Involution Codimensions of Finite Dimensional Algebras and Exponential Growth

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1 Joural of Algebra 222, do: jabr , avalable ole at o Ivoluto Codmesos of Fte Dmesoal Algebras ad Expoetal Growth A. Gambruo Dpartmeto d Matematca e Applcazo, Uersta ` d Palermo, Palermo, Italy E-mal: a.gambruo@upa.t ad M. Zacev Departmet of Algebra, Faculty of Mathematcs ad Mechacs, Moscow State Uersty, Moscow, Russa E-mal: zacev@w.math.msu.su Commucated by Susa Motgomery Receved March 26, 1999 Let F be a feld of characterstc zero ad let A be a fte dmesoal algebra wth voluto over F. We study the asymptotc behavor of the sequece of -codmesos c Ž A,. of A ad we show that ExpŽ A,. lm ' cž A,. exsts ad s a teger. We gve a explct way for computg ExpŽ A,. ad as a cosequece we obta the followg characterzato of -smple algebras: A s -smple f ad oly f ExpŽ A,. dm A Academc Press F 1. INTRODUCTION The purpose of ths paper s to determe for ay fte dmesoal algebra wth voluto the expoetal growth of ts sequece of -codmesos characterstc zero. Let F be a feld of characterstc zero ad let A be a F-algebra wth voluto. Let V be the space of multlear polyomals the varables x, x,..., x, x ad let W Ž A,. 1 1 be the subspace of multlear -polyomal dettes of A; the the codmeso of W Ž A,. s c Ž A,., the th codmeso of A $35.00 Copyrght 2000 by Academc Press All rghts of reproducto ay form reserved.

2 472 GIAMBRUNO AND ZAICEV A celebrated theorem of Amtsur 1 states that f a algebra wth voluto A satsfes a -polyomal detty the A satsfes a ordary Ž o-voluto. polyomal detty. I lght of ths result 7 t was otced that, as the ordary case, f A satsfes a o-trval -polyomal detty the c Ž A,. s expoetally bouded. I 2 a explct expoetal boud for c Ž A,. was exhbted ad ths tur gave a ew proof of Amtsur s theorem where ow the degree of a ordary detty for A s related to the degree of a gve -detty of A. We should also meto that 10 we characterzed fte dmesoal algebras A such that c Ž A,. s polyomally bouded terms of the represetato theory of the hyperoctahedral group H. The asymptotc behavor of c Ž A,. was determed 4 case A M Ž F. k s the algebra of k k matrces over F ad s ether the traspose or the symplectc voluto. It turs out that c ŽM Ž F.,. k C t k 2 for some explct costats C ad t. Ths result was acheved by provg that the -trace codmesos ad the -codmesos are asymptotcally equal ad the combg a result of Loday ad Proces 12 o the -trace dettes of M Ž F. k wth some asymptotc computatos of Regev 13. Here we shall determe the expoetal behavor of the sequece c Ž A,. for ay fte-dmesoal algebra A. To do so, we defe ' ExpŽ A,. lm sup c Ž A,., ExpŽ A,. lm f c Ž A,. ad, case of equalty, ExpŽ A,. ExpŽ A,. ExpŽ A,.. Let c Ž A. deote the usual Ž wthout. th codmeso of the algebra A ad let ExpŽ A. deote the correspodg expoetal growth. I 8 ad 9 t was proved that for ay PI-algebra A, ExpŽ A. exsts ad s a teger. By explotg the methods of 8 we shall prove that f A s fte dmesoal, ExpŽ A,. exsts ad ca be explctly computed: suppose that F s algebracally closed ad wrte A B J, where B B s a maxmal semsmple subalgebra ad J s the Jacobso radcal of A. The Ž Ž. Ž.. Ž. Ž. Exp A, max dm F C1 C t, where C 1,...,Ct are dstct -smple subalgebras of B ad C1 Ž. JC2 Ž. J JCt Ž. 0. Whe F s a arbtrary feld of characterstc zero, amog other cosequeces we shall prove that for a fte dmesoal -smple F-algebra A, ExpŽ A,. dm Z A, where Z Z A s the symmetrc ceter of A. It s worth metog that lght of ths result, the equalty ExpŽM Ž F.,. k ExpŽM Ž F.. dm M Ž F. obtaed k F k 4 s actually characterzg fte dmesoal smple algebras wth voluto of the frst kd.e., Z Z. '

