Weak identities in the algebra of symmetric matrices of order two

TL;DR

This paper characterizes the weak polynomial identities of the Jordan algebra of 2x2 symmetric matrices over a field of characteristic zero, identifying key identities that generate the weak verbal ideal.

Contribution

It explicitly describes the weak identities for the algebra, highlighting the standard degree four identity and the metabelian identity as generators.

Findings

Weak polynomial identities are generated by the standard degree four identity.

The metabelian identity also generates the weak verbal ideal.

Provides a complete description of weak identities for the algebra.

Abstract

We describe the weak polynomial identities of the Jordan algebra of symmetric $2\times 2$ matrices over a field of characteristic zero. The corresponding weak verbal ideal is generated by the standard identity of degree four and the metabelian identity.