Involutions for upper triangular matrix algebras

Abstract

In this paper we describe completely the involutions of the first kind of the algebra UT n ( F ) of n × n upper triangular matrices. Every such involution can be extended uniquely to an involution on the full matrix algebra. We describe the equivalence classes of involutions on the upper triangular matrices. There are two distinct classes for UT n ( F ) when n is even and a single class in the odd case. Furthermore we consider the algebra UT 2 ( F ) of the 2 × 2 upper triangular matrices over an infinite field F of characteristic different from 2. For every involution ∗, we describe the ∗-polynomial identities for this algebra. We exhibit bases of the corresponding ideals of identities with involution, and compute the Hilbert (or Poincaré) series and the codimension sequences of the respective relatively free algebras. Then we consider the ∗-polynomial identities for the algebra UT 3 ( F ) over a field of characteristic zero. We describe a finite generating set of the ideal of ∗-identities for this algebra. These generators are quite a few, and their degrees are relatively large. It seems to us that the problem of describing the ∗-identities for the algebra UT n ( F ) of the n × n upper triangular matrices may be much more complicated than in the case of ordinary polynomial identities.