν by κ′ =
κ , β2
µ′ =
µ , β
ν′ =
νβ − dβ(ξ) . β2
The tensor field A and the function κ of an almost cosymplectic (κ, µ, ν)-space satisfy, among others, the following relations A2 Y = −κ(Y − η(Y )ξ),
dκ(ξ) = 2νκ.
Hence it follows that for an almost cosymplectic (κ, µ, ν)-space M, (a) κ 6 0 at every point of M; (b) if κ = 0 at a certain point of M, then κ vanishes identically on M; (c) κ = 0 if and only if A = 0; √ (d) if κ < 0, then the eigenvalues of A are: 0 of multiplicity 1, and ± −κ each of multiplicity n. The tensor fields ϕ, h, A of an almost cosymplectic (κ, µ, ν)-space satisfy the following system of differential equations Lξ ϕ = 2h,
Lξ h = − 2κϕ + νh − µA,
Lξ A = µh + νA.
An almost cosymplectic (0, µ, ν)-space is locally a product of an open interval and an almost K¨ahlerian manifold. Let M be an almost cosymplectic (κ, µ, ν)-space with κ < 0. Then: (i) the leaves of the canonical foliation F of M are locally flat K¨ahlerian manifolds; (ii) the almost cosymplectic structure of M can be D-homothetically transformed to √ an almost cosymplectic (−1, µ′ , 0)-structure with µ′ = µ/ −κ.
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In view of the above, the local structure of an almost cosymplectic (−1, µ, 0)-space is of special interest. It can be characterized in the following way (see Theorem 2 of [9]). Let M be an almost cosymplectic manifold. Given µ ∈ Rη (M), the following two conditions (I) and (II) are equivalent: (I) M is an almost cosymplectic (−1, µ, 0)-space. e with coordi(II) About any point p ∈ M, there is a neighbourhood U = (−ε, ε) × U 0 1 2n 1 2n nates (x = t, x , . . . , x ), t being a coordinate on (−ε, ε) and (x , . . . , x ) coordinates e . On the set U, the function µ depends on t only, and the structure tensor fields ϕ, on U ξ, η, g can be expressed as X X j ∂ ∂ ϕ= ϕi dxi ⊗ j , ξ = , η = dt, g = dt ⊗ dt + gij dxi ⊗ dxj , ∂x ∂t where the Latin indices take on values 1, 2, . . . , 2n only, the sums are over the repeated indices, ϕji , gij are functions only of t and such that the non-zero components of the fundamental form Φ are the following X Φij = ϕki gkj = +1 if j = i + n, −1 if i = j + n. Moreover, on U the tensor fields A and h can be written as X j X j ∂ ∂ A= Ai dxi ⊗ j , h = hi dxi ⊗ j , ∂x ∂x
where Aj , hj are functions of t only. The components ϕij , Aji , hji satisfy the condition P s j i ji Ai As = δi and the following system of differential equations dϕji = 2 hji , dt
4
dhji = 2 ϕji − µ Aji , dt
dAji = µ hji . dt
(6)
Models of almost cosymplectic (−1, µ, 0)-spaces
Let M1 , M2 , M3 denote the following constant 2n-by-2n matrices In Øn Øn In Øn In M1 = , M2 = , M3 = , Øn −In −In Øn In Øn
(7)
where Øn indicates the zero n-by-n matrix and In the unit matrix of rank n, n ∈ N. Let µ : (a, b) → R be a smooth function, where (a, b) is an open interval such that −∞ 6 a < 0 < b 6 ∞. Let Φ, H, A be functional 2n-by-2n matrices whose entries are smooth functions ϕji , hji , aji : (a, b) → R, 1 6 i, j 6 2n, so that for any t ∈ (a, b), Φ(t) = ϕji (t) , H(t) = hji (t) , A(t) = aji (t) . We assume that the matrices Φ, H, A satisfy the following system of linear differential equations with the given initial conditions Φ′ = 2 H,
H′ = 2 Φ − µ A,
A′ = µ H,
Φ(0) = M2 ,
H(0) = −M3 ,
A(0) = M1 .
