Quantized Universal Enveloping Algebras andq-de Rham Cocycles

210, 630]663 Ž1998. JA987583

JOURNAL OF ALGEBRA ARTICLE NO.

Quantized Universal Enveloping Algebras and q-de Rham Cocycles Abdellah Sebbar Centre de recherches mathematiques, Uni¨ ersite´ de Montreal, ´ C.P. 6128, Succursale Centre Ville, Montreal, ´ Quebec H3C 3J7, and CICMA, Concordia Uni¨ ersity, Montreal, ´ Quebec ´ H3G 1M8, Canada E-mail: [email protected] Communicated by Susan Montgomery Received January 28, 1998

1. INTRODUCTION In this work a close connection is established between certain cohomology spaces of quantized universal enveloping algebras and a twisted q-de Rham ŽJackson]Aomoto. cohomology of configuration spaces. In the Lie algebra case, the idea of such a connection belongs to V. Ginzburg and V. Schechtman. In w4x and w5x, these authors have constructed canonical morphisms between the de Rham homology of certain local systems over configuration spaces and Ext-spaces between Fock-type modules over Kac]Moody and Virasoro Lie algebras. This construction is, in turn, a generalization of the classical Feigin]Fuchs construction. In their study of representation theory of Virasoro algebras, Feigin and Fuchs have discovered a way of obtaining intertwiners between Fock modules over the Virasoro algebra from the top homology of certain one-dimensional local systems. We investigate this connection between the geometry of configuration spaces and the representation theory in the case of quantum groups. The representations considered here are Verma modules over the quantized enveloping algebras of semisimple Lie algebras. The existence and the uniqueness of these modules were established by Lusztig w9x. We consider a family of operators between the Verma modules that satisfy certain difference equations and certain cocycle conditions. These equations are built using a family of q-difference operators that generate a flat connec630 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

q-DE

RHAM COCYCLES

631

tion in a one-dimensional vector bundle over the n-dimensional torus. In fact, these difference operators are the ‘‘differentials’’ of a ‘‘q-de Rham’’ complex of the space of formal algebraic q-differential forms over the n-torus. The homology groups of this complex can be regarded as the homology groups of the n-torus with coefficients in a local system with stalk C. From these data, we construct the canonical ‘‘q-de Rham’’ cocycles, and consequently we obtain the canonical maps between the homology of the local systems and the Ext-spaces between the Verma modules. This paper is organized as follows. Section 2 is concerned with Hopf algebras; we make some constructions and prove some results that we need later. Two key ingredients are introduced, namely a bracket that will have a major role in all of the work, and a cochain complex of Hochschild type that will lead to the Ext-spaces. In Section 3, we treat the simple case of the algebra Uq Ž s l 2 .. And in Section 4 we treat the general case of a semisimple Lie algebra; we define a sequence of certain ¨ ertex operators that, taken together, define a cocycle in a double complex. This double complex is a mix of the difference de Rham complex and the Hochschild cochain complex with coefficients in a Hom-space between two Verma modules. In the course of the work, we found some nice features of the q-deformed picture. We mention one of them: the appearance of the Kashiwara operators ­ i and i ­ of Lusztig in the solution of the main difference equations. I should mention also that the same construction has $ been made in w13x for the quantum affine algebra Uq Žs l 2., with different techniques. The operators are the so-called screening operators, and the representations are the q-analog of the Wakimoto modules. 2. HOPF ALGEBRAS AND THEIR ACTIONS In this section we present some constructions concerning Hopf algebras and their representations and establish some results related to them. All of the algebraic structures will be over the field of complex numbers. Let H be a Hopf algebra, and let D, A, and « denote, respectively, the comultiplication, the antipode, and the counit maps. These maps satisfy the following axioms, in which we use the Sweedler notation for the comultiplication: DŽ x . s

Ý x9 m x0

Ž xgH..

Ž x.

Ý x9 m Ž x0 . 9 m Ž x0 . 0 s Ý Ž x9. 9 m Ž x9. 0 m x0 Ž x.

Ž x.

Ž Coassociativity axiom. . Ž 2.1.

632

ABDELLAH SEBBAR

Ý x9« Ž x0 . s Ý « Ž x9. x0 s x Ž x.

Ž Counit axiom. .

Ž 2.2.

Ž x.

Ý x9 AŽ x0 . s Ý AŽ x9. x0 s « Ž x . .1 Ž x.

Ž Antipode axiom. . Ž 2.3.

Ž x.

Moreover, A is an antihomomorphism, and « ( A s « . The modules considered in this section are algebra left-modules. If M and N are two H-modules, one can define a structure of left module on both M m N and HomŽ M , N . by x ? Ž m m n. s

Ý Ž x9m . m Ž x0 n .

Ž m g M, n g N, x g H . ,

Ž x.

Ž x ? f . Ž m . s Ý x9 f Ž A Ž x0 . m .

Ž m g M , f g HomŽ M , N . , x g H . .

Ž x.

Each vector space carries a structure of H-module through the map « . Therefore the dual space M * s HomŽ M , C. is an H-module, where the action of H is given by

Ž x ? f . Ž m . s Ý « Ž x9 . f Ž A Ž x0 . m . Ž x.

s

Ý f Ž AŽ « Ž x9. x0 . m . Ž x.

s f Ž AŽ x . m .

using Ž 2.2. .

2.1. Composition of Maps If M , N , and P are three H-modules, we wish to factorize the action of elements of H on maps in HomŽ M , P . that are compositions of maps from HomŽ M , N . and HomŽ N , P .. To simplify the notations, we change the superscripts 9 and 0 to numerical subscripts when more than one is involved. The proof of the following lemma was outlined to me by S. Montgomery. LEMMA 2.1.

For e¨ ery x g H, the following relation holds in H m H m H:

Ý x 1 m 1 m x 2 s Ý x 1, 1 m AŽ x 1, 2 . x 2, 1 m x 2, 2 . Ž x.

Ž 2.4.

Ž x.

Proof. We prove the identity by applying the coassociativity of the map D several times. By coassociativity we have for x g H x 1 m x 2, 1 m x 2, 2 s x 1 , 1 m x 1, 2 m x 2 .

q-DE

633

RHAM COCYCLES

Now, applying 1 m D m 1 to the left side, and 1 m 1 m D to the right side, we obtain by coassociativity x 1 m x 2, 1 , 1 m x 2, 1, 2 m x 2, 2 s x 1, 1 m x 1, 2 m x 2, 1 m x 2, 2 . Hence x 1, 1 m A Ž x 1 , 2 . x 2, 1 m x 2, 2 s x 1 m A Ž x 2, 1, 1 . x 2, 1, 2 m x 2, 2 s x 1 m « Ž x 2, 1 . m x 2, 2 s x1 m 1 m x 2 , using Ž2.2. twice. This proves the lemma. PROPOSITION 2.2 ŽComposition lemma.. If M , N , and P are three H-modules, then for e¨ ery f g HomŽ N , P . and for e¨ ery g g HomŽ M , N . and x g H, we ha¨ e x ? Ž f( g. s

Ý Ž x9 ? f . ( Ž x0 ? g . . Ž x.

Proof. The relation can be written as x 1 f Ž g Ž A Ž x 2 . m . . s x 1, 1 f Ž A Ž x 1, 2 . x 2, 1 g Ž A Ž x 2, 2 m . . .

ŽmgM .,

which follows from Ž2.4. in the above lemma after applying 1 m 1 m A to both sides. The composition lemma seems to be just a consequence of the axiomatic definition of the Hopf algebra, especially from the coassociativity. Let us consider the composition map Ž f, g . ª f ( g. It is a bilinear map and therefore induces a linear map (

Hom Ž N , P . m Hom Ž M , N . ª Hom Ž M , P . . Using the action of H on the tensor product and on the Hom space, we can restate the composition lemma as the following result COROLLARY 2.3. The composition map is H-linear. Remark 2.1. The linearity of the composition map is known and proved in the literature only when N Žor both M and P . is finite-dimensional, in which case HomŽ M , N . is isomorphic to M * m N Žsee w8x.. Here we established it for the general case, because all of the representations we will be considering are infinite-dimensional.

634

ABDELLAH SEBBAR

2.2. A Bracket and a Cochain Complex Let H be a Hopf algebra over C with the associated maps as above. We define a bilinear map ² ? , ? : on H m H by ² x, y : s

Ý x9 yAŽ x0 . y « Ž x . y

Ž x, y g H . .

Ž 2.5.

Ž x.

This bracket satisfies the following relations: PROPOSITION 2.4.

For all x, y, z in H, we ha¨ e

Ž1. ² xy, z : s ² x, ² y, z :: q e Ž x .² y, z : q « Ž y .² x, z :. Ž2. « Ž² x, y :. s 0. Ž3. A2 Ž² x, y :. s ² A2 Ž x ., A2 Ž y .:. Proof. The first relation follows since D is an algebra homomorphism and A is an algebra antihomomorphism. The second relation follows from Ž1.3. and from « being an algebra homomorphism. We will prove the third relation: we use the identity DŽ AŽ x .. s ÝŽ x . AŽ x0 . m AŽ x9., the fact that A is an antihomomorphism, and that « Ž AŽ x .. s « Ž x .. We have D Ž A2 Ž x . . s

Ý A2 Ž x . 9 m A2 Ž x . 0 s Ý AŽ AŽ x . 0 . m AŽ AŽ x . 9. s Ž A m A . Ž Ý AŽ x . 0 m AŽ x . 9 . s Ž A2 m A2 . Ž Ý x9 m x0 . s

Ý A2 Ž x9. m A2 Ž x0 . .

