*-polynomial identities of 4x4 upper triangular matrices with the reflection involution

TL;DR

This paper characterizes all *-polynomial identities for a specific subalgebra of 4x4 upper triangular matrices with a reflection involution, expanding understanding of algebraic identities under involution.

Contribution

It provides a complete description of *-polynomial identities for a particular subalgebra of upper triangular matrices with reflection involution.

Findings

Explicit set of *-polynomial identities identified

Subalgebra's identities differ from full matrix algebra

Advances understanding of involution in upper triangular matrices

Abstract

Let $UT_4(F)$ be $4\times 4$ upper triangular matrix algebra over a field $F$ of characteristic zero and let $\mathcal{A}$ be the subalgebra of $UT_4(F)$ linearly generated by $\{\mathbf{e}_{ij}:1 \leq i\leq j \leq 4 \} \setminus \mathbf{e}_{23}$ where $\{\mathbf{e}_{ij} : 1 \leq i\leq j \leq 4\}$ is the standard basis of $UT_4(F)$. We describe the set of all $*$-polynomial identities for $\mathcal{A}$ with the involution defined by the reflection of second diagonal.