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A characterization of algebras with polynomial growth of the codimensions

Authors: A. Giambruno and M. Zaicev
Journal: Proc. Amer. Math. Soc. 129 (2001), 59-67
MSC (1991): Primary 16R10, 16R50; Secondary 16P99
Published electronically: June 21, 2000
MathSciNet review: 1694862
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Let $A$ be an associative algebras over a field of characteristic zero. We prove that the codimensions of $A$ are polynomially bounded if and only if any finite dimensional algebra $B$ with $Id(A)=Id(B)$has an explicit decomposition into suitable subalgebras; we also give a decomposition of the $n$-th cocharacter of $A$ into suitable $S_n$-characters.

We give similar characterizations of finite dimensional algebras with involution whose $*$-codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.

References [Enhancements On Off] (What's this?)

  • [BR] A. Berele and A. Regev, Applications of hook diagrams to P.I. algebras, J. Algebra 82 (1983), 559-567. MR 84g:16012
  • [B] A. Berele, Cocharacter sequences for algebras with Hopf algebra actions, J. Algebra 185 (1996), 869-885. MR 97h:16032
  • [CR] C. W. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, John Wiley and Sons, New York, 1962. MR 90g:16001
  • [DG] V. Drensky and A. Giambruno, Cocharacters, codimensions and Hilbert series of the polynomial identities for $2 \times 2$ matrices with involution, Canadian J. Math. 46 (1994), 718-733.
  • [GR] A. Giambruno and A. Regev, Wreath products and P.I. algebras, J. Pure Applied Algebra 35 (1985), 133-149. MR 86e:16027
  • [GZ] A. Giambruno and M. Zaicev, On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145-155. CMP 99:05
  • [K1] A. Kemer, T-ideals with power growth of the codimensions are Specht, Sibirskii Matematicheskii Zhurnal 19 (1978), 37-48 (Russian), English transl Siberian Math. J.
  • [K2] A. Kemer, Ideals of identities of associative algebras, Transl. Math. Monogr., vol. 87, Amer. Math. Soc., Providence RI, 1988. MR 92f:16031
  • [KR] A. Krakowsky and A. Regev, The polynomial identities of the Grassmann algebra, Trans. AMS 181 (1973), 429-438. MR 48:4005
  • [R] A. Regev, Existence of identities in $A \otimes B$, Israel J. Math. 11 (1972), 131-152. MR 47:3442
  • [Ro] L. H. Rowen, Ring Theory, Academic Press, New York, 1988. MR 89h:16001

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Additional Information

A. Giambruno
Affiliation: Dipartimento di Matematica e Applicazioni, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy

M. Zaicev
Affiliation: Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119899 Russia

Received by editor(s): December 1, 1998
Received by editor(s) in revised form: March 26, 1999
Published electronically: June 21, 2000
Additional Notes: The first author was partially supported by the CNR and MURST of Italy; the second author was partially supported by RFFI, grants 96-01-00146 and 96-15-96050.
Communicated by: Ken Goodearl
Article copyright: © Copyright 2000 American Mathematical Society

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