Therefore, ² A2 Ž x . , A2 Ž y . : s

Ý A2 Ž x9. A2 Ž y . A Ž A2 Ž x0 . . y « Ž A2 Ž x . . A2 Ž y .

s A2 Ž Ý x9 yA Ž x0 . . y « Ž x . A2 Ž y . s A2 Ž ² x, y : . .

Note that ²1, x : s ² x, 1: s 0 for every x g H. And if H is commutative, then ² x, y : s 0 for every x, y g H, whereas if H is cocommutative, then ² x, A Ž y . : s A² x, y : .

q-DE

RHAM COCYCLES

635

Let M be a left H-module. Following w11x, we call m g M an Hin¨ ariant if xm s « Ž x . m for every x g H. If N is another H-module, we set ² x, f : s x ? f y « Ž x . f

x g H,

f g Hom Ž M, N . .

Ž 2.6.

Notice the analogy with the definition of ² x, y :. We say that f is an in¨ ariant if ² x, f : s 0 for every x g H. PROPOSITION 2.5. ha¨ e

For all x, y in H and for all f , c in HomŽ M, N . we

² xy, f : s x ? ² y, f : q « Ž y . ² x, f : . ² x, fc : s

Ý ² x9, f :² x0 , c : q ² x, f :c q f ² x, c : .

Ž 2.7. Ž 2.8.

Ž x.

Proof. The first relation is the same as the relation Ž1. in the above proposition when we substitute f for z. The second relation is a consequence of the composition lemma and the fact that « Ž x . s ÝŽ x . « Ž x9. « Ž x0 ., which follows from Ž2.2.. Remark 2.2. The importance of the bracket ² ? , ? : will appear throughout this work. For Lie algebras, the expression w x, f x defined by w x, f xŽ m. s x f Ž m. y f Ž xm., where f is a homomorphism between two modules, defines a left action of the Lie algebra on the Hom-space. This action is enough to define homological sequences, e.g., Koszul complexes. For our case, we are dealing with associative algebras that need two actions Žleft and right. to define homological sequences, e.g., Hochshild complexes Žsee below.. This explains for the moment the choice of the bracket ² x, f :, which we define as the difference of two actions of x on f . In the case of the universal enveloping algebra, this bracket coincides with the Lie bracket, and for the quantized versions of these algebras, the appearance of the trivial action will emphasize the role of the group-like elements, as will be seen in the next sections. Let M be a H-module, and let us consider the following sequence: C s C Ž H m , M . : 0 ª M ª Hom Ž H , M . ª ??? v

v

ª Hom Ž H mn , M . ª ??? , and the linear map d : Hom Ž H mny1 , M . ª Hom Ž H mn , M . ,

636

ABDELLAH SEBBAR

defined as follows: If f g HomŽ H mny1, M . and x 1 m x 2 m ??? m x n g H mn, then df Ž x 1 , x 2 , . . . , x n . s x 1 ? f Ž x 2 , . . . , x n . ny1

q

Ý Ž y1. i f Ž x 1 , . . . , x iy1 , x i x iq1 , x iq2 , . . . , x n .

is1 n

q Ž y1 . f Ž x 1 , x 2 , . . . , x ny1 . « Ž x n . . One can look at this sequence as a Hochshild complex of the associative algebra H with a left and a right action that commute on the space M . The right action is given by the trivial action, i.e., by « w1x. It follows that Ž C , d . is a cochain complex, i.e., d 2 s 0. One can choose the coefficients of the cochains in HomŽ M , N ., where M and N are two H-modules. Thus we obtain a complex, v

C Ž H , M , N . s Hom Ž H m , Hom Ž M , N . . . v

v

If f g HomŽ M , N ., then d f Ž x . s ² x, f :. Hence, d f s 0 implies that ² x, f : s 0 for all x g H. It follows that the 0th cohomology space is the space of H-invariants. We will see that in the case of quantum groups, the space of invariants coincides with the space of intertwiners. More generally, the cohomology spaces are the Ext-spaces Ext H Ž M , N .. v

3. THE ALGEBRA Uq Ž s l 2 ., INTERTWINERS, AND COCYCLES 3.1. The Main Constructions Let q be a nonzero complex parameter that is not a root of unity. The quantum group Uq s Uq Ž s l 2 . is the associative algebra generated by four variables E, F, K " 1 and the relations KKy1 s Ky1 K s 1,

Ž 3.1.

KE s q EK ,

Ž 3.2.

KF s qy2 FK ,

Ž 3.3.

2

w E, F x s

K y Ky1 q y qy1

.

Ž 3.4.

q-DE

637

RHAM COCYCLES

The associative algebra Uq has the structure of a Hopf algebra. Comultiplication, counit, and antipode are given by DŽ E. s 1 m E q E m K , D Ž F . s Ky1 m F q F m 1, DŽ K "1 . s K "1 m K "1,

« Ž K "1 . s 1

« Ž E . s « Ž F . s 0 and and A Ž E . s yEKy1 ,

AŽ K " 1 . s K . 1 .

A Ž F . s yKF,

If l is a nonzero complex number, M Ž l. will denote the Verma module over Uq with highest weight l. This module is generated by a nonzero vector ¨l satisfying E¨l s 0,

K¨l s q l ¨l .

The action of the generators of Uq on the Verma module M Ž l. is summarized in the following proposition, which one can easily prove by induction. PROPOSITION 3.1.

Let ¨l be the highest weight ¨ ector of M Ž l.. Then

K n F a ¨l s q nŽ ly2 a. F a ¨l

Ž a g N, n g Z . ,

ny1

E n F a ¨l s

Ł w a y k xw l y a q k q 1x F ayn ¨l

Ž a g N, n g N . ,

ks0

where

w ax s

a q y ayq q y qy1

.

With the con¨ ention that F a ¨l s 0 if a - 0. Let l and l9 g C, and let us examine the C-linear map Vn : M Ž l9 y 1 . ª M Ž l y 1 .

Ž n g N. ,

given by Vn Ž F a ¨l9y1 . s F aq n ¨ly1

Ž a g N, n g N . .

By direct computation, we obtain PROPOSITION 3.2. The pairing of Vn with the generators of Uq is gi¨ en by Ži. ² E, Vn :Ž F a ¨l9y1 . s qyl 9q2 aq1 Žw a q n xw l y a y n x y w axw l9 y ax. F aq ny1 ¨ly1 ,

638

ABDELLAH SEBBAR

Žii. ² F, Vn :Ž F a ¨l9y1 . s Ž1 y q l9y lq2 n . F aq nq1 ¨ly1 , Žiii. ² K m , Vn :Ž F a ¨l9y1 . s Žy1 q q mŽ ly l9y2 n. . F aq n ¨ly1 , for e¨ ery n, a g N and m g Z. If a g C, we define a twisted differential da : Cww z xx ª Cww z xx dzrz linearly by da Ž z n . s w n q a x z n

dz z

,

where z is a formal variable. Let M Ž l y 1.ww zy1 xx be the module of Laurent series in zy1 with coefficients in M Ž l y 1. and consider the operator VŽ z. s

Ý Vn zyn y1dz : M Ž l9 y 1. ª M Ž l y 1. w zy1 x nG0

dz z

.

We would like to find a number a g C and an operator V Ž E, z . s

Ý Vn Ž E . zyn : M Ž l9 y 1. ª M Ž l y 1. w zy1 x nG0

such that ² E, V Ž z . : s da V Ž E, z . .

Ž 3.5.

This equation is equivalent to ² E, Vn : s w yn q a x Vn Ž E .

Ž n g N. .

Applying this to F a ¨l9y1 for a nonnegative integer a, we obtain

w yn q a x Vn Ž F a ¨l9y1 . s qyl 9q2 aq1 Ž w a q n xw l y a y n x y w a xw l9 y a x . F aq ny1 ¨ly1 . We look for a number a9 depending on a such that, for every n, we have

w a q n xw l y a y n x y w a xw l9 y a x s w yn q a xw n q a9 x . After multiplication by Ž q y qy1 . 2 , the right side gives yq 2 ny aqa9 y qy2 nq aya9 q q aqa9 q qya ya9 , and the left side gives q 2 nq2 ay l y qy2 ny2 aq l q q 2 ay l9 q qy2 aq l9 q q l q qyl y q l9 y qyl 9 .

q-DE

639

RHAM COCYCLES

Identifying the powers containing 2 n Ž q is not a root of unity., we obtain ya q a9 s 2 a y l; hence a9 s 2 a q a y l. Now identifying the powers containing 2 a, we get a q a9 s 2 a y l9. This gives 2 a s l y l9. We must also have q l q qyl s q l9 q qyl 9, which gives q l s q " l9. If q l s q l9, then we are dealing with the same Verma module, since it is q l that is involved in the highest weight condition. Therefore without loss of generality, we can assume that l s "l9. From now on, we suppose l s yl9; hence a9 s 2 a and a s l. Thus ² E, Vn : s w yn q l x Vn Ž E . , where Vn Ž E . Ž F a ¨ y ly1 . s q lq2 aq1 w n q 2 a x F aq ny1 ¨ly1 .

Ž 3.6.

By doing the same for the other generators F, K, Ky1 , and using Proposition 3.2, we have PROPOSITION 3.3. define operators

For X s E, F, K " 1 and for e¨ ery n g N, one can Vn Ž X . : M Ž yl y 1 . ª M Ž l y 1 .

gi¨ en by Vn Ž E . Ž F a ¨ y ly1 . s q lq2 aq1 w n q 2 a x F aq ny1 ¨ly1 , Vn Ž F . Ž F a ¨ y ly1 . s q ny l Ž q y qy1 . F aq nq1 ¨ly1 , Vn Ž K " 1 . Ž F a ¨ y ly1 . s yq " Žynq l . Ž q y qy1 . F aq n ¨ly1 , and which satisfy ² X , Vn : s w yn q l x Vn Ž X . . Remark 3.1. If we omit the term with « Ž x . in the definition of the pairing ² ? , ? :, VnŽ K " 1 . cannot be defined; at the same time, K " 1 are the only generators for which « is not zero. Next, we need to define the operator VnŽ x . for every x g Uq . PROPOSITION 3.4. Let f be the free associati¨ e algebra generated by E, F, and K " 1. Then for e¨ ery x g f and n g N, there exists an operator VnŽ x .: M Žyl y 1. ª M Ž l y 1., which satisfies ² x, Vn : s w yn q l x Vn Ž x . .

Ž 3.7.

Proof. Assume that for x and y in f, and for every n g N, one can define VnŽ x . and VnŽ y . satisfying Ž3.7.. Then for every n g N one has ² xy, Vn : s x ? ² y, Vn : q « Ž y . ² x, Vn:

using Ž 2.7.

s w yn q l x Ž x ? Vn Ž y . q « Ž y . Vn Ž x . . .

640

ABDELLAH SEBBAR

We set VnŽ xy . s x ? VnŽ y . q « Ž y .VnŽ x .. Then we have ² xy, Vn : s wyn q lxVnŽ xy .. Since, for x, a generator of f, VnŽ x . exists and satisfies Ž3.7., the proposition follows. Now we extend the definition of VnŽ x . to Uq . PROPOSITION 3.5. For e¨ ery x in Uq and for e¨ ery n g N, there exists an operator VnŽ x .: M Žyl y 1. ª M Ž l y 1. satisfying the relation Ž3.7.. Proof. Recall that on f we have Vn Ž xy . s x ? Vn Ž y . q « Ž y . Vn Ž x . .

Ž 3.8.

In view of the above proposition, we need to prove that this relation is compatible with the defining relations of the algebra Uq . For the relation Ž3.1., we have Vn Ž KKy1 . s K ? Vn Ž Ky1 . q « Ž Ky1 . Vn Ž K . s KVn Ž Ky1 . Ky1 q Vn Ž K . . Hence Vn Ž KKy1 . Ž F a ¨ y ly1 . s KVn Ž Ky1 . Ž q 2 aq lq1 F a ¨ y ly1 . q qynq l Ž q y qy1 . F aq n ¨ly1 s Ž yq 2 aq lq1 q ny l Ž q y qy1 . K q qynq l Ž q y qy1 . . F aq n ¨ly1 s 0. On the other hand, it is clear that VnŽ1. s 0. Similar calculations hold for Ky1 K s 1. For the relation Ž3.2., we have Vn Ž KE . Ž F a ¨ y ly1 . s K ? Vn Ž E . Ž F a ¨ y ly1 . q « Ž E . Vn Ž K . Ž F a ¨ y ly1 . s KVn Ž E . Ky1 Ž F a ¨ y ly1 . q 0 s q 2 aq3 ly2 nq3 w n q 2 a x F aq ny1 ¨ly1 and Vn Ž EK . Ž F a ¨ y ly1 . s Ž E ? Vn Ž K . q Vn Ž E . . Ž F a ¨ y ly1 . s yVn Ž K . EKy1 F a ¨ y ly1 q Vn Ž K . Ky1 F a ¨ y ly1 q Vn Ž E . F a ¨ y ly1 .

q-DE

RHAM COCYCLES

641

The coefficient of F aq ny1 ¨ yl y1 is q 2 aq lq1 Ž Ž q y qy1 . qynq l w a xw l q a x q Ž q y qy1 . qynq l w a q n xw l y a y n x q w n q 2 a x . s q 2 aq lq1

1 q y qy1

Ž qynq2 lq2 a y qy3 nq2 ly2 a .

s q 2 aq3 ly2 nq1 w n q 2 a x . Therefore Vn Ž q 2 EK . Ž F a ¨ y ly1 . s q 2 aq3 ly2 nq3 w n q 2 a x F aq ny1 ¨ly1 s Vn Ž KE . Ž F a ¨ y ly1 . . The compatibility with Ž3.3. is checked in the same way. Finally, Vn

ž

K y Ky1 q y qy1 s

/

Ž F a ¨ y ly1 .

1 q y qy1

Ž qyn q l Ž q y qy1 . q q ny l Ž q y qy1 . . F aqn ¨ly1

s Ž q ny l q qynq l . F aq n ¨ly1 . On the other hand: Vn Ž w E, F x . s yVn Ž F . EKy1 q EVn Ž F . Ky1 q Ky1 Vn Ž E . KF y FVn Ž E . . Applying this to Ž q y qy1 . F a ¨ yl y1 , we obtain successively the following factors as coefficients of F aq n ¨ly1:

Ž q 3 aqnq1 y q aq nq1 .Ž q lqa y qyl ya . , Ž q 3 aq2 nq2 y q a .Ž q lyŽ aqnq1. y qyl qŽ aqnq1. . , qyl q2 aq2 nq1 Ž q nq 2 aq2 y qyny2 ay2 . , and

q lq2 aq1 Ž q nq 2 a y qyny2 a . .

Summing these expressions, we get

Ž q y qy1 . Vn Ž w E, F x . F a ¨ y ly1 s Ž qyl qn Ž q y qy1 . q q lyn Ž q y qy1 . . F aq n ¨ly1 , which gives the same expression as VnŽ K y Ky1 ..

642

ABDELLAH SEBBAR

Recall that we have set VŽ z. s

Ý Vn zyn y1dz :

dz

M Ž yl y 1 . ª M Ž l y 1 . w zy1 x

z

nG0

V Ž x, z . s

M Ž l y1 . w zy1 x M Ž y l y1 . ªM

Ý Vn Ž x . zyn :

,

Uq . . Ž xgU

nG0

Using the definition of the twisted differential dl and the previous propositions, we have For e¨ ery x g Uq , there exists an operator

THEOREM 3.6. V Ž x, z . s

Ý Vn Ž x . zyn :

M Ž yl y 1 . ª M Ž l y 1 . w zy1 x

nG0

linearly dependent on x such that ² x, V Ž z . : s dlV Ž x, z . .

Ž 3.9.

Moreo¨ er, for x, y in Uq , we ha¨ e V Ž xy, z . s x ? V Ž y, z . q « Ž y . V Ž x, z . .

Ž 3.10.

3.2. Intertwiners and Cocycles We consider the complex of length one: dl

V : 0 ª V 0 ª V 1 ª 0, v

where V 0 s C w zy1 x ,

V 1 s C w zy1 x

dz z

.

Recall that dl is defined linearly by dl Ž zyn . s w l y n x zyn

dz z

.

The length one is due to the fact that s l 2 has one simple root. From this complex and the cochain complex C introduced in the first section, we construct the following bigraded space: v

C i j s C i j Ž Uq , M Ž yl y 1 . , M Ž l y 1 . . s Hom Ž Uqmi , Hom Ž M Ž yl y 1 . , M Ž l y 1 . m V j . . for i, j ) 0, where V j s 0 for j G 2.

q-DE

RHAM COCYCLES

643

Note that C i j is isomorphic to HomŽ Uqmi m M Žyl y 1., M Ž l y 1. m V . The bigraded space C i j has a natural structure of a bicomplex. The first differential d9: C i j ª C iq1, j is induced by the differential d introduced in the first section. The second differential d0: C i j ª C i, jq1 is induced by the differential dl of V . The operator V Ž z . is an element of C 01 , and we denote it by V 01 Ž z .. The operator V Ž x, z .Ž x g Uq . defines an element V 10 Ž z . of the space C 10 given by j.

v

V 10 Ž z . Ž x . s V Ž x, z . . PROPOSITION 3.7. The elements V 01 Ž z . and V 10 Ž z . satisfy d9V 01 Ž z . s d0 V 10 Ž z . ,

Ž 3.11.

d9V 10 Ž z . s 0.

Ž 3.12.

