On Lie Algebras Generated by Few Extremal Elements

TL;DR

This paper studies Lie algebras generated by up to five extremal elements, revealing their structure via algebraic varieties and showing that maximal cases are nilpotent with extremality maps zero, using computational methods.

Contribution

It characterizes the algebraic varieties of Lie algebras generated by extremal elements for graphs with up to five vertices, demonstrating their structure and nilpotency in maximal cases.

Findings

For graphs with up to 5 vertices, the associated variety is a finite-dimensional affine space.

Maximal-dimensional Lie algebras generated by 5 extremal elements are nilpotent.

The extremality bilinear map must be zero in maximal cases, implying all extremal elements are sandwich elements.

Abstract

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 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.