3 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 473 Fally we remark that to compute the expoetal behavor of arbtrary algebras wth voluto A seems to be ot a easy task at preset; the ordary Ž o-voluto. case by a basc reducto of Kemer Žsee 11. t s eough to study dettes of the Grassma evelope of a fte dmesoal Z2-graded algebra. Ufortuately a smlar reducto s ot avalable for algebras wth voluto at preset. 2. CODIMENSIONS AND COCHARACTERS Throughout F s a feld of characterstc zero ad A s a F-algebra wth voluto. Let A a Aa a 4 ad A a Aaa 4 be the sets of symmetrc ad skew elemets of A, respectvely. We let F² X, : F² x, x, x, x,... : be the free algebra wth volu- to of coutable rak. Recall that fž x, x,..., x, x. F² X, : 1 1 s a -polyomal detty for A f fža, a,...,a, a for all a 1,...,a A. Let IdŽ A,. deote the deal of all -polyomal dettes of A ad let V deote the space of multlear -polyomals x 1, x 1,..., x, x. If we set s x x ad k x x, 1, 2,..., the, sce char F 2, t s useful to wrte V as 4 V Spa w w S, w s or w k, 1,...,. F Ž1. Ž. Let H be the hyperoctahedral group. Recall that H Z2 S s the wreath product of Z 1, 4 2, the multplcatve group of order 2, ad S. We wrte the elemets of H as Ž a,...,a ;. 1, where a Z 2, S. The group H acts o V as follows Žsee. 5 : for h Ž a,...,a ;. 1 H defe hs s, hk k a Ž. Ž. Ž. k Ž. ad the exted ths acto dagoally to V Ž.. Sce V IdŽ A,. s varat uder ths acto, we vew V ŽV IdŽ A,.. as a H -module. Its character Ž A,. s called the th -cocharacter of A ad c Ž A,. Ž A,.Ž 1. dm V V IdŽ A,. F s the th -codmeso of A. Recall that there s a oe-to-oe correspodece betwee rreducble H -characters ad pars of parttos Ž,., where r, r, for all r 0, 1,...,. If, deotes the rreducble H-character correspodg to Ž,., we wrte where m, Ý Ý A, m,,, r0 r r 0 are the correspodg multplctes.

4 474 GIAMBRUNO AND ZAICEV For r 0,...,, we let V Spa w w S,w s for 1,...,r ad r, r Ž1. Ž. w k for r 1,...,4 be the space of multlear polyomals s 1,...,s r, k r1,...,k. To smplfy the otato, the sequel we shall wrte y, y j ad z, z j for symmetrc ad skew symmetrc varables, respectvely, that are depedet from each other. Therefore f we let s1 y 1,...,sryr ad kr1 z,...,k z, the 1 r V the space of multlear polyomals y,..., y, z,..., z. r, r 1 r 1 r It s clear that to study V IdŽ A,. t s eough to study Vr, r IdŽ A,. for all r. If we let Sr act o the symmetrc varables y 1,..., yr ad Sr act o the skew varables z 1,..., z r, we obta a acto of Sr Sr o V ad V ŽV IdŽ A,.. r, r r, r r, r s a left Sr Sr-module. Let Ž A,. be ts character ad let c Ž A,. Ž A,.Ž 1. r, r r, r r, r dm V ŽV IdŽ A,.. F r, r r, r. It s well kow that there s a oe-to-oe correspodece betwee rreducble S S -characters ad pars of parttos Ž,. r r such that r, r. For ay partto of a teger t let be the correspodg rreducble St-character. The s the rreducble S S -character assocated to the par Ž,. r r. The followg result holds THEOREM 1 5, Theorem 1.3. Let A be a PI-algebra wth oluto; the, for all r, Ý Ý A, m ad,, r0 r r Ý A, m. r, r, r r Moreoer ž / c A, Ý cr, r Ž A,.. r r0