(8)
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The following lemma is rather obvious. Lemma 4.1. The system of linear differential equations (8) has the unique solution given on (a, b) by Φ(t) = f1 (t) M1 + f2 (t) M2 + f3 (t) M3 , H(t) = h1 (t) M1 + h2 (t) M2 + h3 (t) M3 ,
(9)
A(t) = a1 (t) M1 + a2 (t) M2 + a3 (t) M3 , where the real functions fi , hi , ai are the unique solutions of the systems of linear differential equations fi′ = 2 hi ,
h′i = 2 fi − µ ai ,
a′i = µ hi ,
i = 1, 2, 3,
satisfying the following initial conditions f1 (0) = 0,
h1 (0) = 0,
a1 (0) = 1,
f2 (0) = 1,
h2 (0) = 0,
a2 (0) = 0,
f3 (0) = 0,
h3 (0) = −1,
a3 (0) = 0.
(10)
Lemma 4.2. For the functional matrices Φ, A, H, we also have Φ2 = −I2n ,
A2 = I2n ,
H2 = I2n ,
AΦ = −ΦA,
HΦ = −ΦH,
AH = −HA,
A = −ΦH,
H = ΦA,
Φ = HA.
(11)
Proof. Let X1 , . . . , X9 be the functional matrices defined by X 1 = Φ2 ,
X2 = A2 ,
X3 = H2 ,
X4 = AΦ + ΦA,
X5 = HΦ + ΦH,
X6 = AH + HA,
X7 = ΦH + A,
X8 = ΦA − H,
X9 = HA − Φ.
(12)
Using (8), one verifies that the functional matrices X1 , . . . , X9 satisfy the following system of linear differential equations X′1 = 2X5 ,
X′2 = µX6 ,
X′3 = 2X5 − µX6,
X′4 = µX5 + 2X6 ,
X′5 = 4X1 + 4X3 − µX4 ,
X′6 = − 2µX2 + 2µX3 + 2X4 ,
X′7 = 2X1 + 2X3 − µX8,
X′8 = µX7 + 2X9 ,
X′9 = − µX2 + µX3 + 2X8
with the initial conditions X1 (0) = −I2n ,
X2 (0) = I2n ,
X3 (0) = I2n ,
X4 (0) = X5 (0) = X6 (0) = X7 (0) = X8 (0) = X9 (0) = Ø2n .
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The above system has exactly one solution which is of the form X1 (t) = −I2n ,
X2 (t) = I2n ,
X3 (t) = I2n ,
X4 (t) = X5 (t) = X6 (t) = X7 (t) = X8 (t) = X9 (t) = Ø2n for any t ∈ (a, b). This applied to (12) gives (11), completing the proof.
Let G be the functional 2n-by-2n matrix defined on (a, b) by G = −ΦM2 .
(13)
Lemma 4.3. The functional matrix G(t) is symmetric and positive-definite for any t ∈ (a, b) and G(0) = I2n . Explicitly, G can be written as −f1 In (f2 + f3 )In G= (14) . −f1 In (f2 − f3 )In Proof. By applying (7) and (9) to (13), we obtain G = − f1 M3 + f2 I2n + f3 M1 , which can be rewritten in the form presented in (14). Consequently, G(t) is symmetric for any t ∈ (a, b), and by (10), it follows that G(0) = I2n . Moreover, using (7) and (9), we find Φ2 = (f1 M1 + f2 M2 + f3 M3 )2 = (f12 − f22 + f32 )I2n , which compared to Φ2 = −I2n (cf. (11)), gives (f2 −f3 )(f2 +f3 ) = 1+f12 . Since f2 (0) = 1, f3 (0) = 0, it follows that f2 + f3 > 0 and f2 − f3 > 0 on (a, b). Consequently, in view of (14), the matrix G(t) is positive-definite for any t ∈ (a, b). Suppose that N = (a, b) × R2n ⊂ R2n+1 . Let (t, x1 , . . . , x2n ) be the coordinate system induced from R2n+1 on N. Define tensor fields ϕ, ξ, η, g on N by X j X ∂ ∂ ϕ= ϕi dxi ⊗ j , ξ = , η = dt, g = dt ⊗ dt + gij dxi ⊗ dxj , (15) ∂x ∂t
where ϕji and gij are the entries of the functional matrices Φ and G.
