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Involution Codimensions of Finite Dimensional Algebras and Exponential Growth

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Involution Codimensions of Finite Dimensional Algebras and Exponential Growth Journal of Algebra 222, 471᎐484Ž. 1999 doi:10.1006rjabr.1999.8016, available online at http:rrwww.idealibrary.com on Involution Codimensions of Finite Dimensional Algebras and Exponential Growth A. Giambruno View metadata, citation and similar papers at core.ac.uk brought to you by CORE Dipartimento di Matematica e Applicazioni, Uni¨ersitaÁ di Palermo, 90123 Palermo, Italy provided by Elsevier - Publisher Connector E-mail: [email protected] and M. Zaicev Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State Uni¨ersity, Moscow, 119899 Russia E-mail: [email protected] Communicated by Susan Montgomery Received March 26, 1999 Let F be a field of characteristic zero and let A be a finite dimensional algebra with involution ) over F. We study the asymptotic behavior of the sequence of n ) ) ) s ) -codimensions cAnnŽ., of A and we show that ExpŽ.A, lim ªϱ 'cAnŽ., exists and is an integer. We give an explicit way for computing ExpŽ.A, ) and as a consequence we obtain the following characterization of )-simple algebras: A is ) ) s ᮊ -simple if and only if ExpŽ.A, dim F A. 2000 Academic Press 1. INTRODUCTION The purpose of this paper is to determine for any finite dimensional algebra with involution the exponential growth of its sequence of )-codi- mensions in characteristic zero. Let F be a field of characteristic zero and let A be an F-algebra with ) ) involution . Let VnŽ.be the space of multilinear polynomials in the UU ) variables x11, x ,..., xnn, x and let WA nŽ., be the subspace of multilin- ) ) ear -polynomial identities of A; then the codimension of WAnŽ., is ) cAnŽ., , the nth codimension of A. 471 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 472 GIAMBRUNO AND ZAICEV A celebrated theorem of Amitsurwx 1 states that if an algebra with involution A satisfies a )-polynomial identity then A satisfies an ordinary Ž.non-involution polynomial identity. In light of this result inwx 7 it was noticed that, as in the ordinary case, if A satisfies a non-trivial )-poly- ) wx nomial identity then cAnŽ., is exponentially bounded. In 2 an explicit ) exponential bound for cAnŽ., was exhibited and this in turn gave a new proof of Amitsur's theorem where now the degree of an ordinary identity for A is related to the degree of a given )-identity of A. We should also mention that inwx 10 we characterized finite dimensional ) algebras A such that cAnŽ., is polynomially bounded in terms of the representation theory of the hyperoctahedral group Hn. ) wx The asymptotic behavior of cAnŽ., was determined in 4 in case s = ) A MFkŽ.is the algebra of k k matrices over F and is either the ) transpose or the symplectic involution. It turns out that cMFnkŽŽ.., , t 2 n nªϱ Cn k for some explicit constants C and t. This result was achieved by proving that the )-trace codimensions and the )-codimensions are asymptotically equal and then combining a result of Loday and Procesiwx 12 ) on the -trace identities of MFkŽ.with some asymptotic computations of Regevwx 13 . Here we shall determine the exponential behavior of the sequence ) cAnŽ., for any finite-dimensional algebra A. To do so, we define nn ExpŽ.A, ) s lim sup cA Ž., ) , Exp Ž.A, ) s lim inf cA Ž., ) ''nnªϱ nªϱ n and, in case of equality, ExpŽ.A, ) s Exp Ž.A, ) s Exp Ž.A, ) . ) Let cAnŽ.denote the usualŽ. without nth codimension of the algebra A and let ExpŽ.A denote the corresponding exponential growth. Inwx 8 and wx 9 it was proved that for any PI-algebra A, ExpŽ.A exists and is an integer. By exploiting the methods ofwx 8 we shall prove that if A is finite dimensional, ExpŽ.A, ) exists and can be explicitly computed: suppose U that F is algebraically closed and write A s B q J, where B s B is a maximal semisimple subalgebra and J is the Jacobson radical of A. Then ExpŽ.A, ) s max dim ŽC Ži.Žq иии qC i.., where C Ži.