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Geometric models of simple Lie algebras via singularity theory

Cheol-Hyun Cho, Wonbo Jeong, Beom-Seok Kim

TL;DR

This work builds a geometric realization of ADE Lie algebras from simple curve singularities by harnessing Milnor fibers, monodromy, and Seifert form. It defines a geometric root system $oldsymbol{ extPhi}_oldGamma$ inside $H_1(M,oundary M;b Z)$ via $(oldsymboleta,oldsymboleta)=2$ and shows it is isomorphic to the classical root system of type $oldGamma$, with a pairing given by the negative symmetrized Seifert form. The authors then construct the Coxeter wheel, geometrically realize all roots as arcs and spokes on the wheel, and provide a completely geometric Lie algebra $ rak g_ ext{geo}$ whose brackets reproduce the corresponding simple Lie algebras, together with a Weyl-group action realized as flips of wheel edges and the Coxeter-element/Kostant correspondence. The approach extends to $n$ variables via Thom–Sebastiani stabilization and folding to embed non-simply-laced algebras, offering a unified geometric framework that connects singularity theory, root systems, and Lie theory, with links to Fukaya categories and quiver representations. The framework yields a vivid, diagrammatic interpretation of Lie brackets and Weyl-group dynamics directly from Milnor-fiber geometry, suggesting further connections to representation theory and low-dimensional topology.

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.

Geometric models of simple Lie algebras via singularity theory

TL;DR

This work builds a geometric realization of ADE Lie algebras from simple curve singularities by harnessing Milnor fibers, monodromy, and Seifert form. It defines a geometric root system inside via and shows it is isomorphic to the classical root system of type , with a pairing given by the negative symmetrized Seifert form. The authors then construct the Coxeter wheel, geometrically realize all roots as arcs and spokes on the wheel, and provide a completely geometric Lie algebra whose brackets reproduce the corresponding simple Lie algebras, together with a Weyl-group action realized as flips of wheel edges and the Coxeter-element/Kostant correspondence. The approach extends to variables via Thom–Sebastiani stabilization and folding to embed non-simply-laced algebras, offering a unified geometric framework that connects singularity theory, root systems, and Lie theory, with links to Fukaya categories and quiver representations. The framework yields a vivid, diagrammatic interpretation of Lie brackets and Weyl-group dynamics directly from Milnor-fiber geometry, suggesting further connections to representation theory and low-dimensional topology.

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.

Paper Structure

This paper contains 35 sections, 39 theorems, 60 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Dynkin diagrams, simple Lie algebras and simple singularities
  3. Known relations between simple Lie algebras and simple singularities
  4. From the theory of simple Lie algebras to simple singularities
  5. From simple singularity to the root system
  6. Basic singularity theory
  7. Main result 1: Geometric root system
  8. Main result 3: Geometric construction of Lie algebra
  9. Extension to $n$-variable cases
  10. Main result 4: Description of Weyl group reflections and the Coxeter element
  11. On non-simply-laced simple Lie algebras
  12. Organization of the paper
  13. Milnor fibers of simple curve singularities
  14. Coxeter wheel from the Milnor fiber
  15. $E_{6}$-singularity
  16. ...and 20 more sections

Key Result

Theorem 1.1

The variation operator $\mathrm{var}$ is a linear isomorphism sending relative cycles to the cycles without boundary in $M$.

Figures (28)

  • Figure 1: Geometric roots and the Coxeter wheels for $E$-types.
  • Figure 2: $A_2$ root systems from Lie theory and geometry.
  • Figure 3: Examples of geometric roots.
  • Figure 4: $\mathfrak{sl}_2$ triple for $A_1$-singularity.
  • Figure 5: Milnor fibers of $A_{k}$ and $D_{k}$-singularities.
  • ...and 23 more figures

Theorems & Definitions (68)

  • Theorem 1.1: AGV2
  • Theorem 1.2: AGV2
  • Definition 1.3: Geometric Cartan subalgebra $\mathfrak{h}_{geo}$
  • Remark 1.4
  • Definition 1.5: Geometric pairing on $\mathfrak{h}_{geo}^*$
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 58 more