Jordan maps and zero Lie product determined algebras

TL;DR

This paper investigates conditions under which certain algebraic maps satisfy specific identities, contributing to understanding zero Lie product determined algebras and properties of Jordan homomorphisms.

Contribution

It establishes new criteria for skew-symmetric bilinear maps in algebras generated by specific commutator subspaces, impacting the study of zero Lie product determined algebras and Jordan homomorphisms.

Findings

Conditions for bilinear maps imply algebraic identities

Application to zero Lie product determined algebras

Proof that Jordan homomorphisms decompose into homomorphisms and antihomomorphisms

Abstract

Let $A$ be an algebra over a field $F$ with {\rm char}$(F)\ne 2$. If $A$ is generated as an algebra by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ is an arbitrary vector space over $F$, the condition that $\Phi(x^2,x)=0 $ for all $x\in A$ implies that $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ for all $x,y,z\in A$. This is applicable to the question of whether $A$ is zero Lie product determined, and is also used in proving that a Jordan homomorphism from $A$ onto a semiprime algebra $B$ is the sum of a homomorphism and an antihomomorphism.