A cohomological proof of Peterson-Kac's theorem on conjugacy of Cartan subalgebras of affine Kac-Moody Lie algebras

TL;DR

This paper presents a cohomological and geometric proof of the conjugacy of Cartan subalgebras in affine Kac-Moody Lie algebras, offering an alternative to previous methods.

Contribution

It introduces a novel cohomological and geometric approach based on reductive group schemes and buildings, differing from Peterson and Kac's original methods.

Findings

Proves conjugacy of Cartan subalgebras using cohomology

Utilizes reductive group schemes and building theory

Provides an alternative proof to existing theorems

Abstract

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings.