Zero product determined Lie algebras

Abstract

A Lie algebra L over a field F is said to be zero product determined (zpd) if every bilinear map f : L × L → 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 sl2 ⋉ V , where V is a simple sl2-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 Lq and 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 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.