Proof. For x g Uq one has, by the definition of d9, d9 Ž V 01 Ž z . . Ž x . s x ? V 01 Ž z . y « Ž x . V 01 Ž z . s ² x, V 01 Ž z . : , and d0 V 10 Ž z .Ž x . s dlV Ž x, z .. Thus, the first relation is simply a consequence of the relation Ž3.9. of Theorem 3.6. And for x, y g Uq , one has d9 Ž V 10 Ž z . . Ž x m y . s x ? V 10 Ž z . Ž y . y V 10 Ž z . Ž xy . q « Ž y . V 10 Ž z . Ž x . s x ? V Ž y, z . y V Ž xy, z . q « Ž y . V Ž x, z . s 0, using the relation Ž3.10. of Theorem 3.6. Let C s C Ž Uq , M Žyl y 1., M Ž l y 1.. denote the simple complex associated with the double complex C , that is, v

v

vv

Cn s

[

C ab

Ž n g Z. .

aqbsn

Its differential d is defined by w1x: a

d N C a b s d9 q Ž y1 . d0 . THEOREM 3.8. The element Ž V 01 Ž z ., V 10 Ž z .. is a 1-cocycle of the complex C . v

Proof. The element Ž V 01 Ž z ., V 10 Ž z .. is in C 1 Ž Uq , HomŽ M Žyl y 1., Ž M l y 1.... Applying d, we get d Ž V 01 Ž z . q V 10 Ž z . . s d9V 01 Ž z . q d0 V 01 Ž z . q d9V 10 Ž z . y d0 V 10 Ž z . .

644

ABDELLAH SEBBAR

And we have d0 V 01 Ž z . s 0, since V is of length one. Using Proposition 3.7, we have d9V 10 Ž z . s 0 and d9V 01 Ž z . s d0 V 10 Ž z .. Hence v

d Ž V 01 Ž z . q V 10 Ž z . . s 0.

Let us consider again the complex V , a monomial zyn g Ker dl if and only if wyn q lx s 0. Since q is not a root of unity, this is equivalent to l s n. Thus the complex V is acyclic if l is not an integer. We assume that l is a nonnegative integer for the rest of this section. Thus the space H 0 Ž V . is a one-dimensional space generated by the function zyl . The space H 1 Ž V . is generated by the class of the form zyl dzrz. If we consider the homology spaces Hi s H i *, then H1 is a one-dimensional space generated by the linear form v

v

v

v

V1 ª C

v ¬ Res zs0 Ž v z l . . The space H

0

is generated by the linear form V0 ª C f Ž z . ¬ Res zs0 f Ž z . z l

ž

dz z

/

.

PROPOSITION 3.9. The operator Res zs0 Ž V 01 Ž z l .. in Hom C Ž M Žy l y1., Ž M l y 1.. is an intertwiner. Proof. From the first section, we know that C 0 is a space of Uqinvariants. We need to show that in fact it coincides with the space of intertwiners in our case. Let f g C 0 , i.e., d f s 0. We need to show that f intertwines with the generators E, F, and K " 1. Let m g M Žyl y 1.; since ² E, f :Ž m. s 0, we have yf Ž EKy1 m. q Ef Ž Ky1 m. s 0. Therefore f Ž EKy1 m. s Ef Ž Ky1 m.. This shows that E intertwines with f Žnote that Ky1 is invertible.. Now, if ² Ky1 , f :Ž m. s 0, then Ky1f Ž Km. y f Ž m. s 0, which shows that K intertwines with f . The intertwining property for Ky1 follows from the invariance of f with K. Finally, if ² F, f :Ž m. s 0, then yKy1f Ž KFm. q Ff Ž m. s 0; since K intertwines with f , we see that F does too. Remark 3.2. The operator Res zs0 Ž V 01 Ž z l ..: M Žyl y 1. ª M Ž l y 1. is the unique Uq-homomorphism sending ¨ y ly1 to F l ¨ly1. Hence it is nontrivial Žnotice that F l ¨ yl y1 is also a singular vector..

q-DE

645

RHAM COCYCLES

From the discussion preceding the proposition, we have the following: COROLLARY 3.10. We ha¨ e H1 Ž V . ( H 0 Ž Uq , Hom Ž M Ž yl y 1 . , M Ž l . . . . v

PROPOSITION 3.11. The operator Res zs0 Ž V 10 Ž z . dzrz . is a nontri¨ ial element of the space Ext 1U qŽ M Žyl y 1., M Ž l y 1... Proof. The operator Res zs0 Ž V 01 z ldzrz . is in the space HomŽ Uq , HomŽ M Žyl y 1., M Ž l.... By Theorem 3.8 it is a 1-cocycle of the algebra Uq with coefficients in the Uq-module Hom C Ž M Žyl y 1., M Ž l y 1..; hence it defines an element of Ext 1U qŽ M Žyl y 1., M Ž l y 1... It is a nontrivial element; indeed, V 10 Ž z . z l

dz z

Ž E. s

Ý q lq1 w n x zynq l nG0

dz z

.F ny1 ¨ly1 .

ŽRecall that VnŽ E .Ž ¨ y ly1 . s q lq1 w n x F ny 1 ¨ly1¨ly1 .. Therefore Res zs0 V 10 Ž z . z l

ž

dz z

/Ž

E . Ž ¨ y ly1 . s q lq1 w l x F ly1 ¨ly1 .

The right side is not zero since l is a nonzero integer and q is not a root of unity. Remark 3.3. The case when q is a root of unity does not present a significant difference with the generic case, except for the fact that the homology spaces are not one-dimensional, and all of the above constructions can be carried out, obtaining an infinite family of linearly independent cocycles w12x.

4. GENERALIZATION TO THE DEFORMATION OF A SEMISIMPLE LIE ALGEBRA 4.1. The Quantum Group Uq Ž g . Root Systems Let Ž a i j .1 F i, jF N be an n = n indecomposable matrix with integer entries such that a i i s 2 and a i j F 0 for i / j, and let Ž d1 , . . . , d N . be a vector with relatively prime entries such that the matrix Ž d i a i j . is symmetric and positive definite. Notice that Ž a i j . is a Cartan matrix of a simple finite-dimensional Lie algebra g. Let h be a Cartan subalgebra, P s  a i ,

646

ABDELLAH SEBBAR

1 F i F N 4 the corresponding root system, and P ks  a ik , 1 F i F N 4 the corresponding coroot system. Ž h, P, P k. is called a realization of g w7x. There exists a nondegenerate symmetric bilinear C-valued form Ž., .. on h satisfying

Ž a ik , h . s ² h, a i : dy1 i

for h g h ,

i s 1, . . . , N.

Here ²., .: is the natural pairing between h* and h. Since Ž., .. is nondegenerate, there is an isomorphism m : h ª h* defined by ² h1 , m Ž h . : s Ž h1 , h .

Ž h1 , h g h . .

This isomorphism induces a symmetric Žnondegenerate. bilinear form Ž., .. on h*. Thus we have

m Ž a ik . s dy1 i ai ,

i s 1, . . . , N,

and

Ž a i , a j . s d i ai j s d j a ji . Let r be the element of h* defined by ² a ik , r : s 1,

i s 1, . . . , N.

Then r satisfies Ž r , a i . s d i . We define also the fundamental reflexions ri , 1 F i F n, of h* by ri Ž l . s l y ² a ik , l:

Ž l g h* . .

In particular, ri Ž a j . s a j y a i j a i . Gaussian Binomial Coefficients Let q be an indeterminate. For n g Z, d g N, we define the q-integer w n x d by

w nx d s

q d n y qyd n q d y qyd

.

If d s 1, we denote it simply by w n x, and we have w n x d s w nd xrw d x. We also set n

w n x d !s Ł w s x d . ss1

And we define the q-binomial coefficients n j

s d

w d x d ??? w n y j q 1 x d w j xd!

for j g Z,

j F n,

q-DE

647

RHAM COCYCLES

and n j

s0

if j ) n.

d

We have n j

s Ž y1 .

j

d

yn q j y 1 j

d

and ny1

n

Ł Ž1 q q 2 s d z . s Ý q d jŽ ny1.

ss0

js0

n j

Ž n G 0. ,

zj

Ž 4.1.

d

where z is another indeterminate. It follows that w nj x d g Zw q, qy1 x. If m and n are in Z and j g N, then mqn j

s d

Ý

q dŽ m lyn k .

kqlsj

m k

d

n l

. d

By putting z s y1 in Ž4.1. and using w n x d s w n xyd for integer n, we obtain n

n j

Ý q d jŽ1yn. js0

s 0.

Ž 4.2.

d

The Drinfeld]Jimbo Algebra Uq w2x, w6x We assume that q is a generic complex number, and we set qi s q d i . We consider the algebra Uq defined by the generators Ei , Fi , K i" 1 Ž1 F i F n. and the relations K i Ky1 s Ky1 i i K i s 1,

Ki Kj s Kj Ki ,

K i Ej s qia i j Ej K i ,

ijF K , K i Fj s qya i j i

Ei , Fj s d i j 1ya i j s Ý Ž y1.

ss0 1ya i j s Ý Ž y1.

ss0

1 y ai j s

di

1 y ai j s

di

K i y Ky1 i qi y qy1 i

Ž 4.3. Ž 4.4.

,

Ž 4.5.

