Polynomial identities in matrix algebras with pseudoinvolution

Abstract

Let $F$ be an algebraically closed field of characteristic zero. In this paper we deal with matrix superalgebras (i.e. algebras graded by $\mathbb{Z}_2$, the cyclic group of order $2$) endowed with a pseudoinvolution. The first goal is to present the classification of the pseudoinvolutions that it is possible to define, up to equivalence, in the full matrix algebra $M_n(F)$ of $n \times n$ matrices and on its subalgebra $UT_n(F)$ of upper-triangular matrices. Along the way we shall give the generators of the $T$-ideal of identities for the algebras $M_2(F)$, $UT_2(F)$ and $UT_3(F)$, endowed with all possible inequivalent pseudoinvolutions.