On superalgebras with pseudoautomorphism of polynomial codimension growth (Q6600692)

Let \(A\) be an associative superalgebra over a field of characteristic zero, endowed with a so-called \textit{pseudoautomorphism}; that is, we are given a linear map \(p:A\to A\) preserving the \(\mathbb{Z}/2\mathbb{Z}\)-grading, moreover, \(p\) satisfies \(p^2(a)=(-1)^{|a|}a\), and \(p(ab)=(-1)^{|a||b|}p(a)p(b)\) for \(a,b\in A_0\cup A_1\). After proving a variant of the Wedderburn-Malcev Therorem for superalgebras with pseudoautomorphism, the author turns to the study of the codimension growth of the polynomial identities in this setup. The main result is a finite list of superalgebras with pseudoautomorphism such that the identities of \(A\) have polynomially bounded codimension growth if and only if for each algebra \(B\) in the list, there exists an identity of superalgebras with prseudoautomorphism satisfied by \(A\) but not by \(B\).