Superidentities for the algebras $UT_2$ and $UT_3$ on a finite field

TL;DR

This paper investigates the polynomial identities related to superidentities of upper triangular matrix algebras over finite fields, specifically focusing on cases where the grading group is rac12 and the matrix size is 2 or 3.

Contribution

It classifies all superidentities for the algebras UT_2 and UT_3 over finite fields with rac12-grading, extending previous classifications to finite field cases.

Findings

Classified superidentities for UT_2 over finite fields.

Extended superidentity classification to UT_3.

Provided explicit descriptions of graded polynomial identities.

Abstract

Let $ F $ be a finite field and consider $ UT_n $ the algebra of $ n\times n $ upper triangular matrices over $ F $. In [1], it was proved that every $ G $-grading is elementary. In [2], the authors classified all nonisomorphic elementary $ G $-gradings. They also described the set of all $G $-graded polynomial identities for $ UTn $ when $ F $ is an infinite field. In [3], was described the all $ G $-graded polynomial identities for $ UT_n $ when $ F $ is a finite field. In this work, we will discuss the case when $G = \mathbb{Z}_2$, $ n=2, 3 $ and $ F $ is a finite field.