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

DOI:
https://doi.org/10.1090/S0002-9947-03-03360-9

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 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: 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).

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

Email:
agiambr@unipa.it

**M. Zaicev**

Affiliation:
Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, Moscow, 119992 Russia

Email:
zaicev@mech.math.msu.su

DOI:
https://doi.org/10.1090/S0002-9947-03-03360-9

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