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Codimension growth and minimal superalgebras

Authors: A. Giambruno and M. Zaicev
Journal: Trans. Amer. Math. Soc. 355 (2003), 5091-5117
MSC (2000): Primary 16R10; Secondary 16P90
Published electronically: July 24, 2003
MathSciNet review: 1997596
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Abstract: A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope $G(A)$ of a finite dimensional superalgebra $A$. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field.

The importance of such algebras is readily proved: $A$ is a minimal superalgebra if and only if the ideal of identities of $G(A)$ is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties $\mathcal{V}$ such that $\exp({\mathcal{V}})=d\ge 2$ and $\exp(\mathcal{U})<d$ for all proper subvarieties ${\mathcal{U}}$ of ${\mathcal{V}}$. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003).

As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.

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Additional Information

A. Giambruno
Affiliation: Dipartimento di Matematica ed 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, 119992 Russia

Keywords: Polynomial identity, T-ideal, superalgebra, variety, growth
Received by editor(s): June 12, 2002
Received by editor(s) in revised form: March 20, 2003
Published electronically: July 24, 2003
Additional Notes: The first author was supported in part by MIUR of Italy.
The second author was partially supported by RFBR, grants 02-01-00219 and 00-15-96128.
Article copyright: © Copyright 2003 American Mathematical Society