Generalized Cartan TypeWLie Algebras in Characteristic Zero

195, 170]210 Ž1997. JA977067

JOURNAL OF ALGEBRA ARTICLE NO.

Generalized Cartan Type W Lie Algebras in Characteristic Zero ˇ D Dragomir Z. ] okovic* ´ Department of Pure Mathematics, Uni¨ ersity of Waterloo, Waterloo, Ontario, N2L 3G1 Canada

and Kaiming Zhao† Institute of Systems Science, Academia Sinica, Beijing 100080, People’s Republic of China Communicated by Robert Steinberg Received July 11, 1996

1. INTRODUCTION This paper is a sequel to our previous paper w1x in which we studied the generalized Witt algebras W Ž A, T, w . over a field F of characteristic 0. We have tried to make this paper independent of w1x, and so, in Section 2, we give a short description of generalized Witt algebras and recall some basic facts about them. In this paper we study some simple subalgebras Wd of simple generalized Witt algebras W Ž A, T, w .. These subalgebras are analogues of the classical infinite dimensional simple Lie algebras of Cartan type W. Let us recall the definition of these classical algebras. Let F be a field of characteristic 0, and t 1 , . . . , t n independent and commuting indeterminates over F. Denote by Pn and Q n the polynomial algebra F w t 1 , . . . , t n x, and the Laurent polynomial algebra F w t 1" 1, . . . , t n" 1 x, respectively. By Wn we denote the Witt algebra, i.e., the Lie algebra of all * Supported in part by the NSERC Grant A-5285. † Supported by Academia Sinica of People’s Republic of China. 170 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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formal vector fields

­

n

Ý fi ­ t

is1

Ž 1.1. i

with coefficients f i g Q n . The subalgebra Wnq of Wn consisting of all vector fields Ž1.1. with polynomial coefficients, i.e., f i g Pn , is known as the general Lie algebra of Cartan type, or Cartan type W. ŽThere are also topologized versions of Wn and Wnq where the coefficients f i are formal Laurent and power series in t 1 , . . . , t n , respectively, and F is the real or complex field.. It is well known that Wn and Wnq are simple Lie algebras. Apart from Wnq, there are three more series of infinite dimensional simple Lie algebras of Cartan type, namely types S, H, and K. These three series will not be used in this paper and we omit their definitions Žsee, e.g., w3, 10x.. Let k be an integer such that 0 F k F n. Then the subspace Wn, k of Wn with basis consisting of all monomials t 1m 1 ??? t nm n

­ ­ ti

,

i s 1, . . . , n

such that m1 , . . . , m k G 0, while m kq1 , . . . , m n are arbitrary integers, is a subalgebra of Wn . Clearly we have Wn, 0 s Wn and Wn, n s Wnq . The Lie algebras Wn, k are examples of simple infinite dimensional Lie algebras belonging to the class W *, recently introduced by J. M. Osborn in his paper w7x. ŽHe has also defined classes S*, H *, and K * corresponding to Lie algebras of Cartan types S, H, and K, respectively.. From our point of view, Osborn’s definition of the class W * is unduly restrictive, and we shall introduce and study a more general class of simple infinite dimensional Lie algebras. For the convenience of the reader, we now give the definition of the class W * Žsee also w7, 9x.. Let D be an abelian group with a fixed direct decomposition D s D 1 = ??? = D n such that each D i is a subgroup of the additive group of F, and furthermore D i s Z for i s 1, . . . , k and Z ; D i ; F for i s k q 1, . . . , n. The group algebra FD has a basis consisting of the monomials t a [ t 1a 1 t 2a 2 ??? t na n ,

a s Ž a1 , . . . , an . ,

where a i g D i and t 1 , t 2 , . . . , t n are independent commuting indeterminates. Define ‘‘partial differentiation’’ operators ­r­ t i by

­ ­ ti

Ž t a . s a i t a ty1 i

i s 1, . . . , n.

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Then the Lie algebra LŽ D, k . consisting of all formal vector fields which can be written as linear combinations of ta

­ ­ ti

a s Ž a1 , . . . , an . g D ,

,

with a 1 , . . . , a k G 0, is a Lie algebra of type W *. The Lie bracket in this algebra is, of course, defined by ta

­ ­ ti

, tb

­ ­ tj

s ta

­tb ­ ­ ti ­ t j

ytb

­ta ­ ­ t j ­ ti

.

Osborn has shown w7x that each algebra in the class W * is simple. For k s 0 we obtain the Lie algebra LŽ D, 0. which is a generalized Witt algebra Žsee Section 2 for the definition.. These simple generalized Witt algebras LŽ D, 0. are not the most general ones for two reasons. First, the maximal torus of LŽ D, 0. is finite dimensional, and second, the abelian group D is of a very special kind. Our generalization of the simple Lie algebras of Cartan type W is given in Section 3 and will not be repeated here. Our definition avoids both restrictions mentioned above. Thus we start with an arbitrary simple generalized Witt algebra W Ž A, T, w . built from an arbitrary Žtorsion-free. nonzero abelian group A, a maximal torus T, which is just a vector space over F, and a nondegenerate pairing w : T = A ª F. In order to define a subalgebra of W Ž A, T, w . of Cartan type W, we make use of a map d: I ª T, dŽ i . s d i , where I is some index set. We require that d satisfies two conditions: the elements d i are linearly independent and that  d i Ž x . [ w Ž d i , x .: x g A4 s Z for all i g I. Such maps d are called admissible. Each admissible map d: I ª T defines a subalgebra Wd of, W Ž A, T, w .. In general the subalgebras Wd are not simple. In fact we show by an example that it may happen that Wd s T. We shall now describe the contents of the paper and highlight our main results. In Section 2 we recall the definition of generalized Witt algebras, and the Kawamoto’s simplicity theorem. In Section 3 we introduce the subalgebras Wd ; W Ž A, T, w . and determine the necessary and sufficient conditions for Wd to be simple. The proof of this simplicity theorem is much harder than the proof of the corresponding result for the algebras belonging to the Osborn’s class W * Žsee w7x.. Starting with Section 4, we assume in the rest of the paper that the Lie algebra Wd is simple. In that case we refer to Wd as an algebra of generalized Cartan type W. In Section 4 we study the derivation algebra

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DerŽWd . of Wd . We first show that this algebra is the sum of the subalgebra Der9ŽWd . of locally inner derivations and the subspace DerŽWd . 0 of derivations of degree 0. If the maximal torus T has finite dimension, then every locally inner derivation of Wd is inner. The main result of this section is the determination of all derivations of Wd of degree 0 ŽTheorem 4.3.. Each admissible map d: I ª T defines not only a subalgebra Wd of the generalized Witt algebra W Ž A, T, w . but also a subalgebra FAq d of the group algebra FA Žsee Section 3.. Furthermore Wd is a free left FAq d -module and FAq d is a left Wd-module. In Section 5, following an idea of Osborn w9x, we show that there is an injective homomorphism C : Aut Ž Wd . ª Aut Ž FAq d . such that u g AutŽWd . and s s C Ž u . satisfy the identities

u Ž fw . s s Ž f . u Ž w . , s Ž w ? f . s u Ž w. ? s Ž f . for all f g FAq d and w g Wd . Each w g Wd acts on FAq d as a derivation, and so we obtain an injective . homomorphism Wd ª DerŽ FAq d . In the case where this homomorphism is an isomorphism, one can show that C is also an isomorphism. In particular Žthis follows also from Osborn’s paper w9x., we have an isomorphism Cn : Aut Ž Wnq . ª Aut Ž Pn . . For n s 2, the structure of the group AutŽ P2 . is well known Žsee w6x and the references mentioned there.. Hence one can use C2 to describe the . structure of AutŽWq 2 . In Section 6 we assume that the index set I is finite, say I s  1, . . . , n4 . Let Td be the subtorus of T spanned by d1 , . . . , d n . The generalized Witt algebra Ww d x s FA m Td is a subalgebra of W s FA m T. We also set Wwqd x s Ww d x l Wd . By mimicking the classical definition, we define the divergence Div: Ww d x ª FAq d . Its kernel, Sw d x , is a subalgebra of Ww d x and we set Swqd x s Sw d x l Wd . We show that both Wwqd x and Swqd x are characteristic subalgebras of Wd . The problem of characterizing all w g Wd that are locally nilpotent on Wd seems to be very hard. We are able to show Žsee Proposition 6.7. that DivŽ w . s 0 for every w g Wwqd x which is locally nilpotent on Wwqd x. We also show that every torus in Wwqd x has dimension at most n.

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In Section 7 we show that H 2 ŽWd , F . s 0 by assuming only that I is not empty. In Section 8 we introduce the principal Z-gradation of Wd and describe the subalgebra ŽWd . 0 of degree 0. We are indebted to Arno van den Essen for supplying the proof of Proposition 6.7 in the classical case Wd s Wnq . We are also grateful to J. M. Osborn for sending us preprints of his papers. Indeed his papers provided the necessary motivation for our work in this area.

2. GENERALIZED WITT ALGEBRA W In this section, for the convenience of the reader, we recall the definition of generalized Witt algebras and some basic facts concerning them. For more details we refer the reader to our paper w1x. Let A be an abelian group, F a field, and T a vector space over F. We denote by FA the group algebra of A over F. The elements t x , x g A, form a basis of this algebra, and the multiplication is defined by t x ? t y s t xqy. We shall write 1 instead of t 0 . The tensor product W s FA mF T is a free left FA-module. We denote an arbitrary element of T by ­ Žto remind us of differential operators.. For the sake of simplicity, we shall write t x­ instead of t x m ­ . We now choose a pairing w : T = A ª F which is F-linear in the first variable and additive in the second one. For convenience we shall also use the notations

w Ž ­ , x . s ²­ , x: s ­ Ž x . for arbitrary ­ g T and x g A. There is a unique F-bilinear map W = W ª W sending Ž t x­ 1 , t y­ 2 . to t x­ 1 , t y­ 2 [ t xqy Ž ­ 1 Ž y . ­ 2 y ­ 2 Ž x . ­ 1 . ,

Ž 2.1.

for arbitrary x, y g A and ­ 1 , ­ 2 g T. It is easy to verify that this map makes W into a Lie algebra. We refer to this algebra W s W Ž A, T, w . as a generalized Witt algebra. The subspaces Wx s t x T, x g A, define an A-gradation of W, i.e., W is the direct sum of the Wx ’s, and w Wx , Wy x ; Wxqy for all x, y g A. It follows from Ž2.1. that adŽ ­ . acts on Wx as a scalar ­ Ž x .. Hence each ­ g T is ad-semisimple, and T is a torus Ži.e., an abelian subalgebra consisting of ad-semisimple elements.. The following theorem is due to Kawamoto w5x. THEOREM 2.1. Suppose that the characteristic of F is 0. The Lie algebra W s W Ž A, T, w . is simple if and only if A / 0 and w is nondegenerate in the

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sense that the conditions ² ­ , x : s 0, ; ­ g T « x s 0

Ž 2.2.

