Coxeter Elements and Root Bases

Details Coxeter Elements and Root Bases Coxeter Elements and Root Bases

2008-11-14

arxiv.org/abs/0811.2324v1

Mathematics

Representation Theory

13 pages, 4 figures

Scientific paper

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.

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