Asymptotic Behaviour of Codimensions of P. I. Algebras Satisfying Capelli Identities

Let A be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence cn(A) is asymptotic to a function of the form ang n, where ∈ N and g ∈ Z. The second author conjectured that if A is any p. i. algebra in characteristic zero, with cocharacter sequence cn(A), then the asymptotic behaviour of this sequence should be given by cn(A) a · n · , where is a non-negative integer, t is an integer or a half-integer, and a belongs to Q[ √ 2π, √ b], for some 0 < b ∈ Z. We call n the exponential part and a · n the rational part of the codimension growth. Giambruno and Zaicev made progress on this conjecture by proving that for some non-negative integer , f1(n) n ≤ cn(A) ≤ f2(n) , where f1(n) and f2(n) are Laurent polynomials, but not necessarily of the same degree. In this paper we make some progress towards verifying the above conjecture by proving the following theorem. Theorem 1. Let A be a p.i. algebra with 1 satisfying a Capelli identity. Then cn(A) a · n · n where, furthermore, g ∈ 12Z. Also, the constant a here is a sum of Selberg–type integrals; see for example Theorem 36 below. This theorem can help to determine whether or not the generating function f(x) = ∑ n≥0 cn(A)x n of the codimensions is algebraic. For example, if g ∈ Z and is negative, then f(x) is not algebraic; see Lemma 3.2 in [1]. This application raises the following further question about Theorem 1: when is g < 0? The techniques of [6] imply the following: Let Aj , j = 1, 2, be p. i. algebras with T–ideals of identities id(Aj) = Ij . This is known to be the case, for example, Received by the editors June 5, 2006. 2000 Mathematics Subject Classification. Primary 16R10.