² ­ , x : s 0, ; x g A « ­ s 0

Ž 2.3.

and hold. As mentioned earlier, W is a free left FA-module. There is also a natural structure of a left W-module on FA, namely such that t x­ ? t y s ­ Ž y . t xqy

Ž 2.4.

for x, y g A and ­ g T. These two module structures are related by the identity

w fu, g¨ x s f Ž u ? g . ¨ y g Ž ¨ ? f . u q fg w u, ¨ x ,

Ž 2.5.

where f, g g FA and u, ¨ g W are arbitrary. The W-module structure on FA gives rise to a homomorphism W ª Der Ž FA .

Ž 2.6.

because each w g W acts on FA as a derivation. Clearly Ž2.6. is also a homomorphism of FA-modules. One can show Žsee w1x. that, if the characteristic of F is 0 and 1 F rank A - `, then DerŽ FA. is a simple Lie algebra and, for suitably chosen T and w , the homomorphism Ž2.6. is an isomorphism. From now on we shall assume that the characteristic of F is 0 and that W s W Ž A, T, w . is a simple Lie algebra. In particular Ž2.2. and Ž2.3. hold, and consequently A is a nonzero torsion-free Žabelian . group. We conclude this section by giving a method of constructing new Lie algebras which generalizes the construction of generalized Witt algebras. We start with a commutative and associative F-algebra B with identity and a Lie algebra g over F. We also assume that a Lie homomorphism r : g ª DerŽ B . is given. For x g g and a g B we write xa instead of r Ž x .Ž a.. A tedious but straightforward computation shows that B m g is a Lie algebra if we define the bracket by

w a m x, b m y x s aŽ xb . m y y b Ž ya . m x q ab m w x, y x . Note that W s W Ž A, T, w . has this form where g s T is an abelian Lie algebra, B s FA is the group algebra of A, and r has the obvious meaning. Every derivation of FA extends uniquely to a derivation of the field of fractions, F Ž A., of FA. Thus we obtain a Lie homomorphism T ª Der F Ž A., and the Lie algebra F Ž A. m T.

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In connection with this construction we propose the following conjecture. Conjecture 1. If W s FA m T is a simple generalized Witt algebra, we conjecture that the Lie algebra F Ž A. m T is also simple.

3. LIE ALGEBRA Wd Recall that W s W Ž A, T, w . denotes a simple generalized Witt algebra over a field F of characteristic 0. Let I be an index set, d: I ª T an injective map, and write d i s dŽ i . for i g I. We say that d is admissible if the following two conditions hold: ŽInd. d i , i g I, are linearly independent; ŽInt. d i Ž A. s Z for all i g I. We assume throughout that an admissible d has been fixed. We set Aq d s  x g A: d i Ž x . G 0, ; i g I 4 , A0d s  x g A: d i Ž x . s 0, ; i g I 4 , A d , i s  x g A: d i Ž x . s y1; d j Ž x . G 0, ; j g I _  i 4 4 , Aad , i s  x g A: d i Ž x . s y1; d j Ž x . s 0, ; j g I _  i 4 4 , A d s Aq d jŽ D i g I Ad , i . . We now introduce some subspaces of W: Wdq s Wd , i s

ž

Ý xgA d, i

Ý

xgA q d

Wx ;

Ft x ? d i ,

/

i g I;

and Wd s Wdq q

Ý Wd , i . igI

In fact all of these subspaces are subalgebras of W. This is obvious for Ž . Ž . Wdq since Aq d is a submonoid of A. If x, y g A d, i , then d i x s d i y s y1 and consequently w t x d i , t y d i x s 0. Hence each Wd, i , i g I, is an abelian subalgebra of W.

GENERALIZED CARTAN ALGEBRAS

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LEMMA 3.1. Wd is a subalgebra of W. q Proof. Let x g Aq d and y g A d, i . Then either x q y g A d or x q y g A d, i and d i Ž x . s 0. In both cases we have

t x­ , t y d i s t xqy Ž ­ Ž y . d i y d i Ž x . ­ . g Wd for all ­ g T. Now let x g A d, i and y g A d, j . We claim that t x d i , t y d j s t xqy Ž d i Ž y . d j y d j Ž x . d i . g Wd .

Ž 3.1.

This is obvious if i s j. So let i / j. Then either d i Ž y . s 0 or x q y g Ž . xqy d j g Wd . Similarly d j Ž x . t xqy d i g Wd , Aq d j A d, j . In both cases d i y t and so Ž3.1. holds. As Wdq is a subalgebra of W, the proof is completed. We also introduce the subalgebra FAq d of FA, which is the span of all elements t x with x g Aq d . Since W is a left FA-module, we can view W q also as a left FAq d -module. Then it is easy to see that the subspaces Wd q and Wd are FA d -submodules of W. q Let x g A d, i and y g Aq d . Then either x q y g A d or x q y g A d, i and d i Ž y . s 0. In both cases we have t x d i ? t y s d i Ž y . t xqy g FAq d . Hence, by restricting the action of W on FA, we can view FA as a left Wd-module, and then FAq d is a Wd-submodule of FA. When d is fixed, and there is no danger of confusion, we shall write Aq, A i , Aai , Wq, Wi , FAq instead of a q q Aq d , A d , i , A d , i , Wd , Wd , i , FA d ,

respectively. EXAMPLE 1. Let A s Z n and let T be the vector space over F with basis  d1 , . . . , d n4 . We define the pairing w : T = A ª F by setting

w Ž d i , e j . s di , j ,

1 F i , j F n,

where  e 1 , . . . , e n4 is the standard basis of Z n. Since w is nondegenerate, the generalized Witt algebra Wn s W ŽZ n, T, w . is simple. In this case we can identify the group algebra FA with the algebra of Laurent polynomials F w t 1" 1 , . . . , t n" 1 x, and Wn with

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DerŽ FA.. Under this identification the element n

Ý Pi d i g Wn is1

is identified with the vector field n

­

Ý t i Pi ­ t

is1

. i

The map d: I ª FA where I s  1, . . . , n4 and dŽ i . s d i s t i Ž ­r­ t i . is admissible and the subalgebra ŽWn .d usually denoted by Wnq , consists of all vector fields n ­ Ý Qi ­ t i is1 with Q i g F w t 1 , . . . , t n x, i g I, i.e., all polynomial vector fields. When F s R or C, this is the classical example of a simple infinite dimensional Lie algebra of Cartan type W, see, e.g., w3x where it is denoted by FWnp o l. In the general case we have T ; Wq; Wd . The subalgebra Wd does not have to be simple. The next example shows that we may have Wd s T. EXAMPLE 2. Let A be the direct sum of countably many copies of Z indexed by integers i G 0. An element x g A will be written as x s Ž x 0 , x 1 , . . . . with x i g Z, almost all 0. We take T to be the vector space with basis ­ i , i G 0. Define ­ Ž x . by requiring that it is linear in ­ g T and by setting

­ 0 Ž x . s y2 x 0 y x 1 y x 2 y ??? , ­ 1Ž x . s x 0 , ­ i Ž x . s 2 x iy2 y x iy1 ,

i G 2.

The generalized Witt algebra W s FA m T is simple. Let I be the set of nonnegative integers and define d: I ª T by dŽ i . s ­ i . Then d is admissible. It is easy to verify that Aqs  04 and A i s B for all i G 0. Hence Wd s T. THEOREM 3.2. The Lie algebra Wd is simple if and only if the following conditions hold: Ži. if ­ g T and ­ Ž x . s 0 for all x g A d , then ­ s 0; Žii. if x g A d , then d i Ž x . s 0 for almost all i g I; Žiii. Aai / B for all i g I.

GENERALIZED CARTAN ALGEBRAS

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Proof. We show first that the conditions are necessary. Hence we suppose that Wd is simple. If ­ g T and ­ Ž x . s 0 for all x g A d , then w ­ , Wd x s 0. As Wd is simple we must have ­ s 0. So Ži. holds. Let V1 be the subspace of Wd spanned by all vectors t x­ g Wd with ­ / 0 and d i Ž x . ) 0 for infinitely many i’s. As T l V1 s 0, we have V1 / Wd . Let t x­ 1 g V1 , ­ 1 / 0, and t y­ 2 g Wd , ­ 2 / 0. Since x, y g A d , d i Ž x q y . ) 0 for infinitely many i’s. It follows that w t x­ 1 , t y­ 2 x g V1 , and so V1 is a proper ideal of Wd . Hence V1 s 0, i.e., Žii. holds. Let I1 s  i g I: Aai s B4 . Since Žii. holds, we can define Nx s

Ý di Ž x . ,

x g Ad .

igI1

As Aai s B for i g I1 , it follows that Nx G 0 for all x g A d . Let V2 be the subspace of Wd spanned by all t x­ g Wd with ­ / 0, x g A d , and Nx ) 0. Let t x­ 1 g V2 , ­ 1 / 0, and t y­ 2 g Wd , ­ 2 / 0. As Nx ) 0 and Ny G 0, we have Nxqy ) 0. Consequently w t x­ 1 , t y­ 2 x g V2 , and so V2 is an ideal of Wd . This ideal is proper because V2 l T s 0. Since Wd is simple, we conclude that V2 s 0, i.e., we have shown that Nx s 0 for all x g Ad. Assume that the vectors d i , i g I1 , span T. Then I s I1. Let T0 ; T be the kernel of the linear function f : T ª F defined by f Ž d i . s 1, i g I. Let x g A d be such that yx g A d and x / 0. Let t x­ 1 , tyx­ 2 g Wd with ­ 1 , ­ 2 / 0. Since the d i span T, we have d i Ž x . / 0 for some i. As x, yx g A d , there exist r, s g I such that d r Ž x . s y1, d s Ž x . s 1, and d i Ž x . s 0 for i g I _  r, s4 . It follows that ­ 1 g Fd r , ­ 2 g Fd s and that tyx­ 2 , t x­ 1 s ­ 2 Ž x . ­ 1 q ­ 1 Ž x . ­ 2 g T0 . Consequently the subspace T0 q

Ý Wi q igI

Ý

xgA q d _ 0 4

Wx

is an ideal of Wd of codimension 1. As Wd is simple, we have a contradiction. Hence we have shown that the vectors d i , i g I1 , span a proper subspace of T. The subspace V3 of Wd spanned by all vectors t x d i g Wd , with i g I1 , is proper. Let t x d i g V3 and t y­ g Wd , ­ / 0. Then x g Aqj A i , i g I1 , and y g A d . If y g A j for some j g I, then we have ­ g Fd j , and consequently w t x d i , t y­ x g V3 . Otherwise y g Aq and since Ny s 0, it follows that d i Ž y . s 0, and so w t x d i , t y­ x s y­ Ž x . t xqy d i g V3 . This shows that V3 is an ideal of Wd . As V3 / Wd , we must have V3 s 0. Consequently I1 s B and so Žiii. holds.

