Graded linear maps on superalgebras

Abstract

Let A be an associative algebra over a fixed field F of characteristic zero. In this paper we focus our attention on those algebras A graded by Z2, the cyclic group of order 2. In this case A is said to be a superalgebra and it can be decomposed in the direct sum of homogeneous subspaces: A=A0⊕A1. The main goal of this paper is to prove tight relations between some graded linear maps that can be defined on superalgebras, namely involutions, superinvolutions and pseudoinvolutions. Along the way, we shall present a classification of the pseudoinvolutions that one can define on the algebra UTn(F) of n×n upper-triangular matrices. In the final part of the paper we shall also give some consequences of these results in the context of the theory of polynomial identities.