Superinvolutions on upper-triangular matrix algebras

Abstract

Let UTn(F) be the algebra of n×n upper-triangular matrices over an algebraically closed field F of characteristic zero. In [18], the authors described all abelian G-gradings on UTn(F) by showing that any G-grading on this algebra is an elementary grading. In this paper, we shall consider the algebra UTn(F) endowed with an elementary Z2-grading. In this way, it has a structure of superalgebra and our goal is to completely describe the superinvolutions which can be defined on it. To this end, we shall prove that the superinvolutions and the graded involutions (i.e., involutions preserving the grading) on UTn(F) are strictly related through the so-called superautomorphisms of this algebra. We shall show that there exist two classes of inequivalent superinvolutions when n is even and a single class otherwise. Along the way, we shall give a complete description of the polynomial identities and the cocharacter sequences of UT2(F) and UT3(F) endowed with all possible superinvolutions.