On Lie algebras generated by few extremal elements

JOURNAL OF ALGEBRA

卷:348

Article

Roozemond, Dan

关键词: Lie algebras;    Extremal elements;    Computational algebra;    Algorithms;    Algebraic geometry;    Computational results;   

DOI  :  10.1016/j.jalgebra.2011.10.015

来源: Elsevier

【 摘 要 】

We give an overview of some properties of Lie algebras generated by at most 5 extremal elements. In particular, for any finite graph Gamma and any field K of characteristic not 2, we consider an algebraic variety X over K whose K-points parametrize Lie algebras generated by extremal elements. Here the generators correspond to the vertices of the graph, and we prescribe commutation relations corresponding to the nonedges of Gamma. We show that, for all connected undirected finite graphs on at most 5 vertices, X is a finite-dimensional affine space. Furthermore, we show that for maximal-dimensional Lie algebras generated by 5 extremal elements. X is a single point. The latter result implies that the bilinear map describing extremality must be identically zero, so that all extremal elements are sandwich elements and the only Lie algebra of this dimension that occurs is nilpotent. These results were obtained by extensive computations with the MAGMA computational algebra system. The algorithms developed can be applied to arbitrary Gamma (i.e., without restriction on the number of vertices), and may be of independent interest. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free