Zero product determined Lie algebras

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 ${\mathfrak{sl}}_2⋉V$, 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.

25 pages

References in corpus (1)

• On non-local representations of the ageing algebra

Cited by in corpus (1)

• Zero action determined modules for associative algebras