Ei1y a i jys Ej Eis s 0

if i / j,

Ž 4.6.

Fi1y a i jys Fj Fi s s 0

if i / j.

Ž 4.7.

648

ABDELLAH SEBBAR

The last two relations are referred to as the Serre relations. There is a unique algebra involution v : Uq ª Uq such that v Ž Ei . s Fi , v Ž Fi . s Ei , v Ž K i . s Ky1 i . The associative algebra Uq has a Hopf algebra structure given by w9x: D Ž Ei . s Ei m 1 q K i m Ei , D Ž Fi . s Fi m Ky1 q 1 m Fi , i DŽ Ki . s Ki m Ki ,

« Ž Ei . s « Ž Fi . s 0, A Ž Ei . s yKy1 i Ei ,

y1 D Ž Ky1 m Ky1 i . s Ki i ,

« Ž K i . s « Ž Ky1 i . s 1,

A Ž Fi . s yFi K i ,

A Ž K i" 1 . s K i. 1 .

For i s 1 . . . N. 4.2. Differentials and Operators The Maps ­ i and i ­ Following w9x, we let f9 be the free algebra with 1 generated by the Fi ’s. Let Zw P x be the root lattice and Nw P x be the submonoid of Zw P x of all linear combinations of elements of P with coefficients in N. For any a s Ýa i a i in Nw P x, we denote by fXa the subalgebra of f9 spanned by monomials Fi1 Fi 2 ??? Fi r such that for any i, the number of occurrences of i in the sequence i1 , i 2 , . . . , i r is equal to a i . Each fXa is a finite-dimensional vector space, and we have a direct sum decomposition f9 s [ a fXa , where a runs over Nw P x. We also have fXa fXa 9 ; fXaq a 9 , 1 g fX0 , and Fi g fXa i . An element x g f9 is said to be homogeneous if it belongs to fXa for some a; we then set < x < s a. We denote by i ­ the linear map i ­ : f9 ª f9 such that i­

Ž 1 . s 0,

i­

Ž Fj . s di , j

for all j

and i­

Ž xy . si ­ Ž x . y q q Ž < x < , a i . x i ­ Ž y .

for all homogeneous x, y. Similarly, we denote by ­ i the linear map ­ i : f9 ª f9 such that

­ i Ž 1 . s 0,

­ i Ž Fj . s d i , j

for all j

and

­ i Ž xy . s q Ž < y < , a i .­ i Ž x . y q x ­ i Ž y .

q-DE

649

RHAM COCYCLES

for all homogeneous x, y. These maps are examples of the so-called Kashiwara operators w9x. If x g fXa , then i ­ Ž x . and ­ i Ž x . are in fXay a i if a i G 1 and i ­ Ž x . s ­ i Ž x . s 0 if a i s 0. In w9x it is shown that the maps i ­ and ­ i stabilize the radical I of a certain bilinear inner product on f9; this radical turns out to contain Ževen to be generated by. the Serre relations. Therefore they are also defined on I . Here we will check directly that i ­ and ­ i conserve the the quotient f9rI Serre relations, because the inner product will not be of any use in this section. PROPOSITION 4.1.

For k, i, and j in  1, . . . , N 4 , i / j, we ha¨ e

1ya i j k­

žÝ žÝ

Ž y1.

s

ss0

1ya i j

­k

ss0

Ž y1.

s

1 y ai j s

di

1 y ai j s

di

Fi1y a i jys Fj Fi s s 0,

Ž 4.8.

Fi1y a i jys Fj Fi s s 0.

Ž 4.9.

/ /

Proof. A simple induction shows that k­

Ž Fjn . s ­ k Ž Fjn . s d k j q jny1 w n x j Fjny 1

Ž n g N. .

It follows that for k / i and k / j, the proposition is clear. If k s i: i­

Ž Fi1y a ys Fj Fis . ij

s q Ž < F i < , a i .i ­ Ž Fi1y a i jys . Fi s q Fi1y a i jys Fji ­ Ž Fi s . s

s q Ž < F i < , a i .qŽ < F j < , a i .k ­ Ž Fi1ya i jys . Fj Fi s q qisy1 w s x i Fi1y a i jys Fj Fisy1 . s

Since < Fi s < s s a i , Ž a i , a i . s 2 d i , < Fj < s a j , and Ž a j , a i . s d i a i j , we have i­

Ž Fi1y a ys Fj Fis . ij

s qis 1 y a i j y s i Fiya i jys Fj Fi s q qisy1 w s x i Fi1ya i jys Fj Fi sy1 . At this point we set a s 1 y a i j , and we have to show that a

Ý Ž y1. ss0

s

a s

di

Ž qis w a y s x i Fiaysy1 Fj Fis q qisy1 w s x i FiaysFj Fisy1 . s 0.

650

ABDELLAH SEBBAR

The left-hand side is equal to ay1

a s

s Ý Ž y1.

ss0

di

w a y s x i qis Fiaysy1 Fj Fis

a

q

a s

di

s Ý Ž y1.

a s

Ý Ž y1.

s

ss1 ay1

s

ss0

w s x i qisy 1 Fiays Fj Fisy1

ay1

y

di

w a y s x i qis Fiaysy1 Fj Fis

a sq1

s Ý Ž y1.

ss0

di

w s q 1 x i q s Fiaysy1 Fj Fis

which is equal to 0 because w as x d i w a y s x i s w s qa 1 x d i w s q 1x i . If k s j: j­

Ž Fi1y a ys Fj Fis . s q Ž < F < , a .­ j Ž Fi1y a ys Fj . Fis s i

ij

j

ij

s q js a i j Fi1y a i j . It follows that j­

žÝ

1 y a i j Ž y1 .

s

ss0 1ya i j

s

žÝ

Ž y1.

ss0

s

1 y ai j s

di

1 y ai j s

di

Fi1y a i jys Fj Fis

/

/

q s a i j Fi1ya i j ,

which is equal to 0 by Ž4.2.. This proves Ž4.8.. The relation Ž4.9. is obtained in the same way. COROLLARY 4.2. The maps ­ i and i ­ extend to well-defined linear maps on the algebra f generated by Fi Ž1 F i F N . satisfying the Serre relations. The following proposition will be useful in all that follows. PROPOSITION 4.3.

For x g f homogeneous and 1 F i F N, we ha¨ e K i x s qyŽ < x < , a i . xK i Ei x s xEi q

K ii ­ Ž x . y ­ i Ž x . qi y qy1 i

Ž 4.10. Ky1 i

Ž 4.11.

q-DE

651

RHAM COCYCLES

Proof. The relation Ž4.10. is clear for x s 1, and for x s Fj it follows from Ž4.4., since Ž< x <, a i . s d i a i j and q Ž < x <, a i . s qia i j. Assume that Ž4.10. is true for homogeneous x9 and x0 in f; then K i Ž x9 x0 . s qyŽ < x 9< , a i . x9K i x0 s qyŽ < x 9
since < x9 x0 < s < x9 < q < x0 < .

Therefore Ž4.10. is true for any homogeneous x g f. The relation Ž4.11. is a version of Proposition 3.1.6 in w9x. We will prove it using Ž4.10.. For x s 1 it is clear, and for x s Fj it follows from Ž4.5., since ­ i Ž x . si ­ Ž x . s d i j and 1

Ž K ii ­ Ž x . y ­ i Ž x . Ky1 i . s di j q y qy1 i

i

K i y Ky1 i qi y qy1 i

.

Assume that Ž4.11. is true for homogeneous x9 and x0 in f; then Ei x9 x0 y x9 x0 Ei s x9Ei x0 q s

1 qi y qy1 i q

s

1 qi y qy1 i

x9 Ž K ii ­ Ž x0 . y ­ i Ž x0 . Ky1 i .

1 qi y qy1 i 1

qi y qy1 i

Ž K ii ­ Ž x9. y ­ i Ž x9. Ky1 i . x0 y x9 x0 Ei

Ž K i i ­ Ž x9. y ­ i Ž x9. Ky1 i . x0

Ž q Ž < x 9< , a .K i xXi ­ Ž x0 . y x9­ i Ž x0 . Ky1 i i

qK ii ­ Ž x9 . x0 y q Ž < x 0 < , a i .­ i Ž x9 . x0 Ky1 i . s

K i i ­ Ž x9 x0 . y ­ i Ž x9 x0 . Ky1 i qi y qy1 i

.

Therefore Ž4.11. is true for x9 x0. This completes the proof. Remark 4.1. Since ­ i and i ­ are linear, the relation Ž4.11. is in fact valid for every x g f.