5 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS GUESSING Exp A, Throughout A wll be a fte dmesoal algebra wth voluto over the feld F. We wrte A B J, where B s a maxmal semsmple subalgebra of A ad J JŽ A. s the Jacobso radcal of A. Clearly J J ad, accordg to 10, Theorem 4 we may assume that also B B s stable uder the voluto. Wrte B B1 Bm where B 1,...,Bm are -smple algebras. Recall that for each 1,...,m ether B s a fte dmesoal smple algebra wth duced voluto or B C C op, where C s a smple homomorphc mage of B ad o C C op s the exchage voluto Ž a, b. Ž b, a. Žsee 14, Proposto We ow defe a teger d dž A. the followg way: we cosder all possble ozero products of the type C JC J C JC 0, 1 2 k1 k where C,...,C are dstct subalgebras from the set B,..., B 4 1 k 1 m ad k 1 Ž f k 1 ths meas that C B for some We the defe d dž A. to be the maxmal dmeso of a subalgebra C1 Ck satsfyg the above equalty. We shall prove that case F s algebracally closed, d cocdes wth ExpŽ A,.. We ote that f B,...,B are ot ecessarly dstct amog the B j s 1 t ad B JB J JB 0 the dmž B B. d Žsee 8, Lemma t 1 t 4. MULTIALTERNATING POLYNOMIALS LEMMA 1. Let t 0, a b d, ad fž y,..., y, z,..., z, x,..., x. 1 a 1 b 1 t be a multlear polyomal alteratg o y,..., y 4 ad o z,..., z 4 1 a 1 b. If y,..., y B, z,..., z B, x,..., x A the fž 1 a 1 b 1 t y 1,..., y a, z 1,..., z, x,..., x. 0. b 1 t Proof. For each 1,...,m, let E E E be a bass of B, where E ad E are sets of symmetrc ad skew elemets, respectvely. The E E E E s a bass of B, where E E ad E E. Sce f s multlear, t s eough to evaluate f o a bass of A. The proof s smlar to the proof of 8, Lemma 3. If all the varables f are bass elemets of B, the, sce BB j 0 for j we wll get a zero value uless all elemets come from oe -smple compoet, say B.I ths case, sce dm B d ad a b d, the ether a E t t or b E t. Sce f s alteratg the y s ad the z s the the value of f wll stll be zero. Therefore, to get a o-zero value of f we should evaluate at least oe varable o elemets of J. Ths case also leads to a zero value of f by the maxmalty of d Žsee detals 8, Lemma 3.. t

6 476 GIAMBRUNO AND ZAICEV For ay partto of a teger t ad a Youg tableau T we let RT ad CT be the subgroups of row ad colum permutatos of T, respec- tvely; the e Ý Ž sg. s a essetal dempotet of FS T R T, CT t ad FS e s a mmal left deal of FS. t T t Let r, r, ad W be a left rreducble S S -module;, r r f T s a tableau of shape ad T s a tableau of shape, the W FS Ž S. e e, where S ad S act o dsjot sets of, r r T T r r tegers. For a partto we deote wth Ž,, the cojugate partto of. REMARK 1. Let r, r, ad W, Vr, r be a left rreducble Sr Sr-module. The there exsts f W, such that Ž f f y 1,..., y 1, y 2,..., y 2,..., z 1,..., z 1, z 2,..., z 2, j ad f s alteratg o each set of arables y,..., y, z,..., z j 4 1 1,1 j 1,1j 1. Proof. Let f0 W, be a o-zero polyomal. The there exst tableaux T, T such that g e e f 0. Deote T T 0 Ý Ý f Ž sg.ž sg. g. CT CT The f s a polyomal wth the prescrbed property. 5. THE UPPER BOUND Recall that for a partto, h 1 s the heght of the correspodg dagram. LEMMA 2. Let r, r, ad W, Vr, r be a left rreducble S S -module. If W V IdŽ A,., the h dm A, h r r, r, r dm A, ad l1 l1 d, where J l1 0. Moreoer dm W, a Ž. r Ž. r for some a 1. l1 l1 Proof. Let f be the polyomal descrbed the prevous remark. 1 Clearly W FSS f. Sce f s alteratg o y,..., y 1 4, r r 1 1, t follows that 1 h dm A. Smlarly, h dm A. Suppose that d. The l1 l1 d for all 1,...,l ad, by the prev- ous lemma, sce f s ot a -detty for A, each set of varables y,..., y, z,..., z 4 1 1, 1 l 1, we must substtute at least oe elemet from J. But J l1 0 leads to f beg a -detty for A.

