Constructing simply laced Lie algebras from extremal elements

TL;DR

This paper introduces a geometric construction of simply laced Lie algebras from extremal elements associated with graphs, revealing their structure and degenerations through algebraic varieties.

Contribution

It constructs an algebraic variety parameterizing extremal-generated Lie algebras for any graph and characterizes the case of Dynkin diagrams, providing a new perspective on Lie algebra construction.

Findings

X is an affine space for Dynkin diagrams

Most points in X correspond to a fixed Lie algebra

Degenerations are represented by points outside the dense subset

Abstract

For any finite graph Gamma and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Gamma is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parameterise Lie algebras isomorphic to a single fixed Lie algebra. If Gamma is of affine type, then this fixed Lie algebra is the split finite-dimensional simple Lie algebra corresponding to the associated finite-type Dynkin diagram. This gives a new construction of these Lie algebras, in which they come together with interesting degenerations, corresponding to points outside the open dense subset. Our results may prove useful for recognising these Lie algebras.