Central polynomials of the second-order matrix algebra with graded involution

Abstract

Let F be an infinite field and M1,1(F) be the algebra of 2×2 matrices over F endowed with non-trivial Z2-grading. We consider the involutions ⁎ defined on M1,1(F) which preserve the homogeneous components of the grading. In this paper, we deal with the ⁎-superalgebra (M1,1(F),⁎) and determine the generators of its ideal of (Z2,⁎)-identities, considering that F has characteristic zero and also, we explicitly construct the generators of its space of central (Z2,⁎)-polynomials, when the characteristic of F is different from 2.