On the structure of Kac-Moody algebras

TL;DR

This paper investigates the structural properties of Kac-Moody algebras, focusing on conditions for nonzero brackets of homogeneous elements and characterizing their solvable and nilpotent subalgebras.

Contribution

It provides new criteria for when brackets of homogeneous elements are nonzero and describes the solvable and nilpotent graded subalgebras of Kac-Moody algebras.

Findings

Criteria for nonzero brackets of homogeneous elements

Description of solvable graded subalgebras

Description of nilpotent graded subalgebras

Abstract

Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak g(A)$: given two homogeneous elements $x,y \in \mathfrak g(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak g(A)$.