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

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.