Star-fundamental algebras: Polynomial identities and asymptotics

Abstract

We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution. To any star-algebra A is attached a numerical sequence cn(A), n ≥ 1, called the sequence of-codimensions of A. Its asymptotic is an invariant giving a measure of the-polynomial identities satisfied by A. It is well known that for a PI-algebra such a sequence is exponentially bounded and exp(A) = limn→∞ n√cn(A) can be explicitly computed. Here we prove that if A is a star-fundamental algebra, (equation presented) where C1 > 0, C2, t are constants and t is explicitly computed as a linear function of the dimension of the skew semisimple part of A and the nilpotency index of the Jacobson radical of A. We also prove that any finite dimensional star-algebra has the same-identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in [J. Algebra 383 (2013), pp. 144-167] we get that if A is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, limn→∞ logn cn(A) exp(A)n exists and is an integer or half an integer.