7 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 477 Let dm A p ad dm A q. From the hook formula for the degrees of the rreducble represetatos of the symmetrc group t follows Ž. pl Ž. r Ž. ql that 1 l1 ad 1 l1 r. Therefore dm W, plql r Ž. Ž. r 1 1. l1 l1 THEOREM 2. If A s a fte dmesoal algebra wth oluto oer F t ad d s the teger defed aboe, the c A, a d, for some costats a, t. Proof. For a partto let us wrte tž. l1. From the prevous lemma we have Ý Ý A, m. r, r, 0t t d r r Hece, for some costat, Ý r r cr, r Ž A,. m, t tž.. 0t t d r r Ý Sce by 3 the multplctes m rem 1 we get, are polyomally bouded, from Theo- ž / ž / b r r Ý r r, r Ý Ý r 1 2 r0 t1t2d r0 t 1, t20 c A, c A, a t t b Ý Ž 1 2. b t1t2d t 1, t20 a t t a dd. 6. CENTRAL POLYNOMIALS The exstece of multalteratg cetral polyomals for k k matrces proved 6 easly leads to the followg LEMMA 3. Let C be a fte dmesoal cetral -smple algebra oer F, dm C p ad dm C q. For all m 1 there exsts a multlear polyomal f fž y1 1,..., yp 1,..., y1 2 m,..., yp 2 m, z1 1,..., z 1 q,..., z1 2 m,..., zq 2 m.

8 478 GIAMBRUNO AND ZAICEV such that Ž. 1 f s alteratg o each set of arables y,..., y 4 1 p,12m, j ad z,..., z j 4 1 q,1j2m; Ž. j j Ž 1 2m 1 2m 2 there exst y C, z C such that f y,..., y, z,..., z. 1 p 1 q 1 C. Proof. Suppose frst that C s smple. By 6 for every m 1 there exsts a multlear polyomal fž x 1,..., x 1,..., x 2m,..., x 2 m. 1 pq 1 pq alterat- g o each set of varables x,..., x 4 1 pq,12m, ad f s a cetral polyomal for C. Sce p q dm C t s clear that f ca be vewed as alteratg o 2 m dsjot sets of symmetrc or skew varables. I case C C1 C1 op wth exchage voluto ad C1 smple, the p q dm C1 ad the polyomal fž y1 1,..., yp 1,..., y1 2 m,..., yp 2 m. fž z1 1,..., z 1 p,..., z1 2 m,..., zp 2 m. s the requred oe. The polyomals foud the prevous lemma wll ow be glued together to fd multalteratg polyomals of sutable degree ovashg a fte dmesoal algebra. LEMMA 4. Let F be algebracally closed ad let A be a fte dmesoal algebra oer F. Let C1JC2J Ck1 JCk 0, where C 1,...,Ck are dstct -smple subalgebras of A ad C1 Ck C1 C k. If p dmž C. 1 C k, q dm C1 C k, the for all m 1 there exsts a multlear polyomal such that f fž y1 1,..., yp 1,..., y1 2 m,..., yp 2 m, z1 1,..., z 1 q,..., z1 2 m,..., zq 2 m, y 1,..., y 2k, z 1,..., z 2k. Ž. 1 f s alteratg o each set of arables y,..., y 4 1 p,12m, j ad z,..., z j 4 1 q,1j2m; Ž. j j 2 there exst y C1 C k, z C1 C k, y A, Ž 1 2m 1 2m z A such that f y,..., y, z,..., z, y,..., y, z,..., z p 1 q 1 2k 1 2k Proof. For every 1,...,k, let p dm C, q dm C, ad let f f y 1,..., y 1,..., y 2 m,..., y 2 m, z 1,..., z 1,..., z 2 m,..., z 2 m,1, p,1, p,1, q,1, q

