%0 Journal Article
%T Growth for Algebras Satisfying Polynomial Identities
%A Amitai Regev
%J ISRN Algebra
%D 2012
%R 10.5402/2012/170697
%X The th codimension of a PI algebra measures how many identities of degree the algebra satisfies. Growth for PI algebras is the rate of growth of as goes to infinity. Since in most cases there is no hope in finding nice closed formula for , we study its asymptotics. We review here such results about , when is an associative PI algebra. We start with the exponential bound on then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of . The representation theory of the symmetric group is an important tool in this theory. 1. Introduction We study algebras satisfying polynomial identities (PI algebra). A natural question arises: give a quantitative description of how many identities such algebra satisfies? We assume that is associative, though the general approach below can be applied to nonassociative PI algebras as well. Denote by the ideal of identities of in the free algebra . If then its dimension is always infinite, hence dimension by itself is essentially of no use here. In order to overcome this difficulty we now introduce the sequence of codimensions. 1.1. Growth for PI Algebras Given , we let denote the multilinear polynomials of degree in , so in the associative case . Identities can always be multilinearized, hence the subset and its dimension give a good indication as to how many identities of degree satisfies. In fact, in characteristic zero the ideal is completely determined by the sequence , but we make no use of that remark in the sequel. To study , we introduce the quotient space and its dimension The integer is the th codimension of . Clearly determines since is known. The study of growth for PI algebra is mostly the study of the rate of growth of the sequence of its codimensions, as goes to infinity. We have the following basic property. Theorem 1.1 (see [1]). In the associative case, is always exponentially bounded. This theorem implies several key properties for PI algebras. And it fails in various nonassociative cases. Various recent results indicate that in general there is no hope to find a closed formula for . Instead, one therefore tries to determine the asymptotic behavior of that sequence, as goes to infinity. We mention here three such results. Recall that for two sequences of numbers, if . (1) The asymptotics for the matrices , see Section 6. Theorem 1.2 (see [2]). When goes to infinity, (2) The integrality theorem of Giambruno-Zaicev, see Section 10. Theorem 1.3 (see [3]). Let be an associative PI -algebra with , then the limit exists and is an integer. We denote , so . (3)
%U http://www.hindawi.com/journals/isrn.algebra/2012/170697/