Polynomial identities of bicommutative algebras, Lie and Jordan elements

TL;DR

This paper explores the polynomial identities in bicommutative algebras, focusing on the properties of Lie and Jordan elements, and provides criteria for identifying these elements within free bicommutative algebras.

Contribution

It constructs a list of identities satisfied by commutator and anti-commutator products in free bicommutative algebras and establishes criteria for Lie and Jordan elements.

Findings

Identified identities satisfied by commutator and anti-commutator in free bicommutative algebras

Provided criteria for elements to be Lie or Jordan in these algebras

Enhanced understanding of algebraic structures in bicommutative systems

Abstract

An algebra with identities $a(bc)=b(ac),$ $(ab)c=(ac)b$ is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan.