Direct sums of zero product determined algebras

TL;DR

This paper characterizes zero product determined algebras using tensor products, proving that their direct sums are zero product determined if and only if each component is, with applications to Lie algebras.

Contribution

It provides necessary and sufficient conditions for an algebra to be zero product determined and applies these to show that certain Lie algebras are zero product determined.

Findings

Direct sum of zero product determined algebras is zero product determined if all components are.

Finite-dimensional reductive Lie algebras over algebraically closed fields are zero product determined.

Parabolic subalgebras of such Lie algebras are zero product determined.

Abstract

We reformulate the definition of a zero product determined algebra in terms of tensor products and obtain necessary and sufficient conditions for an algebra to be zero product determined. These conditions allow us to prove that the direct sum \bigoplus_{i \in I} A_i of algebras for any index set I is zero product determined if and only if each of the component algebras A_i is zero product determined. As an application, every parabolic subalgebra of a finite-dimensional reductive Lie algebra, over an algebraically-closed field of characteristic zero, is zero product determined. In particular, every such reductive Lie algebra is zero product determined.