Superidentities for the algebras $U{T}_2$ and $U{T}_3$ on a finite field

Abstract

Let $F$ be a finite field and consider $U{T}_n$ 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 $U{T}_n$ 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.