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We shall now prove that conditions Ži., Žii., and Žiii. imply that Wd is simple. By using Kawamoto’s theorem, we may assume that I / B. Let V be a nonzero ideal of Wd . Since T ; Wd , V is a homogeneous ideal and so it contains nonzero elements of the form t x­ . Our first claim is that V l T / 0. We choose an element t x­ g V, ­ / 0. Hence x g A d . If x g A i for some i g I, then t x d i g V, tyx d i g Wd , and so t x d i , tyx d i s 2 d i g V l T . Now assume that x f A i for all i g I, i.e., x g Aq. Among all t x­ g V with ­ / 0 and x g Aq, we may assume that we have chosen one, say t x­ , such that Mx s

Ý di Ž x . igI

is minimal. Observe that d i Ž x . G 0 for all x because x g Aq, and d i Ž x . s 0 for almost all i g I by condition Žii.. Assume that d j Ž x . ) 0 for some j g I. By Žiii., we can choose y g Aaj . Then t y d j g Wd and t y d j , t x­ s t xqy Ž d j Ž x . ­ y ­ Ž y . d j . g V . As M xqy s M x q M y s M x y 1. The minimality of M x implies that d j Ž x . ­ s ­ Ž y . d j . Since d j Ž x . ) 0, d j Ž y . s y1, and d j Ž x . ­ Ž y . s ­ Ž y . d j Ž y ., it follows that ­ Ž y . s 0. Hence ­ s 0, and we have a contradiction. We can now conclude that d i Ž x . s 0 for all i g I. If x s 0, then t x­ s ­ g T l V, and our claim holds. If x / 0 we choose ­ 9 g T such that ­ 9Ž x . / 0. Then tyx­ 9 g Wd and w tyx­ 9, t x­ x s ­ 9Ž x . ­ q ­ Ž x . ­ 9 is a nonzero element in T l V. Hence our first claim is proved. Our second claim is that V contains some d i . By using the first claim, we choose a nonzero ­ g T l V. Assume that there exists an i g I and an x g A i such that ­ Ž x . / 0. Since t x d i g Wd and w ­ , t x d i x s ­ Ž x . t x d i , we conclude that t x d i g V. Since tyx d i g Wd , it follows that w t x d i , tyx d i x s 2 d i g V, and our claim holds. It remains to consider the case where ­ Ž x . s 0 for all x in the union of the A i ’s. Then Ži. implies that there exists an x g Aq such that ­ Ž x . / 0. Among all such x’s, choose one for which M x is minimal. Since ­ acts on Wx as the nonzero scalar ­ Ž x ., it follows that Wx ; V. Consider first the case where x g yAai for some i g I. Then tyx d i g Wd and t x d i , tyx d i s 2 d i g V .

GENERALIZED CARTAN ALGEBRAS

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Hence our claim holds. Now assume that x f yAai for all i g I. If d j Ž x . ) 0 for some j g I, then we obtain a contradiction by using the same argument as in the proof of the first claim. Hence we have d i Ž x . s 0 for all i g I. As ­ Ž x . / 0, this implies that W " x ; V. It follows that tyx­ , t x d i s ­ Ž x . d i q d i Ž x . ­ g V for all i g I. Since d i Ž x . s 0 and ­ Ž x . / 0, we have d i g V for all i g I. Hence our second claim is proved. Our last claim is that V s Wd . We fix j g I such that d j g V and choose y g Aaj . Let t x­ g Wd , ­ / 0. In order to prove our claim, we have to show that t x­ g V. From w d j , t x­ x s d j Ž x . t x­ g V, we infer that t x­ g V if d j Ž x . / 0. In particular we have t y d j g V. Now assume that d j Ž x . s 0. If t x d j g Wd , then also t xyy d j g Wd and t y d j , t xyy d j s 2 t x d j g V . As d i Ž x y y . G d i Ž x . for all i g I, we have t xyy­ g Wd . Therefore t xyy­ , t y d j s t x Ž ­ Ž y . d j q ­ . g V . If ­ Ž y . / 0, then t x d j g Wd and we have shown above that this implies that t x d j g V. Consequently the above equation implies that t x­ g V. This completes the proof of the claim and also the proof of the theorem. COROLLARY 3.3. all i g I.

If I is finite, then Wd is simple if and only if Aai / 0 for

Proof. Note that condition Žii. of Theorem 3.2 is automatically satisfied. Hence it suffices to show that Žiii. « Ži. when I is finite. For each i g I choose x i g Aai . For arbitrary x g A, the element ysxq

Ý di Ž x . x i igI

satisfies d i Ž y . s 0 for all i g I. Thus A d generates A as a group. Since the left kernel of w is 0, it follows that the condition Ži. is satisfied. One can show that conditions Ži., Žii., and Žiii. are independent of each other. The following example shows that Žii. and Žiii. do not imply Ži.. The other two independence assertions are easy. EXAMPLE 3. Let A be the direct sum of countably many copies of Z indexed by integers i G 0. We write x g A as x s Ž x 0 , x 1 , . . . . where x i g Z are almost all 0. Let T be the vector space over F with basis ­ i ,

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i G 0. Let w : T = A ª F be the pairing defined by

w Ž ­ 0 , x . s ­ 0 Ž x . s x0 , and i

w Ž ­ i , x . s ­ i Ž x . s x i q Ž y1 . x 0 ,

i G 1.

It is easy to verify that w is nondegenerate, and so W s W Ž A, T, w . is simple. Let I s  1, 2, . . . 4 and let d: I ª T be defined by d Ž i . s di s ­i . Clearly d is admissible, and conditions Žii. and Žiii. are satisfied. On the other hand, condition Ži. does not hold because x g A d implies that x 0 s 0. Assume that W and Wd are simple Lie algebras. It is natural to ask whether condition Žii. remains valid if we replace A d with A in its formulation. The following example shows that this is not so. EXAMPLE 4. Use the same algebra W as in Example 3, except that ­ 0 is now defined by

­0Ž x. s

Ý xi . iG0

Then Wi is simple but for e 0 [ Ž1, 0, 0, . . . . we have d i Ž e 0 . / 0 for all i g I. From now on Žthroughout the paper. we shall assume that Wd is simple, and in that case we refer to Wd as an algebra of generalized Cartan type W. 4. DERIVATIONS OF Wd If I s B, then Wd s W. In that case the derivations of Wd were described in our previous paper w1x. Hence we assume throughout this section that I / B. We say that a derivation D g DerŽWd . is homogeneous of degree y g A if D Ž Wx l Wd . ; Wxqy l Wd holds for all x g A. We denote by DerŽWd . x the subspace of DerŽWd . consisting of all derivations of degree x. In particular DerŽWd . 0 is the space of derivations of degree 0.

GENERALIZED CARTAN ALGEBRAS

183

We say that D g DerŽWd . is inner if D s adŽ w . for some w g Wd , i.e., DŽ ¨ . s w w, ¨ x for all ¨ g Wd . We say that D g DerŽWd . is locally inner if, for every finite set  w 1 , . . . , wn4 ; Wd , there exists w g Wd such that DŽ wi . s w w, wi x for i s 1, . . . , n. The locally inner derivations of Wd form a subalgebra of DerŽWd ., which will be denoted by Der9ŽWd .. The following proposition can be proved exactly in the same manner as Proposition 3.3 in w1x. PROPOSITION 4.1.

Assume that Wd is simple. Then Der Ž Wd . s Der9 Ž Wd . q Der Ž Wd . 0

holds, and, if dim T - `, we ha¨ e Der9ŽWd . s adŽWd .. LEMMA 4.2.

If i g I, then the subspace

Ý

Wd l Wx

xgA d y1Fd i Ž x .F2

generates Wd Ž as a Lie algebra.. Proof. Denote by L i the subalgebra of Wd generated by the above subspace. We use induction on k G 2 to prove that Wd l Wx ; L i if x g A d and d i Ž x . s k. By definition of L i , this is true if y1 F k F 2. Assume now that this assertion holds for some k G 2. Let x g A d be such that d i Ž x . s k q 1. Choose y g Aai . Then x q y g A d and d i Ž x q y . s k. Hence, by induction hypothesis, we have Wd l Wxqy ; L i . We now distinguish two cases. q Case 1: x g Aq d . Then x q y g A d and Wxqy ; Wd . It follows that Wxqy ; L i . Consequently

tyy d i , t xqy­ s t x Ž k ­ q ­ Ž y . d i . g L i

Ž 4.1.

for all ­ g T. For ­ s d i we obtain that t x d i g L i . Now Ž4.1. implies that t x­ g L i for all ­ g T, i.e., Wx s Wd l Wx ; L i . Case 2:

x g A j for some j g I _  i4 . Then tyy d j , t xqy d j g L i , and so t xqy d j , tyy d j s t x d j g L i .

Hence Wd l Wx s Ft x d j ; L i . THEOREM 4.3 ŽWd simple.. If D g DerŽWd . 0 , then there exists m g HomŽ A, F . such that DŽ t x­ . s m Ž x . t x­ for all t x­ g Wd l Wx , x g A d .

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AND ZHAO

Proof. If x g Aq d , then there is a unique linear map Dx : T ª T such that DŽ t x­ . s t x Dx Ž ­ . holds for all ­ g T. If x g A j for some j g I, then there is a unique linear map Dx : Fd j ª Fd j such that DŽ t x d j . s t x Dx Ž d j .. Clearly we have Dx Ž d j . s l x d j ,

x g Aj ,

Ž 4.2.

for some l x g F. If t x­ 1 , t y­ 2 g Wd , then by applying D to t x­ 1 , t y­ 2 s t xqy Ž ­ 1 Ž y . ­ 2 y ­ 2 Ž x . ­ 1 . , we obtain that ² Dx ­ 1 , y :­ 2 y ­ 2 Ž x . Dx ­ 1 q ­ 1 Ž y . Dy ­ 2 y ² Dy ­ 2 , x :­ 1 s ­ 1 Ž y . Dxqy ­ 2 y ­ 2 Ž x . Dxqy ­ 1 .

Ž 4.3.

By setting y s 0, we conclude that ² D 0 ­ 2 , x :­ 1 s 0. Since this holds for all x g Aq d and ­ 1 , ­ 2 g T, we infer that D 0 s 0.

Ž 4.4.

We now fix an i g I and y g Aai . We claim that if x g A d and y1 F d i Ž x . F 2, then there exists l x g F such that Dx Ž ­ . s l x ­ ,

;­ g Wd l Wx .

Ž 4.5.

By Ž4.2., this claim is valid if x g A j for some j g I. Thus we may assume that x g Aq d . Then the proof of the claim will be broken into four cases. Case 1: yx g Aai . By replacing x, y, ­ 1 in Ž4.3. with yx, x, d i , respectively, and by using Ž4.2. and Ž4.4., we obtain that

lyx ­ 2 q lyx ­ 2 Ž x . d i q Dx ­ 2 q ² Dx ­ 2 , x : d i s 0. Hence

Ž Dx q lyx . ­ s y² Ž Dx q lyx . ­ , x: d i holds for all ­ g T. By evaluating both sides at x, we conclude that ²Ž Dx q lyx . ­ , x : s 0, and so Dx Ž ­ . s ylyx ­ ,

;­ g T .

Hence Ž4.5. holds with

l x s ylyx .

Ž 4.6.

185

GENERALIZED CARTAN ALGEBRAS

Case 2: d i Ž x . s 0. Then x q y g A i , and so Dxqy Ž d i . s l xqy d i . By the previous case, we have Dyy Ž ­ . s lyy ­ for all ­ g T. By replacing x, y, ­ 1 in Ž4.3. with x q y, yy, d i , respectively, we obtain that

Ž Dx y l xqy y lyy . Ž ­ 2 y ­ 2 Ž x q y . d i . s 0. By setting ­ 2 s d i , we conclude that Ž Dx y l xqy y lyy .Ž d i . s 0. It follows that

Ž Dx y l xqy y lyy . Ž ­ 2 . s 0,

;­ 2 g T .