652

ABDELLAH SEBBAR

The Operators Vi Ž z . and Vi Ž x, z . The notations used here are those of the previous subsection. Fix l g h* and let M Ž l. denote the Verma module over Uq of highest weight l y r and highest weight vector ¨l satisfying, for i s 1, . . . , N, Ei ¨l s 0, k

K i ¨l s q Ž ly r , a i . ¨l s qi² ly r , a i : ¨l . The fundamental reflexions ri of h* were defined by ri l s l y ² l, a ik: a i . We fix an index i g  1, . . . , N 4 for the remainder of this section, and we consider l9 s ri l. For j s 1, . . . , N, we introduce the following q-numbers associated with an integer n: k

m i"j

Ž n. s

qi" a i j Ž² l , a i ² l,

:yn.

a ik :

y1 m" i j Ž n . s Ž qi y qi .

y1

yn

if n / ² l , a ik : ,

i

qi" a i j y 1

if n s ² l , a ik : ,

2

and

hi j Ž n . s

Ž yn q ² l , a ik : . a ji yn q ² l , a ik :

j

k

l, a j q 1y² j

:q ² l , a ik : a ji

.

i

We have k

l, a j hi j Ž n . s q 1y² j

:q ² l , a ik : a ji

y mq i j y mi j

q j y qy1 j

and lim hi j Ž n . s lim

qª1

qª1

y mq i j y mi j

q j y qy1 j

s a ji .

We treat these expressions as deformations of the entries of the matrix Ž a i j . i, j . For each n G 0, we define an operator Vi, n : M Ž l9. ª M Ž l. by Vi , n Ž x¨l9 . s xFin ¨l

Ž x g f. .

And for each x, a generator of Uq , we define the operator Vi , n Ž x . : M Ž l9 . ª M Ž l . ,

q-DE

653

RHAM COCYCLES

given by Vi , n Ž Ej . Ž x¨l9 . s hi j Ž n . ­ j Ž x . Fin ¨l q x j ­ Ž Fin . ¨l , s hi j Ž n . ­ j Ž x . Fin ¨l q d i j w n x j xFiny 1 ¨l , n Vi , n Ž K j" . Ž x¨l9 . s m " i j Ž n . xFi ¨l ,

Vi , n Ž Fj . s 0, for 1 F j F N. PROPOSITION 4.4.

For x s Ej , Fj , K j" Ž1 F i F N ., one has

² x, Vi , n : s yn q ² l , a ik : i Vi , n Ž x . .

Ž 4.12.

Proof. If x is homogeneous in f, one has ² Ej , Vi , n : Ž x¨l9 . s Ej Vi , n Ž x¨l9 . y K j Vi , n Ž Ky1 j Ej x¨l9 .

Ž 4.13.

and Ej Vi , n Ž x¨l9 . s Ej xFin ¨l s

1 q j y qy1 j

Ž K j j ­ Ž xFin . y ­ j Ž xFin . Ky1 j . ¨l , using Ž 4.11. .

By definition of ­ j and j ­ , this is equal to 1 q j y qy1 j

Ž K j j ­ Ž x . Fin q q Ž < x < , a .K j x j ­ Ž Fin . j

n y1 yq Ž < F i < , a j .­ j Ž x . Fin Ky1 j y x ­ j Ž Fi . K j . ¨l . n

The second and the fourth terms give 1 q j y qy1 j s

žd

w n x j q jny 1 xK j Finy1 ¨l y qy² ly r , a j :di j w n x j q jny1 xFjny1 ¨l / k

ij

1 q j y qy1 j

y2 Ž ny1.q² l y r , a jk : di j j

žq

w n x j q jny1 xFiny1 ¨l

ly r , a j : yqy² d i j w n x j q jny 1 xFjny1 ¨l j k

s

di j qj y

qy1 j

/

ny1.q² l y r , a i : w n x j ž qyŽ y q jny1y² ly r , a i j k

s yn q ² l , a ik : i d i j w n x j xFiny1 ¨l .

k

:

/ xF

i

ny1

¨l

654

ABDELLAH SEBBAR

Using Ž4.10., the second term in Ž4.13. gives K j Vi , n Ž Ky1 j Ej . x¨l9 s s s

1 q j y qy1 j 1 qj y

k

qy1 j

žK

j j­

Ž x . Fin ¨l y q Žj < ­ j Ž x .< , a j .y2 ² l9y r , a j : K j Vi , n ­ j Ž x . ¨l9 /

qy1 j

žK

j j­

² l9y r , a j Ž x . Fin ¨l y qy2 j

1 qj y

y1 K j Vi , n Ž j ­ Ž x . y Ky1 j ­ j Ž x . K j . ¨l9

k

:q ² l y r , a jk :

?qyŽ < F i < , a j .­ j Ž x . Fin ¨l n

s

1 qj y

qy1 ij

žK

j j­

k

l, a j Ž x . Fin ¨l y q 1y² j

/

:q 2 ² l , a ik : a jiyn a ji

­ j Ž x . Fin ¨l ,

/

where we have used the fact that if x is homogeneous, then j ­ Ž x . and ­ j Ž x . are also homogeneous; the fact that Ž< Fin <, a j . s nŽ a i , a j . s nd i a i j s nd j a ji ; and that ² l9, a jk: s ² l , a jk: y ² l , a ik:² a i , a jk: s ² l , a jk: y a ji ² l , a ik: . Summing both terms of Ž4.13., we get ² Ej , Vi , n : Ž x¨l9 . s yn q ² l , a ik :

i

Ž di j w n x j xFiny1 q hi j Ž n . ­ j Ž x . Fin . ¨l .

This proves the proposition for x s Ej . For x s Fj , the proposition is clear, since ² Fj , Vi, n : s 0. And for x s K j we have ² K j , Vi , n : Ž x¨l9 . s K j Vi , n Ky1 j x¨l9 y « Ž K j . Vi , n Ž x¨l9 . k

l9y r , a j : s q Ž < x < , a j . qy² K j xFin ¨l y xFin ¨l j k

s q ja ji Ž² l , a i

ž

:yn.

y 1 xFin ¨l

/

s yn q ² l , a ik : i Vi , n Ž K j" . Ž x¨l9 . . The case x s Ky1 is similar. j

q-DE

655

RHAM COCYCLES

A Cocycle Condition Having defined the operators Vi, nŽ?. for the generators Ej , Fj , K j" , we can also define them for any element of the free associative algebra generated by Ej , Fj , and K j" Ž1 F i F N .. Indeed, assume this is done for two elements x and y; then using Ž2.7., we have ² xy, Vi , n : s x ? ² y, Vi , n : q « Ž y . ² x, Vi , n : ;

Ž 4.14.

therefore, if we set Vi , n Ž xy . s x ? Vi , n Ž y . q « Ž y . Vi , n Ž x . , then ² xy, Vi , n : s yn q ² l , a ik : i Vi , n Ž xy . . It remains to extend this construction to the algebra Uq . PROPOSITION 4.5. For any x in Uq , there is an operator Vi, nŽ x . from M Ž l9. to M Ž l. for any nonnegati¨ e integer n satisfying ² x, Vi , n : s yn q ² l , a ik : i Vi , n Ž x . . Moreo¨ er, for y in Uq , we ha¨ e Vi , n Ž xy . s x ? Vi , n Ž y . q « Ž y . Vi , n Ž x . . Proof. As in the previous section, we need to show that the cocycle condition Ž4.14. leaves the defining relations of Uq invariant. We have Vi , n Ž K k K l . s K k Vi , n Ž K l . Ky1 k . Hence k

y² l9, a k Vi , n Ž K k K l . Ž x¨l . s Ž mq i l Ž n . qk k

s

qiŽynq² l , a i

:q² l , a kk : yn Ž a i , a k .

:.Ž a i k qa i l .

yn q ² l ,

q

y1

a ik : i

n q mq i k Ž n . . xFi ¨l

xFin ¨l ,

which is symmetric in k and l. Therefore, Vi, nŽ K k K l . s Vi, nŽ K l K k .. For Ž . the relation K l Ky1 s Ky1 l l K l , it is straightforward see Section 2 . This shows the compatibility with Ž4.3.. For the relation Ž4.6. we have Vi , n Ž K k Fl . s K k ? Vi , n Ž Fl . q « Ž Fl . Vi , n Ž K k . s 0,

656

ABDELLAH SEBBAR

and for x homogeneous, Vi , n Ž Fl K k . Ž x¨l9 . s Ž Fl ? Vi , n Ž K k . . Ž x¨l9 . s Ž Fl Vi , n Ž K k . K k y Vi , n Ž K k . Fl K k . Ž x¨l9 . n q n s qyŽ < x < , a k .qŽ a k , l9y r . Ž mq i k Ž n . Fl xFi ¨l y m i k Ž n . Fl xFi ¨l .

s 0. Also for x homogeneous we have Vi , n Ž Ek Fl y Fl Ek . Ž x¨l9 . s y Ž Fl ? Vi , n Ž Ek . . Ž x¨l9 . s y Ž Fl Vi , n Ž Ek . K l y Vi , n Ž Ek . Fl K l . x¨lX s yqyŽ < x < , a l .qŽ l9y r , a l . Ž Fl Vi , n Ž Ek . x¨l9 y Vi , n Ž Ek . Fl x¨l9 . . The expression between parentheses is equal to

hi k Ž n . Fl ­ k Ž x . Fin ¨l q d i k w n x k Fl xFiny 1 ¨l y hi k Ž n . ­ k Ž Fl x . Fin ¨l y d i k w n x k Fl xFiny 1 ¨l . Since ­ k Ž Fl x . s q Ž < x <, a k .d k l x q Fl ­ k Ž x ., we obtain Vi , n Ž Ek Fl y Fl Fk . Ž x¨l9 . s d k l q Ž l9y r , a l .hi k Ž n . xFin ¨l . On the other hand,

d k l Vi , n

ž

K k y Ky1 k qk y

qy1 k

/

Ž x¨l9 . s s

dk l qk y qy1 k

dk l qk y qy1 k

Ž Vi , n Ž K k . x¨l9 y Vi , n Ž Ky1 k . x¨l9 . Ž mqi k Ž n . y myi k Ž n . . xFin ¨l

s d k l q Ž l9y r , a k .hi k Ž n . xFin ¨l , which shows the compatibility with the relation Ž4.5.. The verification for the remaining relations is done using the same calculations. For any Uq-module M , we define a shifted differential, dl , i : M w zy1 x ª M w zy1 x

dz z

,

given linearly by dl , i Ž zyn . s yn q ² l , a ik : i zyn

dz z

.

q-DE

657

RHAM COCYCLES

Let us consider the operators `

dz

ns0

z

Ý Vi , n zyn y1dz g HomŽ M Ž ri l. , M Ž l. . w zy1 x

Vi Ž z . s and Vi Ž x, z . s

`

Ý Vi , n Ž x . zyn gHomŽ M Ž ri l. , M Ž l. . w zy1 x

Ž x g Uq . .

ns0

The results of this section can be reformulated in the following: THEOREM 4.6.

