Group gradings on the Jordan algebra of upper triangular matrices

TL;DR

This paper classifies all group gradings on the Jordan algebra of upper triangular matrices over a field, revealing two main types and showing they are uniquely determined by their graded identities.

Contribution

It provides a complete classification of G-gradings on this Jordan algebra, including elementary and mirror type gradings, and establishes their uniqueness via graded identities.

Findings

Two families of gradings identified: elementary and mirror type.

Gradings are classified up to graded isomorphism.

Gradings are uniquely determined by their graded identities.

Abstract

Let G be an arbitrary group and let K be a field of characteristic different from 2. We classify the G-gradings on the Jordan algebra of upper triangular matrices of order n over K. It turns out that there are, up to a graded isomorphism, two families of gradings: the elementary gradings (analogous to the ones in the associative case), and the so called mirror type (MT) gradings. Moreover we prove that the G-gradings on this algebra are uniquely determined, up to a graded isomorphism, by the graded identities they satisfy.