$\mathbb{Z}_2$-graded polynomial identities for the Jordan algebra of $2\times 2$ upper triangular matrices

TL;DR

This paper characterizes all $Z_2$-graded polynomial identities for the Jordan algebra of $2\times 2$ upper triangular matrices over a field with characteristic not 2, providing a basis for the related free algebra.

Contribution

It explicitly describes the $Z_2$-graded polynomial identities and constructs a basis for the free algebra of the Jordan algebra $UJ_2$ with any nontrivial grading.

Findings

Complete set of graded identities for $UJ_2$ identified.

A linear basis for the free graded algebra constructed.

Results applicable to fields of any characteristic not 2.

Abstract

Let $K$ be a field (finite or infinite) of char$(K)\neq 2$ and let $UT_n=UT_n(K)$ be the $n\times n$ upper triangular matrix algebra over $K$. If $\cdot $ is the usual product on $UT_n$ then with the new product $a\circ b=(1/2)(a\cdot b +b\cdot a)$ we have that $UT_n$ is a Jordan algebra, denoted by $UJ_n=UJ_n(K)$. In this paper, we describe the set of all $\mathbb{Z}_2$-graded polynomial identities of $UJ_2$ with any nontrivial $\mathbb{Z}_2$-grading. Moreover, we describe a linear basis for the corresponding relatively free $\mathbb{Z}_2$-graded algebra.