$\mathbb{Z}_2$-graded $*$-polynomial identities and cocharacteres for $M_{1,1}(E)$, $UT_{1,1}(E)$ and $UT_{(0,1,0)}(E)$

Abstract

Let $K$ be a field of characteristic 0, and let $E$ be the infinite-dimensional Grassmann algebra over $K$. We consider $E$ as a $\mathbb{Z}_2$-graded algebra, where the grading is given by the vector subspaces $E_0$ and $E_1$, consisting of monomials of even and odd lengths, respectively. Thus, if $A = A_0 \oplus A_1$ is an associative $\mathbb{Z}_2$-graded algebra, we can consider the $\mathbb{Z}_2$-graded algebra $A \hat{\otimes} E = (A_0 \otimes E_0) \oplus (A_1 \otimes E_1)$. In case both $E$ and $A$ are endowed with superinvolutions, we can define a $\mathbb{Z}_2$-graded involution on $A \hat{\otimes} E$ induced by the respective superinvolutions. In this paper, we consider the $\mathbb{Z}_2$-graded matrix algebras $M_{1,1}(K)$, $UT_{1,1}(K)$, and $UT_{(0,1,0)}(K)$ endowed with superinvolutions. We shall provide a description of the polynomial identities and the cocharacter sequences of $M_{1,1}(K)\hat{\otimes} E$, $UT_{1,1}(K)\hat{\otimes} E$, and $UT_{(0,1,0)}(K)\hat{\otimes} E$, considering these resulting algebras as $\mathbb{Z}_2$-graded algebras with graded involution.