By using Ž4.6., we see that Ž4.5. holds with

l x s l xqy q lyy s l xqy y l y .

Ž 4.7.

Case 3: d i Ž x . s 1. By replacing x and y in Ž4.3. with yy and x q y, respectively, and by using Cases 1 and 2, we obtain that

­ 1 Ž x q y . Dx Ž ­ 2 . q ­ 2 Ž y . Dx Ž ­ 1 . s Ž l xqy q lyy . Ž ­ 1 Ž x q y . ­ 2 q ­ 2 Ž y . ­ 1 . holds for all ­ 1 , ­ 2 g T. In particular, for ­ 1 s ­ 2 s ­ , we obtain that

­ Ž x q 2 y . Ž Dx y l xqy y lyy . Ž ­ . s 0. Note that d i Ž x q 2 y . s y1, and so x q 2 y / 0. The above equation implies that

Ž Dx y l xqy y lyy . Ž ­ . s 0 if ­ Ž x q 2 y . / 0. By linearity, the condition ­ Ž x q 2 y . / 0 can be dropped. Hence Ž4.5. holds with l x as in Ž4.7.. Case 4: d i Ž x . s 2. As d i Ž x q y . s 1, Case 3 gives that Dxqy Ž ­ . s l xqy ­ for ­ g T. By Case 1, we have Dy Ž d i . s l y d i . By setting ­ 2 s d i in Ž4.3., we obtain that 2 Ž Dx y l xqy q l y . ­ 1 s² Ž Dx y l xqy q l y . ­ 1 , y: d i . By evaluating both sides at y, we conclude that ²Ž Dx y l xqy q l y . ­ 1 , y : s 0, and so Dx Ž ­ 1 . s Ž l xqy y l y . ­ 1 for ­ 1 g T. Hence Ž4.5. holds with l x s l xqy y l y . Our first claim is proved. Lemma 4.2 now implies that Ž4.5. holds for all x g A d . Equation Ž4.3. now takes the form L x , y Ž ­ 1 Ž y . ­ 2 y ­ 2 Ž x . ­ 1 . s 0, where L x, y s l xqy y l x y l y .

Ž 4.8.

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AND ZHAO

We now claim that L x, y s 0 whenever x, y, x q y g A d . The hypothesis q Ž . implies that x or y is in Aq d , say x g A d . By 4.4 we have l 0 s 0, and so the claim is true if x s 0 or y s 0. So, we may assume that x, y / 0. Assume first that dim T G 2. Then in Ž4.8. we may choose linearly independent ­ 1 , ­ 2 g T such that ­ 1Ž y . / 0. We conclude that L x, y s 0. Assume now that dim T s 1, and so I s  i4 . By setting ­ 1 s ­ 2 s d i in Ž4.8., we obtain that L x, y d i Ž x y y . s 0. If x / y, then d i Ž x y y . / 0, and so L x, y s 0. If x s y, then we choose z g Aai . Since l z q lyz s 0 by Ž4.6., we have

l2 x s l xqz q l xyz s Ž l x q l z . q Ž l x q lyz . s 2 l x , i.e., L x, x s 0. This completes the proof of our second claim. Every x g A can be written as x s y y z with y, z g Aq d . We define m : A ª F by m Ž x . s l y y l z . Our second claim implies that m is well defined. Clearly m is additive and m Ž x . s l x for x g A d . 5. AUTOMORPHISMS OF Wd Let u g AutŽWd . be fixed. For x g A let Fx s  f g FAq d : u Ž ­ . ? f s ­ Ž x . f , ;­ g T 4 , and let P s  x g A: Fx / 0 4 . Since Wd ? F s 0, we have F ; F0 and so 0 g P. LEMMA 5.1.

FAq d s [x g A Fx .

Proof. It suffices to show that the union of all Fx , x g A, spans FAq d . Ž . Let f g FAq d , f / 0, and choose ­ 0 g T, ­ 0 / 0. Since fu ­ 0 g Wd , we have n

uy1 Ž fu Ž ­ 0 . . s

Ý t x ­i ,

Ž 5.1.

i

is1

where x i g A d are distinct and ­ i g T are nonzero. By applying ad ­ , ­ g T, to both sides of Ž5.1., we obtain that n

Ž u Ž ­ . ? f . u Ž ­ 0 . s Ý ­ Ž x i . u Ž t x ­i . , i

is1

and similarly n

Ž u Ž ­ . k ? f . Ž ­ 0 . s Ý ­ Ž x i . k u Ž t x ­i . , i

is1

k G 0.

Ž 5.2.

GENERALIZED CARTAN ALGEBRAS

187

By choosing ­ such that ­ Ž x i . are distinct for i s 1, . . . , n and by taking k s 0, 1, . . . , n y 1 in Ž5.2., we obtain a system of linear equations to which Cramer’s formulae can be applied. We conclude that there exist f 1 , . . . , f n g FAq d such that

u Ž t x i­ i . s f i u Ž ­ 0 . ,

i s 1, . . . , n.

Ž 5.3.

From Ž5.1. and Ž5.3. we deduce that f s f 1 q ??? qf n .

Ž 5.4.

By applying ad u Ž ­ 9. to both sides of Ž5.3., we obtain that

u Ž ­ 9. ? fi s ­ 9 Ž x i . fi ,

;­ 9 g T ,

i.e., f i g Fx i . Hence Ž5.4. shows that f belongs to the sum of the Fx , x g A. q Ž . Since Wd is simple and Wd ? FAq d / 0, it follows that u T ? FA d / 0. By using Lemma 5.1, we conclude that P /  04 .

LEMMA 5.2. We ha¨ e P ; Aq d and dim Fx s 1 for all x g P. Furthermore, if a nonzero f g Fx is fixed, then for e¨ ery ­ g T there exists a unique ­˜g T such that fu Ž ­ . s u Ž t x­˜. .

Ž 5.5.

Proof. Let x g P and let f, g g Fx be both nonzero. For arbitrary ­ , ­ 9 g T we have

u Ž ­ 9 . , fu Ž ­ . s Ž u Ž ­ 9 . ? f . u Ž ­ . s ­ 9 Ž x . fu Ž ­ . , and so

uy1 Ž fu Ž ­ . . g Wd l Wx s Wd l t x T .

Ž 5.6.

In particular Wd l Wx / 0, and so x g A d . Hence P ; A d . Since P q P ; Ž . Ž . P, we must have P ; Aq d . Note that 5.6 implies 5.5 . It remains to show that dim Fx s 1. By replacing f with g, we see that for each ­ g T there exists a unique ­ˆg T such that gu Ž ­ . s u Ž t x­ˆ. .

Ž 5.7.

If dim T s 1, we choose ­ / 0. Then ­ˆs l­˜ for some l g F*, and Ž5.5. and Ž5.7. imply that g s l f. From now on we assume that dim T ) 1.

-DOKOVIC ´

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AND ZHAO

For arbitrary ­ , ­ 9 g T we have

u t 2 x Ž ­˜Ž x . ­ˆ9 y ­ˆ9 Ž x . ­˜. s u

ž

/

ž

t x­˜, t x­ˆ9

/

s u Ž t x­˜. , u Ž t x­ˆ9 . s fu Ž ­ . , g u Ž ­ 9 . s f Ž u Ž ­ . ? g . u Ž ­ 9. y g Ž u Ž ­ 9. ? f . u Ž ­ . s fg Ž ­ Ž x . u Ž ­ 9 . y ­ 9 Ž x . u Ž ­ . . .

Ž 5.8.

Since dim T ) 1, we can choose ­ , ­ 9 g T such that ­ Ž x . / 0, ­ 9Ž x . s 0, and ­ 9 / 0. Then the right hand side of Ž5.8. is not 0, and so ­˜Ž x . / 0 or ­ˆ9Ž x . / 0. Hence we have shown that there exists ­ 1 g T such that ­˜1Ž x . / 0 or ­ˆ1Ž x . / 0. By replacing ­ and ­ 9 in Ž5.8. with ­ 1 we infer that ­˜1 and ­ˆ1 are linearly dependent. Hence f and g are linearly dependent and so dim Fx s 1. Note that Lemma 5.2 implies that F0 s F. Assume that x g P and f g Fx _  04 . For ­ g T let ­˜g T be such that Ž5.5. holds. Then for t y­ 9 g Wd we have

u Ž w t y­ 9, t x­˜x . s u t xqy Ž ­ 9 Ž x . ­˜y ­˜Ž y . ­ 9 .

ž

/

and

u Ž t y­ 9 . , u Ž t x­˜. s u Ž t y­ 9 . , fu Ž ­ . s Ž u Ž t y­ 9 . ? f . u Ž ­ . q fu Ž w t y­ 9, ­ x . . It follows that

u t xqy Ž ­ 9 Ž x . ­˜y ­˜Ž y . ­ 9 . s Ž u Ž t y­ 9 . ? f . u Ž ­ . y ­ Ž y . fu Ž t y­ 9 . . Ž 5.9.

ž

/

LEMMA 5.3.

If dim T ) 1, then P s Aq d.

Proof. In view of Lemma 5.2, it suffices to show that Aq d ; P. We claim first that x q Aq d; P,

Ž 5.10. Aq d.

for all x g P _  04 . We fix a nonzero f g Fx . Let y g Since dim T ) 1, we can choose ­ / 0 such that ­ Ž y . s 0, and ­ 9 such that ­ 9 and ­˜ are linearly independent. Then Ž5.9. gives that

u t xqy Ž ­ 9 Ž x . ­˜y ­˜Ž y . ­ 9 . s Ž u Ž t y­ 9 . ? f . u Ž ­ . / 0,

ž

/

and so Ž u Ž t y­ 9. ? f g Fxqy _  04 . Hence Ž5.10. holds.

GENERALIZED CARTAN ALGEBRAS

189

Next assume that there exists z g A _  04 , such that d i Ž z . s 0 for all i g I. Let us fix a total ordering F on A. Since t z , tyz g FAq d , by Lemma 5.1, we have m

tz s

tyz s

Ý fi , is1

n

Ý gj js1

where f i g Fx i , g j g Fy j are nonzero and x 1 - ??? - x m , y 1 - ??? - yn . Since f i g j g Fx iqy j and m

n

Ý Ý f i g j s 1 g F0 , is1 js1

we infer that m s n s 1 and y 1 s yx 1. Since f 1 s t z f F s F0 , we have x 1 g P _  04 . By replacing x in Ž5.10. by x 1 and yx 1 , we conclude that Aq d ; P. In the remainder of this proof we assume that for all z g A _  04 there exists an i g I such that d i Ž z . / 0. This implies that each Aai s  a i 4 is a singleton set. We claim that x g P & d i Ž x . ) 0 « x q a i g P.

Ž 5.11.

