Extremal Presentations for Classical Lie Algebras

TL;DR

This paper presents minimal extremal generator presentations for classical Lie algebras of Chevalley type over algebraically closed fields, using simple graph-based relations.

Contribution

It introduces a novel approach to presenting classical Lie algebras via extremal generators and simple graph relations, simplifying their structural understanding.

Findings

Presentations for $C_n$ and $A_n$ Lie algebras using extremal generators.

Relations described by simple graphs, such as paths and triangles.

Provides minimal sets of extremal generators for classical Lie algebras.

Abstract

The long-root elements in Lie algebras of Chevalley type have been well studied and can be characterized as extremal elements, that is, elements $x$ such that the image of $(\ad x)^2$ lies in the subspace spanned by $x$. In this paper, assuming an algebraically closed base field of characteristic not 2, we find presentations of the Lie algebras of classical Chevalley type by means of minimal sets of extremal generators. The relations are described by simple graphs on the sets. For example, for $C_n$ the graph is a path of length $2n$, and for $A_n$ the graph is the triangle connected to a path of length $n-3$.