Proposition 4.4. The quadruple (ϕ, ξ, η, g) defined in (15) is an almost cosymplectic (−1, µ, 0)-space on N. Proof. First, we show that (ϕ, ξ, η, g) is an almost contact metric structure on N. For, in view of (15) and (11), we have ϕ2 = −I + η ⊗ ξ and η(ξ) = 1. We must check that the metric g is compatible with the almost contact structure, that is, g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y )
(16)
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for any vector fields X, Y on N. Since ϕξ = 0 and η(X) = g(ξ, X), it remains to show that (16) holds for X = ∂/∂xi and Y = ∂/∂xj . To do this, using (15), (13) and (11), we find that h ∂ i ∂ i h X r g ϕ i,ϕ j = ϕi grs ϕsj = ΦGΦT = −Φ2 M2 ΦT = M2 ΦT . ∂x ∂x In addition, using relations (7) and (11), we compute h ∂ ∂ i M2 ΦT = −ΦM2 = G = g , . ∂xi ∂xj
Comparing the last two formulas, we see that (16) is satisfied. By (13), (11), we have h ∂ ∂ i h X r i g ϕ i, j = ϕi grj = ΦG = M2 . ∂x ∂x
In view of the above and ϕξ = 0, the fundamental form Φ corresponding to our structure is given by Φ=
n X
α
α+n
(dx ⊗ dx
α+n
− dx
α=1
α
⊗ dx ) = 2
n X
dxα ∧ dxα+n .
α=1
Consequently, dΦ = 0 and (ϕ, ξ, η, g) becomes an almost cosymplectic structure on N. By virtue of (15), the tensor field h = (1/2)Lξ ϕ (cf. (2)) can be written as h=
X
hji dxi ⊗
∂ , ∂xj
where hji =
1 dϕji , 2 dt
so that the components hji are just the entries of the matrix H. Moreover, the tensor field A = −∇ξ = ϕh (cf. (1) and (3)), can be written as A=
X
Aji dxi ⊗
∂ , ∂xj
where Aji =
X
hsi ϕjs ,
so that the components Aji = aji are just the entries of the matrix A = HΦ (cf. 11). Now, since the matrices Φ, H, A satisfy (8), the tensor fields ϕ, h, A satisfy the system (6). Consequently, by the Theorem 2 of [9] (cf. the previous section), the structure is an almost cosymplectic (−1, µ, 0)-space. In the sequel, the almost cosymplectic (−1, µ, 0)-spaces defined in the above will be denoted by N(µ), and they will be called models of almost cosymplectic (−1, µ, 0)-spaces. It can be immediately deduced from Theorem 2 of [9] (cf. the previous section) that: Theorem 4.5. Any almost cosymplectic (−1, µ, 0)-space is locally isomorphic to a certain model space N(µ). In the above theorem, by a local isomorphism is meant a local diffemorphism which preserves the almost cosymplectic structures.
