Zero product determined n-th Schrödinger algebra

Abstract

The n-th Schrödinger algebra schn:=sl2⋉hn is the semi-direct product of the Lie algebra sl2 with the n-th Heisenberg Lie algebra hn. Let K be an algebraically closed field of characteristic zero. A Lie algebra L is zero product determined over K, if for every K-linear space V and every bilinear map ϕ:L×L→V with the property that ϕ(x,y)=0 whenever [x,y]=0, then there exists a linear map f:[L,L]→V such that ϕ(x,y)=f([x,y]) for all x,y∈L. This article shows that the n-th Schrödinger algebra schn is zero product determined. Applying this result, product zero derivations and two-sided commutativity-preserving maps on schn are determined. Furthermore, quasi-derivations, linear anti-commuting maps, quasi-automorphisms and strong commutativity-preserving maps of schn are all obtained.