A class of zero product determined Lie algebras

作者：Wang, Dengyin^*; Yu, Xiaoxiang; Chen, Zhengxin

来源：Journal of Algebra, 2011, 331(1): 145-151.

DOI：10.1016/j.jalgebra.2010.10.037

• Simple Lie algebras
• Parabolic subalgebras
• Zero product determined algebras
• Zero product derivations
• Commutativity-preserving maps

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 phi : L x L -> V, the following condition holds. If phi(x, y) = 0 whenever [x, y] = 0, then there exists a linear map f from [L, L] to V such that phi(x, y) = f([x, y]) for all x, y is an element of 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.

• 出版日期2011-4-1
• 单位福建师范大学; 中国矿业大学（北京）; 江苏师范大学