Geometric models of simple Lie algebras via singularity theory

TL;DR

This paper introduces a geometric construction called the Coxeter wheel that links ADE Dynkin diagrams, root systems, and simple Lie algebras through singularity theory, providing a new geometric perspective.

Contribution

It presents a novel geometric model of simple Lie algebras using Coxeter wheels derived from singularity theory, connecting root systems, Seifert forms, and arcs.

Findings

Coxeter wheels form geometric root systems isomorphic to classical types

The model relates the bilinear form to the Seifert form of singularities

Provides a geometric definition of simple Lie algebras using singularity invariants

Abstract

It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence classes of edges and spokes of a Coxeter wheel form a geometric root system isomorphic to the classical root system of the corresponding type. This wheel is in fact derived from the Milnor fiber of corresponding simple singularities of two variables, and the bilinear form on the geometric root system is the negative of its symmetrized Seifert form. Furthermore, we give a completely geometric definition of simple Lie algebras using arcs, Seifert form and variation operator of the singularity theory.