ℤ2-graded ∗-polynomial identities and cocharacteres for M1,1(E), UT1,1(E) and UT(0,1,0)(E)

Let K be a field of characteristic 0, and let E be the infinite-dimensional Grassmann algebra over K. We consider E as a [Formula: see text]-graded algebra, where the grading is given by the vector subspaces [Formula: see text] and [Formula: see text], consisting of monomials of even and odd lengths, respectively. Thus, if [Formula: see text] is an associative [Formula: see text]-graded algebra, we can consider the [Formula: see text]-graded algebra [Formula: see text]. In case both E and A are endowed with superinvolutions, we can define a [Formula: see text]-graded involution on [Formula: see text] induced by the respective superinvolutions. In this paper, we consider the [Formula: see text]-graded matrix algebras [Formula: see text], [Formula: see text] and [Formula: see text] endowed with superinvolutions. We shall provide a description of the polynomial identities and the cocharacter sequences of [Formula: see text], [Formula: see text] and [Formula: see text], considering these resulting algebras as [Formula: see text]-graded algebras with graded involution.