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The models N (µ), µ = constant
In this section, we describe the models N(µ) of almost cosymplectic (−1, µ, 0)-spaces, µ being a constant function. Here, we can assume that a = −∞ and b = +∞, so that the models N(µ) are defined on N = R2n+1 . We show that the almost cosymplectic (−1, µ, 0)-structure of such a model is left invariant with respect to a Lie group action on N. 1. Model N(µ) with µ = constant, |µ| < 2. p Let us define an auxiliary constant ω by ω = 1 − µ2 /4. Using Lemma 4.1, we claim that the functional matrices Φ, H and A fulfilling (8) are given by 2µ(1 − cosh(2ωt)) 4 cosh(2ωt) − µ2 sinh(2ωt) M + M − M3 , 1 2 4 − µ2 4 − µ2 ω µ sinh(ωt) sinh(2ωt) M1 + M2 − cosh(2ωt) M3 , H(t) = − 2ω ω 2µ(cosh(2ωt) − 1) µ sinh(2ωt) 4 − µ2 cosh(2ωt) A(t) = M1 + M2 − M3 2 2 4−µ 4−µ 2ω Φ(t) =
(17)
for any t ∈ R. In this case, the functional matrix G defined in (13) is of the form G(t) =
4 cosh(2ωt) − µ2 sinh(2ωt) 2µ(cosh(2ωt) − 1) I2n − M1 + M3 2 4−µ ω 4 − µ2
(18)
for any t ∈ R. Denote by (t, xα , y α = xα+n ) (1 6 α 6 n) the Cartesian coordinates in N and consider ∂ , ∂t sinh(ωt) ∂ µ sinh(ωt) ∂ Xα = cosh(ωt) + − , α ω ∂x 2ω ∂y α µ sinh(ωt) ∂ sinh(ωt) ∂ Yα = + cosh(ωt) − 2ω ∂xα ω ∂y α X0 =
as a basis of vector fields on N. With respect to this basis, the almost cosymplectic structure (ϕ, ξ, η, g) defined in (15) has the simplest shape: by virtue of (17) and (18), it is given by ϕX0 = 0,
ϕXα = Yα ,
ϕYα = −Xα ,
ξ = X0 ,
η = dt,
g(Xi, Xj ) = δij ,
(19)
for any 1 6 α 6 n and 0 6 i, j 6 2n. Thus, the Cartesian space N = R2n+1 endowed with this structure becomes the desired model N(µ) with |µ| < 2. Let G1 (ω) (ω being related to µ as above, thus 0 < ω 6 1) be the Lie group of which the underlying manifold is N and the multiplication is given by (t1 , uα1 , v1α ) ∗ (t2 , uα2 , v2α ) = (t1 + t2 , uα1 + eωt1 uα2 , v1α + e−ωt1 v2α )
(20)
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for any (t1 , uα1 , v1α ), (t2 , uα2 , v2α ) ∈ R2n+1 . One sees that this is just the Lie group of matrices of the form ωt 1 1 Øn U e In u v Ø , U = ... , V = ... , −ωt e I V n n Ø1,n Ø1,n 1 un vn
where Ø1,n is the 1-by-n zero matrix.
Proposition 5.1. Let N(µ) be the model of almost cosymplectic (−1, µ, 0)-space with µ = constant, |µ| < 2. Then N(µ) admits the Lie group structure G1 (ω) with respect to which its almost cosymplectic (−1, µ, 0)-structure is left invariant. Proof. Let us change globally the Cartesian coordinates (t, xα , y α) of N into new coordinates (t, uα , v α ) by assuming xα =
t = t,
1+ω α µ α u − v , 2 4
yα = −
µ α 1+ω α u + v . 4 2
Next choose
e0 = ∂ , X eα = eωt ∂ , Yeα = e−ωt ∂ X ∂t ∂uα ∂v α as a new basis of vector fields on N. These vector fields are left invariant with respect to e0 , X eα , Yeα ) and (X0 , Xα , Yα ) are related by the Lie group action (20). The bases (X e 0 = X0 , X
e0 , X0 = X
e α = 1 + ω Xα − µ Y α , X 2 4 1 e µ X α = Xα + Yeα , ω 2ω(1 + ω)
µ 1+ω Yeα = − Xα + Yα , 4 2 µ eα + 1 Yeα . Yeα = X 2ω(1 + ω) ω
Therefore and by (19), the structure (ϕ, ξ, η, g) can be given in the following way e0 = 0, ϕX
eα = ϕX
e0 , X e0 ) = 1, g(X
µ e 1 1 e µ e e0 , η = dt, Xα + Yeα , ϕYeα = − X Yα , ξ = X α − 2ω ω ω 2ω eα , X eα ) = g(Yeα , Yeα) = 1 + ω , , g(X eα, Yeα ) = − µ(1 + ω) , g(X 2 4
ei , X ej ) = 0 otherwise. Thus, the components of ϕ, ξ, η, g with respect to the basis and g(X e0 , X eα , Yeα are constant. Hence it follows that the structure (ϕ, ξ, η, g) is left invariant X with respect to the Lie group action (20). Remark 5.2. In the paper [4], there was defined a wider class of solvable non-nilpotent Lie groups G(k1 , . . . , kn ) admitting left invariant non-cosymplectic almost cosymplectic structures. The importance of this class stems from the fact that these Lie groups admit discrete subgroups for which the coset spaces are compact and inherit almost cosymplectic structures. It is worth noticing that in the present paper the Lie group appearing above, belongs to this class; indeed, this group is in fact the Lie group G(k1 , . . . , kn ) from [4] with k1 = . . . = kn = ω.
