Polynomial identities, central polynomials and cocharacters of $M_2(F)$ with transpose superinvolution

Abstract

Let $F$ be a field of characteristic zero and consider $M_2(F),$ the algebra of $2\times 2$ matrices over $F,$ with canonical ${\mathbb{Z}}_2$-grading and endowed with transpose superinvolution. In this paper, we present the generators of the $T_2^{\ast }$-ideal of $\ast$-identities and of the $T_2^{\ast }$-subspace of central $\ast$-polynomials of $M_2(F).$ As a consequence, we determine the sequences of $\ast$-codimensions and $⟨n⟩$-cocharacters of $M_2(F)$. In particular, we prove that the $n$-th $\ast$-codimension grows like $4^n{n}^{-1/2}$.

16 pages, comments are welcome!