A class of zero product determined Lie algebras

JOURNAL OF ALGEBRA

卷:331

Article

Wang, Dengyin¹  Yu, Xiaoxiang²  Chen, Zhengxin³ 

[1] China Univ Min & Technol, Dept Math, Xuzhou, Jiangsu, Peoples R China

[2] Xuzhou Normal Univ, Sch Math & Comp Sci, Xuzhou, Peoples R China

[3] Fujian Normal Univ, Sch Math & Comp Sci, Fuzhou, Peoples R China

关键词: Simple Lie algebras;    Parabolic subalgebras;    Zero product determined algebras;    Zero product derivations;    Commutativity-preserving maps;   

DOI  :  10.1016/j.jalgebra.2010.10.037

来源: Elsevier

【 摘 要 】

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. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free