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In a similar manner, we will describe the next two models. 2. Model N(µ) p with µ = constant, |µ| > 2. Let ω = µ2 /4 − 1. In this case, the matrices Φ, H, A, G are given by
µ2 − 4 cos(2ωt) sin(2ωt) 2µ(cos(2ωt) − 1) M + M2 − M3 , 1 2 2 µ −4 µ −4 ω µ sin(2ωt) sin(2ωt) H(t) = − M1 + M2 − cos(2ωt) M3, 2ω ω 2µ(1 − cos(2ωt)) µ sin(2ωt) µ2 cos(2ωt) − 4 A(t) = M + M − M3 , 1 2 µ2 − 4 µ2 − 4 2ω µ2 − 4 cos(2ωt) sin(2ωt) 2µ(1 − cos(2ωt)) G(t) = I2n − M1 + M3 . 2 µ −4 ω µ2 − 4 Φ(t) =
(21)
(22)
Let us consider ∂ , ∂t sin(ωt) ∂ µ sin(ωt) ∂ − , Xα = cos(ωt) + α ω ∂x 2ω ∂y α µ sin(ωt) ∂ sin(ωt) ∂ Yα = + cos(ωt) − 2ω ∂xα ω ∂y α X0 =
as a basis of vector fields on N = R2n+1 . By virtue of (21) and (22), the almost cosymplectic structure (15) is given here by ϕX0 = 0,
ϕXα = Yα ,
ϕYα = −Xα ,
ξ = X0 ,
η = dt,
g(Xi, Xj ) = δij ,
(23)
for any 1 6 α 6 n and 0 6 i, j 6 2n. Thus, N endowed with this structure becomes the desired model N(µ) with |µ| > 2. Let G2 (ω) (ω being related to µ as above, ω > 0) be the Lie group for which the underlying manifold is N = R2n+1 and the multiplication is given by (t1 , uα1 , v1α ) ∗ (t2 , uα2 , v2α) = (t1 +
t2 , uα1
+
uα2
(24) cos(ωt1 ) −
v2α
sin(ωt1 ), v1α
+
uα2
sin(ωt1 ) +
v2α
cos(ωt1 ))
for any (t1 , uα1 , v1α ), (t2 , uα2 , v2α) ∈ R2n+1 . G2 (ω) is isomorphic to the group of complex matrices of the form 1 1 √ √ u v ωt −1 In U + −1 V . e .. , V = ... . , U = Ø1,n 1 un vn Proposition 5.3. Let N(µ) be the model of almost cosymplectic (−1, µ, 0)-space with µ = constant, |µ| > 2. Then N(µ) admits the Lie group structure G2 (ω) with respect to which its almost cosymplectic (−1, µ, 0)-structure is left invariant.