Ž,...,C i. are distinct iF1 tii1 t )-simple subalgebras of B and C Ži. JC Ži. J иии JC Ži. / 0. When F is an 12 ti arbitrary field of characteristic zero, among other consequences we shall prove that for a finite dimensional )-simple F-algebra A, ExpŽ.A, ) s q q q s dim Z A, where Z ZAŽ. is the symmetric center of A. It is worth ) s mentioning that in light of this result, the equality ExpŽŽ..MFk , s wx ExpŽŽ..MFkFdim MFk Ž.obtained in 4 is actually characterizing finite dimensional simple algebras with involution of the first kind i.e., Zqs Z. CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 473 Finally we remark that to compute the exponential behavior of arbitrary algebras with involution A seems to be not an easy task at present; in the ordinaryŽ. non-involution case by a basic reduction of Kemer Ž seewx 11. it is enough to study identities of the Grassmann envelope of a finite dimen- sional Z2-graded algebra. Unfortunately a similar reduction is not avail- able for algebras with involution at present. 2. CODIMENSIONS AND COCHARACTERS Throughout F is a field of characteristic zero and A is an F-algebra U U with involution ). Let Aqs Äa g Aa< s a 4 and Ays Äa g Aa< sya 4 be the sets of symmetric and skew elements of A, respectively. ) s UU We let FX²:², Fx11, x , x 22, x , . .: be the free algebra with involu- ) UUg ) tion of countable rank. Recall that fxŽ 11, x ,..., xnn, x .²:FX, is a ) UUs g -polynomial identity for A if faŽ 11, a ,...,ann, a . 0 for all a1,...,an A. Let IdŽ.A, ) denote the ideal of all )-polynomial identities of A and ) ) U let VnŽ.denote the space of multilinear -polynomials in x11, x ,..., xn, U s q U s y U s xni. If we set s xixiiand k xixi, i 1, 2, . , then, since / ) char F 2, it is useful to write VnŽ.as ) s иии <␴ g s s s VnFŽ. Span Ä4w␴ Ž1. w␴ Žn. Sni, w s ior w ik i, i 1,...,n . s ; Let Hn be the hyperoctahedral group. Recall that Hn Z2 Sn is the s ) wreath product of Z2 Ä41, , the multiplicative group of order 2, and Sn. ␴ g ␴ g We write the elements of Hn as Ž.a1,...,ani; , where a Z2 , Sn. ) wx s ␴ g The group Hnnacts on V Ž.as follows Ž see 5.Ž : for h a1,...,an; . s s a␴ Ži. s " Hnidefine hs s␴ Ži., hki k␴ Ži. k␴ Ži. and then extend this action ) ) l ) diagonally to VnnŽ.. Since V Ž.Id ŽA, .is invariant under this action, ) r ) l ) ␹ ) we view VnnŽ.ŽV Ž.Id ŽA, ..as an Hn-module. Its character nŽ.A, is called the nth )-cocharacter of A and ) s ␹ ) s ) r ) l ) cAnnŽ., Ž.Ž.A, 1 dim FVnn Ž.Ž.Ž.Ž.V Id A, is the nth )-codimension of A. Recall that there is a one-to-one correspondence between irreducible ␭ ␮ ␭ & ␮ & y Hn-characters and pairs of partitions Ž., , where r, n r, for s ␹ all r 0, 1, . , n.If ␭, ␮ denotes the irreducible Hn-character corre- sponding to Ž.␭, ␮ , we write n ␹ ) s ␹ nŽ.A, ÝÝm␭, ␮␭, ␮ , rs0 ␭&r ␮&nyr G where m␭, ␮ 0 are the corresponding multiplicities. 474 GIAMBRUNO AND ZAICEV For r s 0,...,n, we let s иии <␴ g s s Vr , nyr SpanÄw␴ Ž1. w␴ Žn. Sni,w s ifor i 1,...,r and s s q wiik for i r 1,...,n4 be the space of multilinear polynomials in s1,...,srr, k q1,...,kn. j j To simplify the notation, in the sequel we shall write yii, y and z ii, z for symmetric and skew symmetric variables, respectively, that are indepen- s s s dent from each other. Therefore if we let s11y ,...,srry and k rq1 s z1,...,knnz yr , then s Vr , nyr the space of multilinear polynomials in y1 ,..., yr , z1 ,..., znyr . ) l ) l It is clear that to study VnrŽ.Id ŽA, .it is enough to study V , nyr IdŽ.A, ) for all r. If we let Sr act on the symmetric variables y1,..., yrnand S yr act = on the skew variables z1,..., znyrrn, we obtain an action of S S yr on r l ) = Vr, nyrrand V , nyrrŽŽ..V , nyrrnId A, is a left S S yr-module. Let ␺ ) ) s ␺ ) s r, nyrrŽ.A, be its character and let cA, nyrrŽ., , nyr Ž.Ž.A, 1 r l ) dim FrV , nyrrŽŽ..V , nyr Id A, . It is well known that there is a one-to-one correspondence between = ␭ ␮ irreducible SrnS yr-characters and pairs of partitions Ž., such that ␭ & r, ␮ & n y r. For any partition ␯ of an integer t let ␹␯ be the ␹ m ␹ corresponding irreducible St-character. Then ␭␮is the irreducible = ␭ ␮ SrnS yr-character associated to the pair Ž., . The following result holds THEOREM 1wx 5, Theorem 1.3 . Let A be a PI-algebra with in¨olution; then, for all r F n, n ␹ ) s ␹ nŽ.A, ÝÝm␭, ␮␭, ␮ and rs0 ␭&r ␮&nyr ␺ ) s ␹ m ␹ r , nyr Ž.A, Ý m␭, ␮␭Ž. ␮. ␭&r ␮&nyr Moreo¨er n n cAŽ., ) s cAy Ž., ) . nrÝ ž/r , n r rs0 CODIMENSIONS OF FINITE DIMENSIONAL ALGEBRAS 475 3. GUESSING ExpŽ.A, ) Throughout A will be a finite dimensional algebra with involution over the field F. We write A s B q J, where B is a maximal semisimple subalgebra of A and J s JAŽ.is the Jacobson radical of A.
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