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Codimension growth and minimal superalgebras |
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| Authors: | A. Giambruno M. Zaicev |
| Affiliation: | Dipartimento di Matematica ed Applicazioni, Università di Palermo, Via Archirafi 34, 90123 Palermo, Italy ; Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992 Russia |
| 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  of a finite dimensional superalgebra  . 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: 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. |
| Keywords: | Polynomial identity T-ideal superalgebra variety growth |
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is a minimal superalgebra if and only if the ideal of identities of 
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 
such that 
and 
for all proper subvarieties 
of 
. 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).