9 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 479 be the polyomal costructed the prevous lemma. We let f A A A A x f x x f x f x x f, 1 2m 1 2m k1 k1 k1 k k where A meas alterato o the p varables y,..., y,..., y, 1, 1 1, p k,1 1..., y ad A meas alterato o the q varables z,..., z,..., k, p 1, 1 1, q k 1 zk,1,..., zk, q k. We ote that each polyomal f correspods to a par of tableaux Ž P, Q., where P s a 2m p rectagle, Q s a 2m q rectagle, ad the varables each colum of P Ž resp. Q. are alteratg. Now, f correspods to the par Ž P, Q. obtaed by glug the rectagles P Ž resp. Q. oe o top of the other ad by alteratg the varables the colums; hece Aj s alterato o the varables the jth colum of P ad A j s alterato o the varables the jth colum of Q. Sce C JC J C JC 0, let c C Ž 1 k., b,...,b J 1 2 k1 k 1 k1 be such that cbcb bk1ck 0. t t For every 1,...,k, let y C, z C be such that, j, j ž / 1 2m 1 2m f y,1,..., y, p, z,1,..., z, q 1 C. Notce that sce CC j 0 for j, whe evaluatg the varables y ad z o the C s, alterato o the colums of the rectagle P Ž resp. Q. ca be replaced wth alterato o the colums of each subrectagle P Ž resp. Q.. Hece ž / 1 2m 1 2m f y 1,1,..., y k, p, z 1,1,..., z k, q, c 1,...,c k, b 1,...,b k k k1 Ž p! p!q! q!. c f b c f b c f b c f 2 m 1 k 1 k k1 k1 k1 k k Ž p! p!q! q!. cbc b c 0. 2 m 1 k 1 k k1 k We may clearly assume that c 1, b 1,...,b k1, ck A A ; suppose that k1 of them are symmetrc ad k2 of them are skew, k1 k2 2k 1. The Ž 1,1 k, p 1,1 k, q 1 k 1 k. 1 2m 1 2m f f y,..., y, z,..., z, y,..., y, z,..., z y y z z k 1 2k k 1 2k 1 2 k k 1 2 does ot vash A ad s the desred polyomal.

10 480 GIAMBRUNO AND ZAICEV 7. THE LOWER BOUND We are ow a posto to fd the lower boud for the expoetal growth of the -codmesos. THEOREM 3. Let A be a fte dmesoal algebra wth oluto oer the algebracally closed feld F ad let d be the teger defed Secto 3. b The c A, a d, for some costats a, b. Proof. Let A B J ad C 1,...,Ck dstct -smple subalgebras of B such that C Ž. 1JC2J Ck1 JCk 0. Let p dm C1 C k, q dmž C. 1 C k ; the d p q. Let 2 d 4k ad dvde 4k by 2 d; the there exst m, t such that 4k 2md t, 0 t 2 d. Thus 2mŽ p q. 4k t. Wrte s 2mp 2k t, 0 t 2 d, s 2mq 2k. If f s the polyomal of the prevous lemma of total degree 2mp 2mq 4k, the the polyomal g fy2 k1 y2kt Vs, s ad g s ot a -detty for A Žwe may make sutable substtutos for the extra varables: ote that f has o-zero value whch belogs to C1 J JC k, hece we ca take y y k1 2kt C k We let the group G S2 mp S2 mq act o g by lettg S2 mp act o the symmetrc varables y l ad S2 mq act o the skew varables z l. Let M be the G-submodule of Vs, s geerated by g. By complete reducblty M cotas a rreducble G-submodule of the form W FGe e Ž g., T T, for some tableaux T ad T, where 2mp ad 2mq. Let l be the legth of the frst row of. Note that for every S, Ž g. 2 mp s stll alteratg o 2m dsjot sets of symmetrc varables, ad Ý acts by symmetrzg l R T varables. It follows that f l Ž. 2m the et Ž g. 0, a cotradcto. Smlarly for l Ž.. Therefore l 2m ad l 2m. Suppose ow that has heght h p. From the prevous lemma we kow that f Ž hece g. takes a o-zero value o A after replacg the j varables y wth sutable elemets from C C1 C k. Sce dm C p, the polyomal e Ž g. T vashes o A sce t s alteratg o p 1 varables. Hece h p. Smlarly h q. We have proved that M cotas a rreducble G-submodule of the form W, where ŽŽ 2m. p. ad ŽŽ 2m. q., are two rectagles. b b 2 mp 2 mq Recall that as m, deg deg a 2mp 2mq p q for some egatve costats b, b ad some a. It follows that c A, dm W c p 2 mp q 2 mq. s, s,

