Zero product determined Lie algebras

TL;DR

This paper investigates which Lie algebras are zero product determined (zpd), characterizing classes of zpd and non-zpd Lie algebras, and exploring implications for commutativity-preserving maps.

Contribution

It provides a classification of important Lie algebras as zpd or non-zpd, including the Galilei algebra and quantum tori, and demonstrates applications in linear map studies.

Findings

Galilei Lie algebra is zpd iff dim V = 2 or odd

Quantum torus and affine Lie algebras are zpd

Aging Lie algebra is non-zpd

Abstract

A Lie algebra $L$ over a field $\mathbb{F}$ is said to be zero product determined (zpd) if every bilinear map $f:L\times L\to \mathbb{F}$ with the property that $f(x,y)=0$ whenever $x$ and $y$ commute is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra $\mathfrak{sl}_2\ltimes V$, where $V$ is a simple $\mathfrak{sl}_2$-module, is zpd if and only if $\dim V =2$ or $\dim V$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $\mathcal{L}_q$ and $\mathcal{L}^+_q$, the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most $3$, while the class of non-zpd Lie algebras includes the ($4$-dimensional) aging Lie algebra $\mathfrak {age}(1)$ and all Lie algebras of dimension more than $3$ in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of a zpd Lie algebra by using it in the study of commutativity preserving linear maps.