Conjugacy theorems for loop reductive group schemes and Lie algebras

TL;DR

This paper establishes conjugacy theorems for extended affine Lie algebras using cohomological and geometric methods rooted in reductive group scheme theory, extending classical results from finite and affine cases.

Contribution

It introduces a new cohomological and geometric approach to conjugacy problems in extended affine Lie algebras, generalizing known results for affine and finite-dimensional Lie algebras.

Findings

Proves conjugacy of Cartan subalgebras in extended affine Lie algebras.

Develops a cohomological framework based on reductive group schemes.

Extends classical conjugacy theorems to higher nullity cases.

Abstract

The conjugacy of split Cartan subalgebras in the finite dimensional simple case (Chevalley) and in the symmetrizable Kac-Moody case (Peterson-Kac) are fundamental results of the theory of Lie algebras. Among the Kac-Moody Lie algebras the affine algebras stand out. This paper deals with the problem of conjugacy for a class of algebras --extended affine Lie algebras-- that are in a precise sense higher nullity analogues of the affine algebras. Unlike the methods used by Peterson-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