11 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 481 Therefore we have ž / ž / c A, c A, Ý c Ž A, r r, r s s, s. r0! c 2 mp 2 mq p q. s! Ž s.! Recallg that s 2mp 2k t ad s 2mq 2k, hece ad by Strlg formula we get! Ž 2mp 2mq.! s! Ž s.! Ž 2mp.!2mq! 2 mp2mq Ž 2mp 2mq. 2 mp 2 mq 2 mp2mq 2 mp 2 mq c A, p q p q Ž 2mp. Ž 2mq. Ž4 kt. where a p q s a costat. a Ž p q. a d, 8. CONSEQUENCES We record two mmedate cosequeces of Theorem 2 ad Theorem 3. COROLLARY 1. If A s a fte dmesoal algebra wth oluto oer the feld F, the ExpŽ A,. exsts ad s a teger dm F A. ŽNote that the procedure of algebrac closure of the groud feld does ot affect the th codmeso.. COROLLARY 2. Let A be a fte dmesoal algebra wth oluto oer the algebracally closed feld F ad let B B be a maxmal semsmple Ž Ž. Ž. subalgebra of A. The Exp A, max dm C C. F 1 t, where C1 Ž.,...,Ct Ž. are dstct -smple subalgebras of B ad C1 Ž. JC2 Ž. J JCt Ž. 0. Let F be the algebrac closure of the feld F. For a algebra wth voluto A over F we let Z ZŽ A. be the ceter of A ad let Z Z A Z A be the symmetrc ceter of A. Next two results gve a exact estmate of ExpŽ A,. the case of smple or semsmple algebras.

12 482 GIAMBRUNO AND ZAICEV COROLLARY 3. Let A be a fte dmesoal -smple algebra wth oluto oer F. The ExpŽ A,. dm A. Proof. Suppose frst that A s smple. If Z Z the Z : F A F Z F, Z F Z 1 where F F ad A F s a cetral smple algebra over F wth duced Z voluto. Moreover Z : F dm A dm A dm A F Z : F dm A F. Z F F F F Z From Corollary 2 t follows that dm A dm Ž A F. ExpŽ Z F Z A F F,. ExpŽ A,.. I case Z Z,.e., s a voluto of the secod kd, the Z Z wth Z s a quadratc exteso of Z ad Z F F F wth exchage voluto. Hece Z A F A Z Z Z Z F A Z F F A Z F A Z F wth voluto gve by Ž a f b f. t follows that ž 1 / Z : F Z : F b f a f. As above A F A F A F Z Z F 1 Z : F Z Z 1 A F A F, where, for 1,..., Z : F, F F ad A Z F A Z F s a -sm- ple algebra wth exchage voluto. As the prevous case t follows that dm Z A dm F A Z F A Z F Exp A F F, Exp A,. Suppose ow that A s ot smple. The A C C op, where C s smple ad s the exchage voluto. I ths case sce Z ZŽ A. ZC, we get ZC : F op A F F ž C ZŽC. F/ ž C ZŽC. F /, 1

13 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 483 Ž op. ZŽC. ZŽC. where F F ad C F C F s -smple over F wth exchage voluto. Thus as before we get dm Z A dm F A Z F A Z F Exp A F F, Exp A,. COROLLARY 4. Let A be a fte dmesoal semsmple algebra wth oluto oer F. If A A s the decomposto of A to -smple subalgebras, the Exp A, max dm Z A, where Z Z A A s the symmetrc ceter of A. Proof. We have A F Ž A F. F F ad by the prevous corollary AF F B1 B t, where B1 Bt are -smple algebras cetral over F ad t Z : F. It follows that ExpŽ A,. ExpŽ A F F,. max dm B ; sce dm F 1 Z A dm F B 1, the cocluso of the corollary ow follows. COROLLARY 5. Let A be a fte dmesoal algebra wth oluto oer F. The Exp A, dm A f ad oly f A s -smple ad F Z. F Proof. I lght of Corollary 3 we oly eed to show that ExpŽ A,. dm F A mples A -smple ad F Z. Let A A F F wth duced voluto. The dm A dm A ExpŽ A,. ExpŽ A,. F F. If A s lpotet, the ExpŽ A,. 0, a cotradcto. Hece A cotas a max- mal semsmple subalgebra B B ad by the prevous corollary dm F A ExpŽ A,. dm F B for a sutable -smple subalgebra B of B. Hece A B s -smple. Ths mples that A s -smple ad by Corollary 3 dm A Exp A, dm A mples F Z. F Z ACKNOWLEDGMENT The frst author was partally supported by CNR ad MURST of Italy; the secod author was partally supported by RBRF grats ad REFERENCES 1. S. A. Amtsur, Idettes rgs wth voluto, Israel J. Math. 7 Ž 1969., Y. Bahtur, A. Gambruo, ad M. Zacev, G-dettes o assocatve algebras, Proc. Amer. Math. Soc. 127 Ž 1999., A. Berele, Cocharacter sequeces for algebras wth Hopf algebra actos, J. Algebra 185 Ž 1996.,

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