Classificação das graduações da álgebra de Jordan das matrizes triangulares superiores e suas relações com as identidades polinomiais graduadas

Abstract

This work deals the classification of group gradings on the Jordan algebra UJn, obtained from the associative algebra UTn(K) of upper triangular matrices, equipped with the Jordan product. The study is situated with in the context of Polynomial Identities Theory, in which gradings play a central role in understanding graded identities and in the structural description of PI-algebras. In this context, we analyze the classification of gradings of UJn based on the results according to which every G-gradings of UJn, over infinite fields of characteristic different from 2, belongs to one of two classes: elementary or mirror type (MT). Moreover, two gradings are isomorphic if and only if they satisfy the same graded identities. Since relatively little is known about the behavior of the graded identities of this algebra, this work seeks to consolidate and systematize the existing results, contributing to a broader understanding of a recent and still developing theory.