Choose ­ g T such that ­ Ž a i . s 0 and ­˜ and d i are linearly independent. By setting y s a i and ­ 9 s d i in Ž5.9., we obtain that

u t xqa i d i Ž x . ­˜y ­˜Ž a i . d i s Ž u Ž t a i d i . ? f . u Ž ­ . / 0,

ž

ž

/

and so u Ž t a i d i . ? f g Fxqa i _  04 . This proves our second claim.  4 Ž . Now let y g Aq d _ 0 be arbitrary. Let x g P be chosen so that d i x G Ž . d i y for all i g I and ns

Ý di Ž x . igI

is minimal. ŽIt follows from Ž5.10. that such an x exists.. Assume that x / y and choose i g I such that d i Ž x . ) d i Ž y .. By Ž5.11. we have x q a i g P, which contradicts the choice of x. Hence x s y, and so y g P. For ­ g T let K Ž ­ . s  x g A: ­ Ž x . s 04 . LEMMA 5.4. If dim T ) 1, then for each x g Aq d there exists a unique f x g Fx such that f x u Ž ­ . s u Ž t x­ . holds for all ­ g T.  4 Proof. Let us fix x, y g Aq d and f g Fx _ 0 . By Lemma 5.2, for each ­ g T there exists a ­˜g T Žunique. such that Ž5.5. holds. Clearly the map T ª T sending ­ ª ­˜ is an injective linear map.

190

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AND ZHAO

The equality Ž5.9. is valid for arbitrary ­ , ­ 9 g T. That equality implies that ­ Ž y . s 0 if and only if ­˜Ž y . s 0. Since y g Aq d is arbitrary, we conclude that K Ž ­ . s K Ž ­˜. ,

;­ g T .

Ž 5.12.

Let ­ , ­ 9 g T be arbitrary. Choose a, b g F, not both zero, such that Ž a­ q b­ 9.Ž y . s 0. By applying Ž5.12. to a­ q b­ 9 instead of ­ , we conclude that Ž a­˜q b­˜9.Ž y . s 0. Hence

­ Ž y.

­˜Ž y .

­ 9Ž y .

­˜9 Ž y .

s 0,

and consequently there exists cŽ x, y, f . g F* such that

­˜Ž y . s c Ž x, y, f . ­ Ž y . ,

;­ g T .

Ž 5.13.

 4 We claim that cŽ x, y, f . is independent of y g Aq d_ 0 . For any z g A let ˆ z: T ª F be the linear function defined by ˆ zŽ­ . s ­ Ž z .. Since dim T ) 1, we can choose z g Aq such that y and ˆ ˆz are d linearly independent. In order to prove our claim, it suffices to show that cŽ x, y, f . s cŽ x, z, f . when ˆ y and ˆ z are linearly independent. In that case we can choose ­ 1 , ­ 2 g T such that

­ 1 Ž y . s ­ 2 Ž z . s 0,

­ 1 Ž z . s ­ 2 Ž y . s 1.

By Ž5.12. and Ž5.13., we have c Ž x, y, f . s ­˜2 Ž y . s ­˜2 Ž y q z . s c Ž x, y q z, f . , c Ž x, z, f . s ­˜1 Ž z . s ­˜1 Ž y q z . s c Ž x, y q z, f . , and so our claim is proved. We conclude that there is a constant cŽ x, f . g F* such that

­˜Ž y . s c Ž x, f . ­ Ž y . ,

;­ g T , ; y g A,

i.e., ­˜s cŽ x, f . ­ for all ­ g T. If f x s cŽ x, f .y1 f, then Ž5.5. implies that f x u Ž ­ . s u Ž t x , ­ . holds for all ­ g T. The uniqueness of f x is obvious. The following theorem is a generalization of Theorem 3.1 of w9x.

191

GENERALIZED CARTAN ALGEBRAS

THEOREM 5.5 ŽWd simple.. There exists a unique homomorphism C : Aut Ž Wd . ª Aut Ž FAq d . such that

u Ž fw . s C Ž u . Ž f . u Ž w . ,

Ž 5.14.

CŽ u . Ž w ? f . s u Ž w. ? CŽ u . Ž f .

Ž 5.15.

and

hold for all u g AutŽWd ., f g FAq d , and w g Wd . Furthermore C is injecti¨ e. Proof. We assume first that dim T ) 1. Given u g AutŽWd ., we define Ž x . s f x , x g Aq the linear map s on FAq d by setting s t d , where f x g Fx is defined as in Lemma 5.4. Hence we have

u Ž t x­ . s f x u Ž ­ . ,

x g Aq d , ­ g T.

Ž 5.16.

As f x / 0 for x g Aq d , Lemmas 5.1 and 5.3 imply that s is bijective. . We claim that s g AutŽ FAq d or equivalently that f x f y s f xqy ,

; x, y g Aq d .

Ž 5.17.

If ­ , ­ 9 g T then f x u Ž ­ . , f y u Ž ­ 9. s f x f y Ž ­ Ž y . u Ž ­ 9. y ­ 9 Ž x . u Ž ­ . . and

w t x­ , t y­ 9 x s t xqy Ž ­ Ž y . ­ 9 y ­ 9 Ž x . ­ . . By applying u to the last equation and by using Ž5.16., we conclude that

Ž f x f y y f xqy . Ž ­ Ž y . u Ž ­ 9. y ­ 9 Ž x . u Ž ­ . . s 0. Since f 0 s 1, Ž5.17. holds if x s 0 or y s 0. If x / 0 then we can choose linearly independent ­ , ­ 9 g T such that ­ 9Ž x . / 0. Hence the above equation implies that Ž5.17. is valid. We now claim that if t y­ g Wd and x g Aq d then

u Ž t y­ . ? f x s ­ Ž x . f xqy .

Ž 5.18.

Assume that x q y f Aq d . Then y, x q y g A i for some i g I and consequently d i Ž x . s 0. Since t y­ g Wd , we have ­ g Fd i , and so ­ Ž x . s 0. Although f xqy is not defined when x q y f Aq d , we should interpret ­ Ž x . f xqy as 0.

-DOKOVIC ´

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AND ZHAO

In order to prove Ž5.18., we consider two cases. Case 1:

Ž y . Ž . y g Aq d . Then u t ­ s f y u ­ and so

u Ž t y­ . ? f x s f y u Ž ­ . ? f x s ­ Ž x . f x f y s ­ Ž x . f xqy . Case 2: y g A i for some i g I and ­ s d i . We apply formula Ž5.9. with f s f x and ­ 9 s d i . Then ­˜s ­ by Lemma 5.4 and we obtain

u Ž t xqy Ž d i Ž x . ­ y ­ Ž y . d i . . s Ž u Ž t y d i . ? f x . u Ž ­ . y ­ Ž y . f x u Ž t y d i . . We choose ­ g T _  04 such that ­ Ž y . s 0 and obtain

Ž u Ž t y d i . ? f x . u Ž ­ . s d i Ž x . u Ž t xqy­ . .

Ž 5.19.

Hence if d i Ž x . s 0, then Ž5.18. holds. Assume now that d i Ž x . / 0. Then Ž . Ž xqy­ . s f xqy u Ž ­ . and d i Ž x . ) 0 and so x q y g Aq d . By 5.16 we have u t so Ž5.18. follows from Ž5.19.. Hence our second claim is proved. We now define C Ž u . s s . In order to verify Ž5.14. and Ž5.15., we may y assume that f s t x , x g Aq d , and w s t ­ . Then

s Ž w ? f . s s Ž t y­ ? t x . s ­ Ž x . s Ž t xqy . s ­ Ž x . f xqy , and, by using Ž5.18.,

u Ž w . ? s Ž f . s u Ž t y­ . ? f x s ­ Ž x . f xqy . Hence Ž5.15. holds. In order to prove Ž5.14., it suffices to check that

u Ž fw . ? f z s s Ž f . u Ž w . ? f z Ž . holds for all z g Aq d . By using 5.18 we obtain that

u Ž fw . ? f z s u Ž t xqy­ . ? f z s ­ Ž z . f xqyqz , and

s Ž f . u Ž w . ? f z s s Ž t x . u Ž t y­ . ? f z s f x ­ Ž z . f yqz s ­ Ž z . f xqyqz . Hence Ž5.14. holds. The condition Ž5.14. uniquely determines C Ž u .. Indeed if we take Ž . f s t x , x g Aq d , and w s ­ g T, then 5.14 becomes

u Ž t x­ . s C Ž u . Ž t x . u Ž ­ . .

GENERALIZED CARTAN ALGEBRAS

193

Ž . Hence Lemma 5.4 implies that C Ž u .Ž t x . s f x for all x g Aq d , i.e., C u s s. If u , u 9 g AutŽWd ., then

uu 9 Ž t x­ . s u Ž C Ž u 9 . Ž t x . u 9 Ž ­ . . s C Ž u . Ž C Ž u 9 . Ž t x . . uu 9 Ž ­ . , and so C Ž uu 9. s C Ž u . C Ž u 9.. If C Ž u . is identity, then Ž5.15. implies that w ? f s u Ž w . ? f, i.e., Ž u Ž w . y Ž . w . ? FAq d s 0. As Wd is simple, we must have u w s w, i.e., u s 1. Hence C is an injective homomorphism. It remains to consider the case dim T s 1. The set I is either empty or a singleton, say I s  14 . Assume first that I s B. Then Wd s W and Aq d s A. Since dim T s 1 and W is simple, if ­ 0 g T _  04 then ­ 0 : A ª F is injective. For the sake of simplicity we shall identify x g A with ­ 0 Ž x . g F. By w1, Theorem 4.2x every u g AutŽW . has the form

u Ž t x­ 0 . s x Ž x . t l xly1­ 0 , where x g HomŽ A, F*. and l g F* is such that l­ 0 Ž A. s ­ 0 Ž A.. Then we define C Ž u . s s by

s Ž t x . s x Ž x. t lx,

x g A.

It is not hard to verify that C is a homomorphism and that it satisfies Ž5.14. and Ž5.15.. Finally let I s  14 . Then d1: A ª Z is an isomorphism. We can identify Wd with the Lie algebra of polynomial vector fields P Ž t .Ž drdt ., P Ž t . g F w t x. Under this identification d1 s t Ž drdt .. The elements e i s t iq1 Ž drdt ., w x i G y1, form a basis of Wd . Note that FAq ds F t . The set of w g Wd such that adŽ w . is locally nilpotent Žresp. locally finite. is Fey1 Žresp. Fey1 q Fe 0 .. Furthermore adŽ w . is semisimple if and only if w s ae0 q bey1 with a / 0. Each m g F determines an automorphism um s expŽ m adŽ ey1 .. of Wd . Since umŽ e0 . s e0 q m ey1 , we see that each ad-semisimple element of Wd is conjugate under AutŽWd . to some ae0 , a g F*. Each l g F* defines another automorphism u l of Wd such that lŽ . u e i s li e i , i G y1. By using the above facts, it is not hard to show that every u g AutŽWd . has the form u s um u l. We now define C Ž u . s s by i

s Ž t i . s li Ž t q m . ,

i G 0.

Then C is a homomorphism satisfying Ž5.14. and Ž5.15..