For any x, y in Uq and i g  1, . . . , N 4 , we ha¨ e ² x, Vi Ž z . : s dl , i Vi Ž x, z . ,

Ž 4.15.

Vi Ž xy, z . s x ? Vi Ž y, z . q « Ž y . Vi Ž x, z . .

Ž 4.16.

4.3. q-de Rham Cocycles In this section we fix an element w of the Weyl group of the Lie algebra g. We assume that w s ri a ??? ri1 is a reduced decomposition of w into fundamental reflexions. We also fix an element l g h*, and we consider the following elements of h*:

l p s ri py 1 ??? ri1 l

Ž 1 F p F a. .

y1 xx Let A s Cww zy1 , and let V p , 1 F p F a, be the free A-module 1 , . . . , za generated by  dz j1rz j1 n ??? n dz j prz j p, 1 F j1 - ??? - j p F a4 . We consider the sequence of A-modules

V : 0 ª V 0 ª V 1 ª ??? ª V a ª 0, v

and we define a linear map d0: V k ª V kq 1 as follows. If y1 h s f Ž zy1 1 , . . . , za .

dz j1 z j1

H ??? H

dz jk z jk

,

Ž 4.17.

where f is a monomial in A, and if n p is the exponent of zy1 in f, then p a

d0 h s

Ý ps1

: yn p q ² l p , a ik p

ip

y1 f Ž zy1 1 , . . . , za .

dz p zp

H

dz j1 z j1

H ??? H

dz jk z jk

.

658

ABDELLAH SEBBAR

Using the notations of the previous subsection, this can be expressed as a

d0 h s

Ý

y1 dl p , i p f Ž zy1 1 , . . . , za .H

dz j1 z j1

ps1

H ??? H

dz jk

.

z jk

We extend d0 by linearity to any h in V k . PROPOSITION 4.7. We ha¨ e d0 2 s 0, so Ž V , d . is a complex. v

Proof. Let form given by

y1 y1 . f Ž z . s f Ž zy1 be a monomial, and let h be a 1 , z2 , . . . , z a Ž4.17.. Then

a

d0 h s

: yn p q ² l p , a ik p

Ý ps1

ip

f Ž z.

dz p zp

H

dz j1 z j1

H, . . . , H

z jk z jk

and a

d0 2 s

: yn p q ² l p , a ik p

Ý ps1 a

?

ž

: yn s q ² l s , a ik s

Ý ss1 a

s

ip

a

žÝ Ý

is

f Ž z.

: yn p q ² l p , a ik p

ps1 ss1

: ? yn s q ² l s , a ik s

is

f Ž z.

dz s zs

H

dz p zp

H ??? H

dz jk z jk

/

ip

dz s zs

H

dz p zp

/

H

dz j1 z j1

H ??? H

dz jk z jk

s0 For each p s 1, . . . , a, we consider the operators defined previously: Vp Ž z p . : M Ž l pq1 . ª M Ž l p . zy1 p

zy1 p

and Vp Ž x, z p . : M Ž l pq1 . ª M Ž l p .

zy1 p

Ž x g Uq .

We recall from Section 1 that we have the following complex: Hom Ž Uqm , Hom Ž M Ž wl . , M Ž l . . . , v

q-DE

659

RHAM COCYCLES

and the differential d9 of this complex is defined on the cochains as follows: d9f Ž x 1 , . . . , x n . s x 1 ? f Ž x 2 , . . . , x n . ny1

q

Ý Ž y1. i f Ž x 1 , . . . , x iy1 , x i x iq1 , x iq2 , . . . , x n .

is1 n

q Ž y1 . f Ž x 1 , . . . , x ny1 . « Ž x n . . We consider the double complex C s Hom Ž Uqm , Hom Ž M Ž wl . , M Ž l . m V vv

v

v

..

( Hom Ž Uqm m M Ž wl . , M Ž l . m V . . v

v

The first differential for this double complex is d9; the second differential is the shifted de Rham differential d0 defined above. In the following, we will construct an a-cocycle in the simple complex associated with C . For x 1 , x 2 , . . . , x m in Uq and p1 , p 2 , . . . , pm being an increasing sequence in  1, . . . , a4 , we define G Ž x 1 , . . . , x m ; p1 , . . . , pm . as equal to the following expression: vv

Ý V1Ž z1 . ??? Vp y1Ž z p y1 . Vp Ž xX1 , z p . xY1 1

1

1

1

Ž x.

? Vp 1q1 Ž z p 1q1 . ??? Vp 2y1 Ž z p 2y1 . Vp 2Ž xX2 , z p 2 . xY2

ž

? ??? Vp mŽ xXm , z p m . xYm Ž Vp m q1 Ž z p m q1 . ??? Vi aŽ z a . . ???

ž

/ / dz

1

ˆ p 1 H ??? H dz ˆ p m H ??? H dz a , H dz where ˆ means omission, and the summation is over all of the terms involved in the comultiplication of x 1 , . . . , x m in the Sweedler notation. In each summand, we consider initially the composition V1Ž z1 .V2 Ž z 2 . ??? VaŽ z a ., and for each pk we substitute Vp kŽ z p k . by Vp kŽ xXk , z p k . xYk ? Ž . . . , where xYk acts on all of the remaining factors to the right, if there are any. For each m, 0 F m F a, we define the operators V m , aym g Hom Ž Uqmm , Hom Ž M Ž wl . , M Ž l . m V ay m . . as follows: V m , aym Ž x 1 , . . . , x m . s Ž y1 . ?

m Ž mq1 .r2

Ý

1Fp 1- ??? -p m Fa

Ž y1.

p 1 q ??? qp m

G Ž x 1 , . . . , x m ; p1 , . . . , pm . .

Ž a.

660

ABDELLAH SEBBAR

To clarify these rather complicated expressions, we will give some examples: V 0, a s V1 Ž z1 . ??? Va Ž z a . dz1 H ??? dz a . V a, 0 Ž x 1 , . . . , x n . s

V1 Ž xX1 , z1 . xY1

Ý Ž x 1 . , . . . , Ž x ay1 .

? Ž V2 Ž xX2 , z 2 . xY2 Ž ??? Ž Vay1 Ž xXay1 , z ay1 . xYay1 ? Va Ž x a , z a . . ??? . . . For a s 2: V 0, 2 s V1 Ž z1 . V2 Ž z 2 . dz1 H dz 2 V 1, 1 Ž x . s

Ý Ž V1Ž x9, z1 . x0 ? V2 Ž z 2 . dz2 y V1Ž z1 . V2 Ž x, z 2 . dz1 . Ž x.

V

2, 0

Ž x1 , x 2 . s

Ý V1Ž xX1 , z1 . xY1 ? V2 Ž x 2 , z 2 . . Ž x 1.

For a s 3: V 0, 3 s V1 Ž z1 . V2 Ž z 2 . V3 Ž z 3 . dz1 H dz 2 H dz 3 . V 1, 2 Ž x . s

Ý V1Ž x9, z1 . x0 ? Ž V2 Ž z 2 . V3 Ž z 3 . . dz2 H dz3 Ž x.

y Ý V1 Ž z1 . V2 Ž x9, z 2 . x0 ? V3 Ž z 3 . dz1 H dz 3 Ž x.

q V1 Ž z1 . V2 Ž z 2 . V3 Ž x, z 3 . dz1 H dz 2 . V 2, 1 Ž x 1 , x 2 . s

Ý

V1 Ž xX1 , z1 . xY1 ? Ž V2 Ž xX2 , z 2 . xY2 ? V3 Ž z 3 . . dz 3

Ž x 1. , Ž x 2 .

y Ý V1 Ž xX1 , z1 . xY1 ? Ž V2 Ž z 2 . V3 Ž x 2 , z 3 . . dz 2 Ž x 1.

q Ý V1 Ž z1 . V2 Ž xX1 , z 2 . xY1 ? V2 Ž x 2 , z 3 . dz1 . Ž x 1.