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Proof. Change the Cartesian coordinates (t, xα , y α) of N into (t, uα , v α ) by t = t,
xα = uα +
µ+2 α v , 2ω
y α = uα −
µ+2 α v . 2ω
Next choose e0 = ∂ , X ∂t
eα = cos(ωt) ∂ + sin(ωt) ∂ , X ∂uα ∂v α
∂ ∂ Yeα = − sin(ωt) α + cos(ωt) α ∂u ∂v
as a new basis of vector fields on N. These vector fields are left invariant with respect to e0 , X eα , Yeα ) and (X0 , Xα , Yα ) are related by the Lie group action (24). The bases (X e 0 = X0 , X
e0 , X0 = X
e α = Xα + Y α , X 1 e µ−2 e Xα = Xα + Yα , 2 2ω
µ+2 Yeα = (Xα − Yα ), 2ω 1 e µ−2 e Yα = Xα − Yα . 2 2ω
Therefore and by (23), the structure (ϕ, ξ, η, g) can be described in the following way µ−2 e µ+2 e e0 , η = dt, Yα , ϕYeα = Xα , ξ = X 2ω 2ω e0 , X e0 ) = 1, g(X eα , X eα ) = 2, g(Yeα , Yeα) = 2 µ + 2 , g(X µ−2
e0 = 0, ϕX
eα = − ϕX
ei , X ej ) = 0 otherwise. Thus, the structure (ϕ, ξ, η, g) is left invariant with respect and g(X to the Lie group action (24). 3. Model N(µ) with µ = ±2. Denoting the sign of µ by ε, we can write µ = 2ε. In this case, we have Φ(t) = −2εt2 M1 + (1 + 2t2 ) M2 − 2t M3 ,
(25)
H(t) = − 2εt M1 + 2t M2 − M3 , A(t) = (1 − 2t2 ) M1 + 2εt2 M2 − 2εt M3 , G(t) = (1 + 2t2 ) I2n − 2t M1 + 2εt2 M3 .
(26)
Let us consider X0 =
∂ , ∂t
Xα = (1 + t)
∂ ∂ − εt α , α ∂x ∂y
Yα = εt
∂ ∂ + (1 − t) α α ∂x ∂y
as a basis of vector fields on N = R2n+1 . By virtue of (25), (26), the almost cosymplectic structure (15) is given here by ϕX0 = 0,
ϕXα = Yα ,
ϕYα = −Xα ,
ξ = X0 ,
η = dt,
g(Xi, Xj ) = δij ,
(27)
for any 1 6 α 6 n and 0 6 i, j 6 2n. Thus, N endowed with this structure becomes the model N(µ) with µ = ±2. Let G3 be the Lie group for which the underlying manifold is N = R2n+1 and the multiplication is given by (t1 , uα1 , v1α ) ∗ (t2 , uα2 , v2α ) = (t1 + t2 , uα1 + uα2 , v1α + t1 uα2 + v2α )
(28)
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for any (t1 , uα1 , v1α ), (t2 , uα2 , v2α ) ∈ R2n+1 . G3 is isomorphic to the Lie group of matrices of the form 1 t V 0 1 U , U = [u1 . . . un ], V = [v 1 . . . v n ], Øn,2 In
where Øn,2 is n-by-2 zero matrix.
Proposition 5.4. Let N(µ) be the model of almost cosymplectic (−1, µ, 0)-space with µ = ±2. Then N(µ) admits the Lie group structure G3 with respect to which its almost cosymplectic (−1, µ, 0)-structure is left invariant. Proof. Change the Cartesian coordinates (t, xα , y α) of R2n+1 into (t, uα , v α ) by t = t,
xα = uα + v α ,
y α = −ε v α .
Next choose
eα = ∂ + t ∂ , Yeα = ∂ e0 = ∂ , X X ∂t ∂uα ∂v α ∂v α as a new basis of vector fields on R2n+1 . These vector fields are left invariant with respect e0 , X eα , Yeα ) and (X0 , Xα , Yα ) are related by to the Lie group action (28). The bases (X e 0 = X0 , X e0 , X0 = X
e α = Xα , X eα , Xα = X
Yeα = Xα − εYα , eα − εYeα . Yα = ε X
Therefore and by (27), the structure (ϕ, ξ, η, g) can be described in the following way e0 = 0, ϕX e α = εX eα − εYeα, ϕYeα = 2εX eα − εYeα , ξ = X e0 , η = dt, ϕX e0 , X e0 ) = 1, g(X eα , X eα ) = 1, g(Yeα, Yeα ) = 2, g(X eα , Yeα) = 1 g(X
ei , X ej ) = 0 otherwise. Thus, the structure (ϕ, ξ, η, g) is left invariant with respect and g(X to the Lie group action (28). Remark 5.5. The Lie group G3 belongs to the class of the so-called generalized Heisenberg groups, which admit almost cosymplectic structures. Some of them were studied in [5, 6]. However, our structures are different from those obtained in the cited papers.
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