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194 COROLLARY 5.6. morphism

AND ZHAO

Suppose that Wd is simple and that the natural homo-

r : Wd ª Der Ž FAq d . is an isomorphism. Then C is also an isomorphism, and Cy1 Ž s . Ž w . s ry1 Ž sr Ž w . sy1 . . for w g Wd and s g AutŽ FAq d . Proof. This is the same as the proof of Theorem 3.7 in w9x. Let Td be the torus of T having  d i : i g I 4 as its basis. Then the subspace Ww d x s FA mF Td is a subalgebra of W. If wd : Td = A ª F is the restriction of w , then Ww d x s W Ž A, Td , wd . . Note that wd may be degenerate Ževen when w is not.. So, in general, Ww d x is not simple even when W is simple. We also introduce another important subalgebra, namely, Wwqd x s Ww d x l Wd . If V is a module over a Lie algebra g and r : g ª g l Ž V . the corresponding representation, we say that w g g is locally nilpotent on V if r Ž w . is a locally nilpotent operator. For w g Wd it is natural to consider local nilpotence on Wd Žfor the adjoint representation . as well as local nilpotence on the Wd-module FAq d. LEMMA 5.7. Let w g Wd . If w is locally nilpotent on Wd , then it is also locally nilpotent on FAq d . The con¨ erse holds in the following two cases: Ži. w g Wwqd x and I is finite; Žii. FAq d is finitely generated. Proof. Assume that w is locally nilpotent on Wd . In order to prove that w is also locally nilpotent on FAq d , we may assume that w / 0. Let f g FAq be arbitrary. There exists n ) 0 such that Žad w . n Ž fw . s 0. By d using the formula Ž2.5., this can be rewritten as Ž r Ž w . n f . w s 0, where r denotes the representation of Wd on FAq d . Since w / 0, we conclude that r Ž w . n f s 0. This proves our first assertion. In order to prove the second assertion, we suppose now that r Ž w . is locally nilpotent.

195

GENERALIZED CARTAN ALGEBRAS

Assume first that Ži. holds. For each i g I, choose an x i g yAai . Then 0 xi FAq d s FA d w t : i g I x .

Ž 5.20.

If t x­ g Wd , we have to show that Žad w . n Ž t x­ . s 0 for some n ) 0. Since n

n

Ž ad w . Ž t x­ . s

Ý Ž y1. k ks0

n k

ž / Ž rŽ

w.

nyk x

t

. Ž ad w . k ­

and r Ž w . is locally nilpotent, it suffices to show that Žad w . k­ s 0 for sufficiently large k. As I is finite, there exists m ) 0 such that r Ž w . m t x i s 0 for all i g I. Since k

S s  r Ž ­ . r Ž w . t x i : i g I, 0 F k - m4 is a finite set, there exists r ) 0 such that r Ž w . r S s 0. If N s m q r y 1, then N

r Ž Ž ad w . ­ . t x i s

N

Ý Ž y1. k ks0

N k

ž / rŽ

w.

Nyk

k

r Ž ­ . r Ž w . t xi s 0

for all i g I. Indeed, if k G m, then r Ž w . k t x i s 0, and otherwise N y k G r and so r Ž w . Ny k S s 0. Since FA0d is r Ž ­ .-invariant and w g Wwqd x , we have r ŽŽad w . N­ . FAq d s 0. Ž .N It follows from Ž5.20. that r ŽŽad w . N­ . FAq d s 0, and so ad w ­ s 0. Assume now that Žii. holds. Let S be a finite set of generators of FAq d. Choose m ) 0 such that r Ž w . m S s 0. Let ¨ g Wd be arbitrary. Since my1

S9 s

D r Ž ¨ . r Ž w. kS

ks0

is a finite set, there exists n ) 0 such that r Ž w . nS9 s 0. If N s m q n y 1, then N

r Ž Ž ad w . ¨ . S s

N

Ý Ž y1. k ks0

N k

ž / rŽ

w.

Nyk

k

r Ž ¨ . r Ž x . S s 0.

ŽŽ . N ¨ . s 0, i.e., Žad w . N ¨ s 0. Since S generates FAq d , we have r ad w We now describe two types of automorphisms u of Wd , and compute the corresponding automorphism s s C Ž u . of FAq d .

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196

AND ZHAO

EXAMPLE 5. For any x g HomŽ A, F*. we can define u g AutŽWd . by

u Ž w . s x Ž x . w, It is easy to see that s Ž t

x.

w g Wx l Wd .

s x Ž x . t for x g Aq d. x

EXAMPLE 6. Let ¨ g Wd be locally nilpotent on Wd . Then u s expŽad ¨ . is an automorphism of Wd . By Lemma 5.7, ¨ is also locally nilpotent on Ž . Ž Ž .. FAq d , i.e., r ¨ is locally nilpotent. We claim that s s exp r ¨ . We have to verify that Ž5.14. and Ž5.15. hold. We shall carry out this verification only for Ž5.14.. From Ž2.5. it follows that k

k

Ž ad ¨ . Ž fw . s

Ý is0

k i

ž / Ž rŽ

i

¨ . f . Ž ad ¨ .

kyi

w

holds for f g FAq d and w g Wd . Consequently we have

s Ž f . u Ž w . s Ž e r Ž ¨ . f . Ž e ad ¨ w . s

`

Ý is0

s s

`

Ý

1 i! 1

ks0

k!

`

1

Ý ks0

i

rŽ¨. f ?

k!

`

Ý js0

k

Ý is0

k i

ž / Ž rŽ

1 j!

j Ž ad ¨ . w

i

¨ . f . Ž ad ¨ .

kyi

w

k

Ž ad ¨ . Ž fw .

s u Ž fw . . The verification of Ž5.15. is similar. 6. SUBALGEBRAS Sw d x AND Swqd x We define the di¨ ergence as the linear map Div: Ww d x ª FA such that Div Ž t x d i . s Ž 1 q d i Ž x . . t x

Ž 6.1.

for all x g A and i g I. It is straightforward to verify that the following classical formulae are valid, Div w u, ¨ x s u ? Div Ž ¨ . y ¨ ? Div Ž u .

Ž 6.2.

and Div Ž fu . s u ? f q f Div Ž u . , where u, ¨ g Ww d x and f g FA are arbitrary.

Ž 6.3.

GENERALIZED CARTAN ALGEBRAS

197

We remark that in the classical case, i.e., when Ww d x s Wn Žsee Example 1., our definition of divergence coincides with the ordinary divergence of vector fields. We set Sw d x s kerŽDiv.. It follows from Ž6.2. that Sw d x is a subalgebra of Ww d x. We also introduce the subalgebra Swqd x s Sw d x l Wd . We recall that Td denotes the subspace of T spanned by the d i ’s. If I s B, then Td s 0, and so Ww d x s 0. Throughout this section we shall assume that I is finite and nonempty. LEMMA 6.1. Ža. Žb.

For w g Wd the following are equi¨ alent:

w g Wwqd x; w ? FA0d s 0.

Proof. It is obvious that Ža. « Žb.. Assume that Žb. holds. We consider first the case where w s ­ g T. Let x g A be arbitrary. For each i g I we choose an x i g Aai and set x0 s x q

Ý di Ž x . x i . igI

Since d i Ž x j . s yd i j , it follows that x 0 g A0d , and so ­ ? t x 0 s 0. Hence we have

¦­ q Ý ­ Ž x . d , x;s 0 i

i

igI

for all x g A. Since w is nondegenerate, we conclude that

­ s y Ý ­ Ž x i . d i g Td . igI

Hence Ža. holds in this case. In the general case we have ws

Ý

t x­x ,

xgA d

where ­x g T Žand ­x g Fd i , if x g A i .. Of course, ­x s 0 for almost all x. If y g A0d , then w ? t y s 0 by hypothesis, i.e.,

Ý xgA d

­x Ž y . t xqy s 0.

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198

AND ZHAO

This implies that ­x Ž y . s 0 for all y g A0d , and so ­x ? FA0d s 0. By the special case above, this implies that ­x g Td . Since this is true for all x g A d , we conclude that w g Wwqd x. Since the invertible elements of FA are of the form at x , a g F*, x g A, we deduce that the invertible elements of the algebra FAq d have the form at x where a g F* and x g A0d . A subalgebra of Wd is a characteristic subalgebra if it is mapped onto itself by every automorphism of Wd . The following proposition generalizes Proposition 3.10 of w9x. PROPOSITION 6.2 Ž I finite.. The subalgebra Wwqd x is characteristic in Wd . . Žsee Theorem Proof. Let u g AutŽWd . and let s s C Ž u . g AutŽ FAq d 5.5.. By using the characterization of the invertible elements of FAq d , we conclude that s Ž FA0d . s FA0d . If w g Wwqd x and x g A0d , then w ? t x s 0 and Ž5.16. implies that

u Ž w . ? s Ž t x . s s Ž w ? t x . s 0. Hence u Ž w . ? FA0d s 0, and Lemma 6.1 implies that u Ž w . g Wwqd x. We have shown that the condition is also sufficient. The idea for the proof of the following lemma is taken from Osborn’s paper w9x. LEMMA 6.3.

If u g AutŽWd . and x g Aar , r g I, then Div u Ž t x d r . s 0.

Proof. For each i g I we choose x i g Aai and we assume that x r s x. Then the elements wi s t x i d i , i g I, form a basis of Wwqd x as a free FAq d -module. Hence we have

u Ž w. s

Ý f i j wj jgI

Ž . with uniquely determined coefficients f i j g FAq d . Let s s C u and set g i s s Ž tyx i . ,

g i j s wi ? g j ,

for i, j g I. The product of the matrices Ž f i j . and Ž g jk . is the identity matrix because

Ý f i j g jk s Ý f i j wj ? g k s u Ž wi . ? s Ž tyx . k

jgI

jgI

s s Ž wi ? tyx k . s d i k .

GENERALIZED CARTAN ALGEBRAS

199

In particular it follows that g s detŽ g i j . is invertible in FAq d , and so g g FA0d . We have gf i j s Gi j where Gi j is the cofactor of g ji in the matrix Ž g i j .. Since g g FA0d , we have wi ? g s 0 for all i. Hence we have g Div u Ž t x i d i . s g Div Ý f i j wj s g Ý wj ? f i j jgI

s

jsI

Ý wj ? Ž gfi j . s Ý wj ? Gi j . jgI

Ž 6.4.

jgI

We consider Gi j as the determinant of the matrix Mi j obtained from Ž g i j . by replacing its jth row by the row consisting of zeros except for the ith coordinate which is 1. Then wj ? Gi j s

Ý det MiŽjk . , kgI

MiŽjk .

where is the matrix obtained from Mi j by applying wj to the elements of the kth row. Hence Ž6.4. gives g Div u Ž t x i d i . s

Ý

det MiŽjk . .

Ž 6.5.

j, kgI

Since w wj , w k x s 0 and g i j s wi g j , it is easy to see that det MiŽjk . q det MiŽkj. s 0 holds for all i, j, k g I. Hence Ž6.5. implies that Div u Ž t x i d i . s 0 for each i g I. In particular for i s r we obtain Div u Ž t x d r . s 0. PROPOSITION 6.4 Ž I finite.. The formula Div u Ž w . s C Ž u . Ž Div Ž w . .

Ž 6.6.

is ¨ alid for all w g Wwqd x and u g AutŽWd .. Proof. By Proposition 6.2 we have u Ž w . g Wwqd x , and so the left hand side of Ž6.6. makes sense. For each i g I we choose an x i g Aai . The . elements t x i d i , i g I, form a basis of Wdq Žas a free FAq d -module . Hence each w g Wdq can be uniquely written as ws

Ý fi t x di i

igI

with f i g

FAq d .

