Codimensions of algebras with additional structures

Let $A$ be an associative algebra endowed with automorphism or antiautomorphism $\varphi$ of order $\leq 2.$ One associates to $A,$ in a natural way, numerical sequence $c^\varphi_n(A),$ $n=1, 2, \ldots$, called the $\varphi$-codimensions which is main tool for quantitative investigation polynomial identities satisfied by $A$. In \cite{GLM} it was proved that such eventually nondecreasing case antiautomorphism. Here we prove still holds and present some recent results about asymptotics $c^\varphi_n(A)$.