Some results on $\ast$-minimal algebras with involution

Abstract

Let $(A,\ast)$ be a $\ast$-PI algebra with involution over a field of characteristic zero and let $c_m(A,\ast)$ denote its $m$-th $\ast$-codimension. Giambruno and Zaicev, in [\textit{Involution codimension of finite dimensional algebras and exponential growth}, J. Algebra \textbf{222} (1999), 471--484], proved that, if $A$ is finite dimensional, there exists the $\lim_{m \to +\infty} \sqrt[m]{ c_m(A,\ast)}$, and it is an integer, which is called the \emph{$\ast$-exponent} of $A$. As a consequence of the presence of this invariant, it is natural to introduce the concept of \emph{$\ast$-minimal algebra}. Our goal in this paper is to move some steps towards a complete classification of $\ast$-minimal algebras