Set s s C Ž u .. By Ž5.14. we have

u Ž w. s

Ý s Ž fi . u Ž t x di . . i

igI

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200

AND ZHAO

By using Lemma 6.3 and the formulae Ž5.15. and Ž6.3., we find that Div u Ž w . s

u Ž t x i d i . ? s Ž f i . q s Ž f i . Div u Ž t x i d i .

Ý igI

s

Ý s Ž t x d i ? f i . s s Ž DivŽ w . . . i

igI

COROLLARY 6.5. The subalgebra Swqd x is characteristic in Wd . PROPOSITION 6.6 Ž I finite.. The subalgebra Swqd x is generated by the Wi ’s, i g I. Proof. Clearly each Wi is contained in Swqd x. Let L be the subalgebra of generated by the Wi ’s. It follows from Ž6.1. that Div is a homogeneous map of degree 0. Consequently Sw d x and Swqd x are homogeneous subalgebras of W. Hence in order to prove that L s Swqd x , it suffices to show that every homogeneous element t x­ g Swqd x belongs to L. We have x g A d . If x f Aq d, then x g A i for some i g I and ­ g Fd i . In that case t x­ g Wi and so t x­ g L. Hence we need only consider the case where x g Aq d . Since ­ g Td , we have

Swqd x

­s

c i g F.

Ý ci di , igI

From DivŽ t x­ . s 0, we obtain that

Ý c i Ž1 q d i Ž x . . s 0. igI

Choose k g I such that c k / 0. Then t x­ s

Ý tx i/k

ž

ci di y

1 q di Ž x . 1 q dk Ž x .

ci dk

/

and each summand on the right hand side has divergence 0. Hence it suffices to consider only the case when exactly two coefficients c i are nonzero. So we assume that

­ s cr dr q cs d s ,

r / s,

c r Ž 1 q d r Ž x . . q c s Ž 1 q d s Ž x . . s 0.

Ž 6.7.

We shall prove that t x­ g L by induction on d r Ž x . G 0. We choose y g Aar and z g Aas . Assume first that d r Ž x . s 0. Since t xqyyz d r g Wr and t zyy d s g Ws , we have t xqyyz d r , t zyy d s s t x Ž d s y Ž 1 q d s Ž x . . d r . g L. Since d r Ž x . s 0, Ž6.7. implies that t x­ g L.

GENERALIZED CARTAN ALGEBRAS

201

Now let d r Ž x . ) 0. By induction hypothesis, we have t xqyyz Ž Ž 2 q d s Ž x . . d r y d r Ž x . d s . g L. Since t zyy d s g Ws , it follows that t xqyyz Ž Ž 2 q d s Ž x . . d r y d r Ž x . d s . , t zyy d s s Ž 2 q d s Ž x . . t x Ž Ž 1 q d s Ž x . . d r y Ž 1 q d r Ž x . . d s . g L. In view of Ž6.7., we conclude again that t x­ g L. The problem of characterizing elements w g Wd that are locally nilpotent on Wd is very hard. It is wide open even in the case of the algebras Wnq Žsee Example 1.. The proof given below that DivŽ X . s 0, for locally nilpotent X g Wnq , was communicated to us by A. van den Essen. PROPOSITION 6.7. Assume that W and Wd are simple and that I is finite and nonempty. If w g Wwqd x is locally nilpotent on Wwqd x , then DivŽ w . s 0. Proof. We shall consider first the case of the classical algebra Wnq . Thus we assume that X g Wnq is locally nilpotent on Wnq or, equivalently Žsee Lemma 5.7., on the polynomial algebra Pn s F w t 1 , . . . , t n x. Let us write n

Xs

­

Ý fi ­ t

is1

f i g Pn .

, i

Let Pnq 1 s F w t 1 , . . . , t nq1 x s Pnw t nq1 x where t nq1 is a new indeterminate. Since X is locally nilpotent on Pn , the vector field t nq1 X is locally nilpotent on Pnq 1. Consequently the map u s expŽ t nq1 X . is an automorphism of the algebra Pnq 1. We have

u Ž t i . s t i q t nq1 X Ž t i . q s t i q t nq1 f i q

1 2!

1 2!

2 t nq1 X 2 Ž t i . q ???

2 t nq1 X Ž f i . q ???

for 1 F i F n and u Ž t nq 1 . s t nq1. Since the determinant of the Jacobian J Ž u . of u is a nonzero constant, the coefficient of t nq 1 in the expansion of det J Ž u . g Pnw t nq1 x must be 0. As the last row of J Ž u . is Ž0, . . . , 0, 1. and

­u Ž t i . ­ tj

s d i j q t nq1

­ fi ­ tj

q ??? ,

i , j F n,

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202

AND ZHAO

it is easy to see that the coefficient of t nq 1 in det J Ž u . is n

Div Ž X . s

Ý is1

­ fi ­ ti

.

Now we consider the general case. Thus let w g Wwqd x be locally nilpotent on Wwqd x. Since d i Ž x . s 0 for all x g A0d , we have Wwqd x ? FA0d s 0. Consequently we can view Wwqd x as a Lie algebra over the group algebra FA0d . Let F be the field of fractions of FA0d and let W s F mF A 0d Wwqd x

Ž 6.8.

be the Lie algebra over F obtained by extension of scalars. Let n s < I < and denote by Wnq the Lie algebra of polynomial vector fields n

Xs

Ý ks1

fk

­

f k g F w t1 , . . . , t n x .

,

­ tk

Let x i g A i be fixed for each i g I. Then it is easy to check that the F-linear map c : Wnq ª W defined by

ž

c t 1k 1 ??? t nk n

­ ­ tj

/

s 1 m t x jyÝ k i x i d j

is an isomorphism of Lie algebras. If X g W is such that c Ž X . s 1 m w, then X is locally nilpotent on Wnq and so DivŽ X . s 0. Consequently DivŽ w . s 0. PROPOSITION 6.8. Let W and Wd be simple and n s < I < - `. If S is a torus in Wwqd x , then dim S F n. Proof. By extending scalars, we may assume that the torus S is split. We consider first the case where W s Wn and Wd s Wnq. The algebra q Wn admits the Z-gradation such that each t i Žresp. ­r­ t i . has degree 1 Žresp. y1.. We denote by WnqŽ k ., k G y1, the homogeneous component of Wnq of degree k, and by p k : Wnqª Wnq Ž k . the canonical projection. We have Wnqs

Ý

Wnq Ž k . ,

kGy1

WnqŽ 0. is a subalgebra of Wnq isomorphic to g l Ž n, F ., and Wnq Ž y1 . s

n

­

Ý F ­t

is1

i

GENERALIZED CARTAN ALGEBRAS

203

is the standard irreducible module of WnqŽ 0.. We also set Wnq w k x s

[ Wnq Ž 0. . iGk

Since Wnqw 1x contains no nonzero semisimple elements, we have S l s 0. Let S0 s S l Wnqw 0x and choose a subtorus S9 of S such that S s S0 [ S9. Since S0 is diagonalizable on WnqrWnq w0x, we conclude that p 0 Ž S0 . is diagonalizable on WnqŽ y1.. Since wp 0 Ž S0 ., py1Ž S9.x s 0, and the action of p 0 Ž S0 . on WnqŽ y1. is faithful, we must have Wnqw 1x

dim p 0 Ž S0 . F dim Wnq Ž y1 . y dim py1 Ž S9 . . Hence dim S s dim p 0 Ž S0 . q dim py1 Ž S9 . F n. Now let us consider the general case. Let F be the field of fractions of FA0d and let W be defined by Ž6.8.. Let i : Wwqd x ª W be the canonical injection and S s Fi Ž S .. Clearly S is a torus in W . Since W , Wnq Žsee the proof of the previous proposition., the first part of the proof shows that dim F S F n. In order to complete the proof, it suffices to show that r [ dim F S F dim F S .

Ž 6.9.

Since Wwqd x has trivial center, the weights of S in Wwqd x span the dual space S* of S. Hence we can choose weights l1 , . . . , l r g S* and w 1 , . . . , wr g S such that l i Ž wj . s d i j . So, there exist nonzero ¨ 1 , . . . , ¨ r g Wwqd x such that w wi , ¨ j x s d i j ¨ j . If a i g F are such that a1 m w 1 q ??? qa r m wr s 0, then r

0s

Ý

a i m wi , 1 m ¨ j s a j m ¨ j

is1

implies that a j s 0. Since j s 1, . . . , r is arbitrary, we conclude that the elements 1 m w 1 , . . . , 1 m wr are linearly independent over F , and so Ž6.9. holds.

204

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AND ZHAO

7. VANISHING OF H 2 ŽWd , F . In w1x we have shown that, for a simple generalized Witt algebra W, the second cohomology group H 2 ŽW, F . is 0 if the maximal torus T ; W has dimension greater than 1, and is 1-dimensional when T has dimension 1. In this section we shall prove the following result. THEOREM 7.1. Let W be a simple generalized Witt algebra, d: I ª T an admissible map such that I / B and Wd is simple. Then H 2 ŽWd , F . s 0. Proof. Let c : Wd = Wd ª F be a 2-cocycle, i.e., a skew-symmetric bilinear form such that

c Ž w u, ¨ x , w . q c Ž w ¨ , w x , u . q c Ž w w, u x , ¨ . s 0 for all u, ¨ , w g Wd . By setting u s t x­ 1 , ¨ s t y­ 2 , and w s t z­ 3 , we obtain that

­ 1 Ž y . c Ž t xqy­ 2 , t z­ 3 . y ­ 2 Ž x . c Ž t xqy­ 1 , t z­ 3 . q ­ 2 Ž z . c Ž t yq z­ 3 , t x­ 1 . y ­ 3 Ž y . c Ž t yq z­ 2 , t x­ 1 . q ­ 3 Ž x . c Ž t zqx­ 1 , t y­ 2 . y ­ 1 Ž z . c Ž t zqx­ 3 , t y­ 2 . s 0.

Ž 7.1.

By setting y s z s 0, we obtain that

­ 3 Ž x . c Ž t x­ 1 , ­ 2 . s ­ 2 Ž x . c Ž t x­ 1 , ­ 3 . .

Ž 7.2.

 4 Ž . For x g Aq d _ 0 choose ­ x g T such that ­ x x s y1, and, if x g A i Ž .  4 for some i g I, set ­x s d i . Thus ­x x s y1 for all x g Aq d _ 0 . We also set ­ 0 s 0. Let f : Wd ª F be the linear function such that f Ž t x­ . s c Ž t x­ , ­x .. The 2-cocycle c f defined by c f Ž u, ¨ . s f Žw u, ¨ x. is a coboundary. By using Ž7.2. we see that

c f Ž t x­ 1 , ­ 2 . s y­ 2 Ž x . f Ž t x­ 1 . s c Ž t x­ 1 , ­ 2 . holds for x g A d _  04 . By replacing c with the cohomologous 2-cocycle c y c f , we may assume that

c Ž t x­ 1 , ­ 2 . s 0,

x g A d _  04 .

Hence, for z s 0, Ž7.1. gives that

c Ž t x­ 1 , t y­ 2 . s 0,

for x q y / 0.

Ž 7.3.

For x g A d l ŽyA d . we get

c x Ž ­ 1 , ­ 2 . s c Ž t x­ 1 , tyx­ 2 . .

Ž 7.4.

