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 $c_n^*(A)$, $n\ge 1$, 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 $\exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)}$ can be explicitly computed. Here we prove that if $A$ is a star-fundamental algebra, \begin{equation*} C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n, \end{equation*} where $C_1>0,C_2, t$ 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, $\lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n}$ exists and is an integer or half an integer.