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 A is attached a numerical sequence c n ∗ ( A ) c_n^*(A) , n ≥ 1 n\ge 1 , called the sequence of ∗ * -codimensions of A A . Its asymptotic is an invariant giving a measure of the ∗ * -polynomial identities satisfied by A A . It is well known that for a PI-algebra such a sequence is exponentially bounded and exp ∗ ⁡ ( A ) = lim n → ∞ c n ∗ ( A ) n \exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)} can be explicitly computed. Here we prove that if A A is a star-fundamental algebra, C 1 n t exp ∗ ⁡ ( A ) n ≤ c n ∗ ( A ) ≤ C 2 n t exp ∗ ⁡ ( A ) n , \begin{equation*} C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n, \end{equation*} where C 1 > 0 , C 2 , t C_1>0,C_2, t are constants and t t is explicitly computed as a linear function of the dimension of the skew semisimple part of A A and the nilpotency index of the Jacobson radical of A 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 A is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so, lim n → ∞ log n ⁡ c n ∗ ( A ) exp ∗ ⁡ ( A ) n \lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n} exists and is an integer or half an integer.