205

GENERALIZED CARTAN ALGEBRAS

If x g A i for some i g I, then t x­ 1 g Wd implies that ­ 1 g Fd i . Similarly, if x g yA i for some i g I, then tyx­ 2 g Wd implies that ­ 2 g Fd i . If x g A0d , then ­ 1 and ­ 2 in Ž7.4. may be arbitrary elements of T. In view of Ž7.3., the 2-cocycle c is uniquely determined by the system of functions c x defined by Ž7.4.. We now assume that x, y, z g A d l ŽyA d . and that x q y q z s 0. Then Ž7.1. becomes

c xqy Ž ­ 1 Ž y . ­ 2 y ­ 2 Ž x . ­ 1 , ­ 3 . q c x Ž ­ 1 , ­ 2 Ž x q y . ­ 3 q ­ 3 Ž y . ­ 2 . y c y Ž ­ 2 , ­ 3 Ž x . ­ 1 q ­ 1 Ž x q y . ­ 3 . s 0.

Ž 7.5.

Assume that I / B. Choose i g I and x g Aai . Set y s 0, z s yx, and ­ i s d i in Ž7.5. to obtain that c 0 Ž ­ 2 , ­ 3 y ­ 3 Ž x . d i . s 0. Since ­ 2 , ­ 3 g T are arbitrary, we conclude that c 0 s 0. Let x, y g A i l ŽyA d . for some i g I. By setting ­ 1 s ­ 2 s d i and ­ 3 s ­ in Ž7.5., we obtain

cx Ž di , ­ Ž y . di y 2 ­ . s cy Ž di , ­ Ž x . di y 2 ­ . .

Ž 7.6.

By replacing ­ with ­ q d i , we infer that ci [ cx Ž di , di . is independent of the choice of x g A i l ŽyA d .. Now Ž7.6. implies that the linear function l i : T ª F defined by

li Ž ­ . s cx Ž di , ­ . q

1 2

ci ­ Ž x .

Ž 7.7.

is also independent of the choice of x g A i l ŽyA d .. By setting ­ s d i , we obtain that

li Ž di . s

1 2

ci .

Ž 7.8.

Now let i, j g I be distinct. Choose x g Aai and y g Aaj . By setting ­ 1 s d i , ­ 2 s d j , and ­ 3 s ­ in Ž7.5., we obtain

cx Ž di , ­ Ž y . d j y ­ . s c y Ž d j , ­ Ž x . d i y ­ . . By replacing ­ with ­ q kd i q ld j , where k and l are arbitrary integers, and by comparing the coefficients of k and l on both sides, we obtain that

c x Ž d i , d j . s c jr2,

c y Ž d j , d i . s c ir2,

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AND ZHAO

and consequently

cx Ž di , ­ . y

1 2

cj ­ Ž y . s cy Ž d j , ­ . y

1 2

ci ­ Ž x . .

By recalling the definition Ž7.7. of the linear functions l i , we conclude that l i is independent of i g I. We shall write l instead of l i from now on. Thus we have

c x Ž d i , ­ . s lŽ ­ . y

1 2

ci ­ Ž x .

Ž 7.9.

for all x g A i l ŽyA d . and ­ g T. Now let x g A0d _  04 . Choose an i g I and y g Aai . Then also x q y g a A i . By setting ­ 2 s d i and ­ 3 s ­ in Ž7.5., we obtain

­ 1 Ž y . c xqy Ž d i , ­ . q c x Ž ­ 1 , ­ Ž y . d i y ­ . s c y Ž d i , ­ Ž x . ­ 1 q ­ 1 Ž x q y . ­ . . By using Ž7.9., we can rewrite this equation as

c x Ž ­ 1 , ­ . q lŽ ­ 1 . ­ Ž x . q lŽ ­ . ­ 1Ž x . s ­ Ž y . c x Ž ­ 1 , d i . q

1 2

ci ­ 1Ž x . .

Ž 7.10. By replacing y with x q y, we conclude that

­ Ž x . cx Ž ­ 1 , di . q

1 2

c i ­ 1 Ž x . s 0.

By choosing ­ g T such that ­ Ž x . / 0, we conclude that

cx Ž ­ 1 , di . q

1 2

c i ­ 1 Ž x . s 0.

Hence Ž7.10. gives that

c x Ž ­ 1 , ­ . s ylŽ ­ 1 . ­ Ž x . y lŽ ­ . ­ 1 Ž x .

Ž 7.11.

holds for x g A0d and ­ , ­ 1 g T. We now extend the linear function l: T ª F to obtain the linear function, again denoted by l, on Wd such that lŽWx l Wd . s 0 for all x g A _  04 . It follows from Ž7.3., Ž7.8., Ž7.9., and Ž7.11. that c Ž u, ¨ . s lŽw u, ¨ x. for all u, ¨ g Wd . Hence c is a coboundary and the proof is completed.

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207

8. PRINCIPAL GRADATION OF Wd For x g A d , there are only finitely many indices i g I such that d i Ž x . / 0 Žsee Theorem 3.3.. Hence we can define the map deg: A d ª Z by deg Ž x . s

Ý di Ž x . . igI

For each k g Z we define the subspace ŽWd . k of Wd by

Ž Wd . k s

Ý

Wd l Wx .

xgA d degŽ x .sk

It is immediate from the definition of Wd that ŽWd . k s 0 for k - y1. It is also clear that

Ž Wd . k , Ž Wd . l ; Ž Wd . kql for all k, l g Z, and Wd s

Ý Ž Wd . k . kGy1

We shall refer to this Z-gradation of Wd as its principal gradation. Our first objective is to describe the structure of the subalgebra ŽWd . 0 . For each i g I, we fix an element x i g Aai . Recall that Td is the subtorus of T spanned by all d i , i g I. We also choose a direct decomposition T s Td [ T 9. The subspace W9 s FA0d ? T 9 of ŽWd . 0 is a subalgebra. If

w 9: T 9 = A0d ª F is the restriction of w , then W9 , W Ž A0d , T 9, w 9 . . For i, j g I set e i j s t x jyx i d j . As an FA0d-module, ŽWd . 0 is free with a basis consisting of the e i j ’s and an F-basis of T 9.

208

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Thus we have ŽWd . 0 s W9 [ L where Ls

Ý

FA0d ? e i j .

i , jgI

Although the e i j ’s depend on the choice of the x i , it is easy to see that L does not. In fact we have L s Ž Wd . 0 l Ww d x s Ž Wd . 0 l Wwqd x . In particular, L is a subalgebra of ŽWd . 0 . Note that

w e i j , e k l x s d jk e i l y d l i e k j . Hence L is isomorphic to the finitary general linear Lie algebra over the ring FA0d . The elements of this general linear Lie algebra are matrices over FA0d whose rows and columns are indexed by I and which have only finitely many nonzero entries. PROPOSITION 8.1. Let W s W Ž A, T, w . and Wd be simple, and let L and W9 be as defined abo¨ e. Then Ža. L is an ideal of ŽWd . 0 and ŽWd . 0rL , W9; Žb. ŽWd .y1 is a simple ŽWd . 0-module; Žc. If < I < - ` and A0d / 0, then W9 is a simple Lie algebra. Proof. Ža. Let t x­ g W9, i.e., x g A0d and ­ g T 9. For f g FA0d and i, j g I we have t x­ , fe i j s t x Ž ­ ? f q ­ Ž x j y x i . f . e i j g L. As ŽWd . 0 s L q W9, it follows that L is an ideal of ŽWd . 0 . Žb. Let V be a nonzero ŽWd . 0-submodule of ŽWd .y1. Since T ; ŽWd . 0 , V is a homogeneous subspace of Wd with respect to the A-gradation. It follows that V contains an element t x d i , x g Aai . For arbitrary y g A0d we have t y d i g ŽWd . 0 , and so t x d i , t y d i s t xqy d i g V . It follows that FA0d ? t x i d i ; V. Since ei j , t xdi s t x j d j g V , we also have FA0d ? t x j d j ; V for all j g I. Hence V s ŽWd .y1.

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209

Žc. Since < I < - `, we may assume that T 9 s  ­ g T : ­ Ž x i . s 0, ; i g I 4 . We need only show that w 9: T 9 = A0d ª F is nondegenerate. Assume that w ŽT 9, x . s 0 for some x g A0d . Since w ŽTd , x . s 0, we have w ŽT, x . s 0. As w is nondegenerate, we have x s 0. Hence the right kernel of w 9 is 0. Assume now that w Ž ­ , A0d . s 0 for some ­ g T 9. Since w Ž ­ , x i . s 0 for all i g I and A is generated by A0d and the x i ’s, we conclude that w Ž ­ , A. s 0. As w is nondegenerate, we must have ­ s 0. Hence also the left kernel of w 9 is 0. The following example shows that the condition < I < - ` is necessary for the simplicity of W9. EXAMPLE 7. Let A be the direct sum of countably many copies of Z indexed by integers i G 0. We write x g A as x s Ž x 0 , x 1 , . . . . where x i g Z are almost all 0. Let T be the vector space over F with basis ­ i , i G 0. Let w : T = A ª F be the pairing defined by

w Ž ­0 , x. s ­0Ž x. s

Ý xi , iG0

and

w Ž ­ i , x . s ­ i Ž x . s x iy1 y x i ,

i G 1.

It is easy to verify that w is nondegenerate, and so W s W Ž A, T, w . is simple. Let I s  0, 1, . . . 4 and let d: I ª T be defined by d Ž i . s d i s ­ iq3 . As d is admissible and satisfies the conditions of Theorem 3.2, Wd is simple. We have T s Td [ T 9 where T 9 is spanned by ­ 0 , ­ 1 , and ­ 2 . Since w Ž ­ 0 y ­ 1 y 2 ­ 2 , A0d . s 0, the restriction

w 9: T 9 = A0d ª F is degenerate. Hence W9 s FA0d ? T 9 is not simple. REFERENCES ˇ D 1. D. Z. ] okovic ´ and K. Zhao, Derivations, isomorphisms, and second cohomology of generalized Witt algebras, Trans. Amer. Math. Soc., in press. 2. A. S. Dzhumadildaev, Central extensions of infinite-dimensional Lie algebras, Funktsional Anal. i Prilozhen. 26 Ž1992., 21]29.

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3. D. B. Fuks, Cohomology of infinite dimensional Lie algebras, Moscow, 1984. 4. I. Kaplansky, Seminar on simple Lie algebras, Bull. Amer. Math. Soc. 60 Ž1954., 470]471. 5. N. Kawamoto, Generalizations of Witt algebras over a field of characteristic zero. Hiroshima Math. J. 16 Ž1986., 417]426. 6. H. Kraft, Algebraic automorphisms of affine space, in ‘‘Topological Methods in Algebraic Transformation Groups,’’ pp. 81]105, Birkhauser, Boston, 1989. ¨ 7. J. M. Osborn, New simple infinite dimensional Lie algebras of characteristic 0, J. Algebra 185 Ž1996., 820]835. 8. J. M. Osborn, Derivations and isomorphisms of Lie algebras of characteristic 0, preprint. 9. J. M. Osborn, Automorphisms of the Lie algebras W * in characteristic 0, Canad. J. Math. 49 Ž1997., 119]132. 10. I. M. Singer and S. Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 Ž1965., 1]114.