Coxeter Elements and Root Bases

TL;DR

This paper demonstrates how Coxeter elements induce root bases in Lie algebras of types A, D, E, providing a combinatorial construction where structure constants are derived from quiver paths, linking to Ringel and Peng-Xiao's work.

Contribution

It introduces a new combinatorial method to construct root bases of Lie algebras using Coxeter elements and quivers, connecting to existing algebraic frameworks.

Findings

Root bases can be derived from Coxeter elements.

Structure constants are given by paths in a quiver.

Connects combinatorial construction to Ringel and Peng-Xiao's frameworks.

Abstract

Let g be a Lie algebra of type A,D,E with fixed Cartan subalgebra h, root system R and Weyl group W. We show that a choice of Coxeter element C gives a root basis for g. Moreover we show that this root basis gives a purely combinatorial construction of g, where root vectors correspond to vertices of a certain quiver $Gammahat$, and show that with respect to this basis the structure constants of the Lie bracket are given by paths in $Gammahat$. This construction is then related to the constructions of Ringel and Peng and Xiao.