Polynomial identities with involution for the algebra of 3 $\times$ 3 upper triangular matrices

TL;DR

This paper characterizes polynomial identities and central polynomials with involution for the algebra of 3x3 upper triangular matrices over a field, focusing on cases with characteristic not equal to 2.

Contribution

It provides a complete description of *-central polynomials and *-identities for UT_3 over fields with specific characteristics, extending understanding of involution-related polynomial identities.

Findings

Describes all *-central polynomials for UT_n when n≥3 and p≠2.

Identifies all *-polynomial identities for UT_3 over infinite fields with p>2.

Abstract

Let $\mathbb{F}$ be a field of characteristic $p$, and let $UT_n(\mathbb{F})$ be the algebra of $n \times n$ upper triangular matrices over $\mathbb{F}$ with an involution of the first kind. In this paper we describe: the set of all $*$-central polynomials for $UT_n(\mathbb{F})$ when $n\geq 3$ and $p\neq 2$ ; the set of all $*$-polynomial identities for $UT_3(\mathbb{F})$ when $\mathbb{F}$ is infinite and $p>2$.