Three-point Lie algebras and Grothendieck's dessins d'enfants

TL;DR

This paper introduces and classifies three-point Lie algebras related to the complex projective line minus three points, using Grothendieck's dessins d'enfants, and investigates their Cartan subalgebra conjugacy.

Contribution

It defines and classifies three-point Lie algebras via dessins d'enfants, extending the theory of affine Kac-Moody algebras to new geometric contexts.

Findings

Classification of three-point Lie algebras using dessins d'enfants

Analysis of conjugacy classes of Cartan subalgebras in these algebras

Connection between algebraic structures and geometric dessins

Abstract

We define and classify the analogues of the affine Kac-Moody Lie algebras for the ring corresponding to the complex projective line minus three points. The classification is given in terms of Grothendieck's dessins d'enfants. We also study the question of conjugacy of Cartan subalgebras for these algebras.