${\mathbb{Z}}_2$-graded $\ast$-polynomial identities and cocharacteres for $M_{1,1}(E)$, $U{T}_{1,1}(E)$ and $U{T}_{(0,1,0)}(E)$

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 $M_{1,1}(K)$, $U{T}_{1,1}(K)$, and $U{T}_{(0,1,0)}(K)$ endowed with superinvolutions. We shall provide a description of the polynomial identities and the cocharacter sequences of $M_{1,1}(K)\hat{\otimes }E$, $U{T}_{1,1}(K)\hat{\otimes }E$, and $U{T}_{(0,1,0)}(K)\hat{\otimes }E$, considering these resulting algebras as ${\mathbb{Z}}_2$-graded algebras with graded involution.