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 $T_2<sup>*$-ideal of $<em>$-identities and of the $T_2^</em>$-subspace of central $<em>$-polynomials of $M_2(F).$ As a consequence, we determine the sequences of $</em>$-codimensions and $\langle n\rangle$-cocharacters of $M_2(F)$. In particular, we prove that the $n$-th $*$-codimension grows like $4<sup>nn<sup>{-1/2}$.