A class of zero product determined Lie algebras

Abstract

Let L be a Lie algebra over a field F. We say that L is zero product determined if, for every F-linear space V and every bilinear map φ : L × L → V , the following condition holds. If φ ( x , y ) = 0 whenever [ x , y ] = 0 , then there exists a linear map f from [ L , L ] to V such that φ ( x , y ) = f ( [ x , y ] ) for all x , y ∈ L . This article shows that every parabolic subalgebra p of a (finite-dimensional) simple Lie algebra defined over an algebraically closed field is always zero product determined. Applying this result, we present a method different from that of Wang et al. (2010) [9] to determine zero product derivations of p , and we obtain a definitive solution for the problem of describing two-sided commutativity-preserving maps on p .