V 3, 0 Ž x 1 , x 2 , x 3 . s

Ý

V1 Ž xX1 , z1 . xY1 ? Ž V2 Ž xX2 , z 2 . . xY2 ? V3 Ž x 3 , z 3 . .

Ž x 1. , Ž x 2 .

PROPOSITION 4.8. We ha¨ e Ž1. Ž2.

d0 V 0, a s d9V a, 0 s 0. d9V k, ayk s Žy1. k d0 V kq1, ayky1 for k s 0, . . . , a y 1.

q-DE

RHAM COCYCLES

661

Proof. We will prove Ž1.; Ž2. is proved in the same way after changing the right side using the relation Ž4.15. of Theorem 4.6. The fact that d0 V 0, a s 0 is clear, since V is of length a. By definition of d9 we have v

d9V a, 0 Ž x 1 , . . . , x a , x aq1 . s x 1 ? V a, 0 Ž x 2 , . . . , x aq1 .

Ž a.

a

Ý Ž y1. k V a, 0 Ž x 1 , . . . , x k x kq1 , . . . , x aq1 .

q

Ž b.

ks1

q Ž y1 .

aq 1

« Ž x aq 1 . V a, 0 Ž x 1 , . . . , x a . .

Ž c.

Using the composition lemma, we have

Ž a. s

xX1 ? V1 Ž xX2 , z1 . xY1 ? Ž xY2 ? w ??? x . ,

Ý Ž x 1. , . . . , Ž x a .

where the terms in w ??? x are the same as in V a, 0 Ž x 2 , . . . , x aq1 ., except for the first factor. Hence

Ž a. s

Ý

Ž xX1 ? V1Ž xX2 , z1 . .Ž Ž xY1 xY2 . ? w ??? x . .

Ž x 1. , . . . , Ž x a .

Using Theorem 4.6, the summand corresponding to k in Ž b ., for k - a, is equal to k Ž y1. V1 Ž xX1 , z1 . xY1

? Ž V2 Ž xX2 , z 2 . xY2 ? ??? xYky1 ? Ž Ž xXk ? Vk Ž xXkq1 , z k . q « Ž xXkq1 . Vk Ž xXk , z k . . xYk xYkq1 ? ??? . . Using the counit axiom, this is equal to k Ž y1. V1 Ž xX1 , z1 . xY1

? Ž V2 Ž xX2 , z 2 . xY2 ? ??? xYky1 ? Ž xXk ? Vk Ž xXkq1 , z k . Ž xYk xYkq1 . ? Ž ??? . . . q Ž y1 . Vk Ž xX1 , z1 . xY1 Ž ??? x ky1 ? Ž Vk Ž xXk , z k . Ž xYk x kq1 . ??? . . . k

Using the composition lemma one more time in the second term, we obtain k Ž y1. V1 Ž xX1 , z1 . xY1

? Ž V2 Ž xX2 , z 2 . xY2 ? ??? xYky1 ? Ž xXk ? Vk Ž xXkq1 , z k . Ž xYk xYkq1 . ? Ž ??? . . . q Ž y1 . Vk Ž xX1 , z1 . xY1 k

? Ž ??? x ky 1 ? Ž Vk Ž xXk , z k . xYk ? Ž xXkq1 ? Ž ??? . xYkq1 ? . ??? . . .

662

ABDELLAH SEBBAR

The term corresponding to k s a in Ž b . is a

Ž y1. V a, 0 Ž x 1 , . . . , x a x aq1 . , which is equal to

Ž y1.

a

Ý V1Ž xX1 , z1 . xY1 ? Ž ??? Ž xYay1 x a . ? Vaq1Ž x aq1 , z a . ??? . a

q Ž y1 . « Ž x aq 1 . V a, 0 Ž x 1 , . . . , x a . . It follows that all the terms in Ž b . cancel, except the first and the last, and it is clear that the first term cancels with Ž a. and the last one cancels with Ž c .. As a consequence of the preceding results, we have the following: THEOREM 4.9. The element V s Ž V 0, 1, . . . , V a, 0 . is an a-cocycle in the simple complex associated with the double complex C Ž Uq , HomŽ M Ž wl., M Ž l. l V ... v

v

And we have THEOREM 4.10. The cocycle V induces linear maps, f m : H m Ž V . * ª Ext Uayq m Ž M Ž wl . , M Ž l . . v

0 F m F a.

Proof. Let fm : H m Ž V . ª C be a linear form. We extend f to the space Z m Ž V . of cocycles up to homotopy. We obtain a linear form fm : Z m Ž V . ª C. Then we extend fm to V m by taking it to be zero on a complement of Z m in V m. Let x 1 , . . . , x aym be elements of Uq , and set v

v

v

Fm s V ay m , m Ž x 1 m ??? m x aym . g Hom Ž M Ž wl . , M Ž l . m V m . . We consider the composition Fm

6

M Ž wl.

M Ž l. m V m Idm f m

6

M Ž l. m C ( M Ž l.. The map obtained is then an element of Hom Ž Uqm ay m , Ž Ž . Ž ... Hom M wl , M l . Since V is an a-cocycle by the preceding theorem, we have d9V aym, m s Žy1. aym d0 V aymq1, my1 ; therefore d9ŽŽ Id m fm .( Fm . s 0. Meanwhile, fm was chosen up to homotopy, satisfying d0( fm s 0; it follows that the resulting map is a cochain in the complex C Ž U q , H om Ž M Ž w l . , M Ž l ... . And since d 0 V a y m , m s Žy1. ay my1 d9V aymy1, mq1 Žby Proposition 4.8., the class of this cochain modulo coboundaries does not depend on the choice of fm up to homotopy, because the above relation implies that the image of fm ( d0 is d9ŽŽ Id m Ž fm ( d0 ..( Fm ., and hence is a coboundary. Since the cohomology spaces of the cochain complex C Ž Uq , HomŽ M Ž wl., M Ž l... are the Ext-spaces Ext U qŽ M Ž wl., M Ž l.., the theorem is proved. v

v

v

q-DE

663

RHAM COCYCLES

:, To illustrate this theorem, we assume that the numbers ² l p , a ik p p s 1, . . . , a, are nonnegative integers. The homology space H a Ž V . is one-dimensional Ž q is not a root of unity., generated by Žthe image of. the linear form r g V a* defined by v

k

k

r Ž h . s Res z as0 ??? Res z 1s0 Ž z1² l1 , a i1 : ??? z a² l a , a i a :h . . The element f aŽ r ., in Hom U qŽ M Ž wl., M Ž l.., is the unique intertwiner k k between M Ž wl. and M Ž l., sending ¨ wl to Fi²a l a , a i a :q1 ??? Fi²1 l1 , a i1 :q1 ¨l . ACKNOWLEDGMENTS I thank Vadim Schechtman for his suggestions and help, Anthony Knapp for his advice, and Susan Montgomery for a helpful comment.

REFERENCES 1. H. Cartan and S. Eilenberg, ‘‘Homological Algebra,’’ Princeton Mathematics Series 19, Princeton Univ. Press, Princeton, NJ, 1956. 2. V. G. Drinfeld, Hopf algebras and the quantum Yang]Baxter equation, So¨ . Math. Dokl. 32 Ž1985., 254]258. 3. B. Feigin and D. Fuchs, Representations of the Virasoro algebra, in ‘‘Representations of Lie Groups and Related Topics’’ ŽA. M. Vershik and D. P. Zhelobenko, Eds.., pp. 465]554, Gordon and Breach, New York, 1990. 4. V. Ginzburg and V. Schechtman, Bosonization and chiral Lie cohomology, preprint. 5. V. Ginzburg and V. Schechtman, Screenings and a universal Lie-de Rham cocycle, ‘‘Kirillov’s Seminar on Representation Theory,’’ American Mathematical Society Translations, Series 2, Vol. 181, pp. 1]34, Amer. Math. Soc., Providence, RI, 1998. 6. M. Jimbo, A q-difference analogue of UŽ g . and the Yang]Baxter equation, Lett. Math. Phys. 10 Ž1985., 63]69. 7. V. Kac, ‘‘Infinite Dimensional Lie Algebras,’’ 3rd ed., Cambridge Univ. Press, Cambridge, 1990. 8. C. Kassel, ‘‘Quantum Groups,’’ Springer-Verlag, New York, 1995. 9. G. Lusztig, ‘‘Introduction to Quantum Groups,’’ Progress in Mathematics, Vol. 110, Birkhauser, Boston. ¨ 10. G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc., 3Ž1. Ž1990., 257]296. 11. S. Montgomery, Hopf algebras and their actions, CBMS Regional Conf. Ser. in Math. 82 Ž1993.. 12. A. Sebbar, ‘‘Quantum Groups, Screening Operators and q-de Rham Cocycles,’’ Ph.D. thesis, SUNY Stony Brook, June 1997. $ 13. A. Sebbar, Quantum affine algebra Uq Žs l 2 ., screening operators, and q-de Rham cocycles, Comm. Math. Phys., to appear. 14. M. E. Sweedler, ‘‘Hopf Algebras,’’ Benjamin, New York, 1969.