Codimension growth and minimal superalgebras

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.