Torsors, Reductive Group Schemes and Extended Affine Lie Algebras

TL;DR

This paper explores the structure of torsors related to multiloop algebras, connecting them to extended affine Lie algebras through reductive group schemes and local field theories.

Contribution

It provides a detailed description of torsors for multiloop algebras, linking them to reductive group schemes and extended affine Lie algebras.

Findings

Characterization of torsors for multiloop algebras

Connection between multiloop algebras and extended affine Lie algebras

Bridge established between algebraic structures and reductive group theory

Abstract

We give a detailed description of the torsors that correspond to multiloop algebras. These algebras are twisted forms of simple Lie algebras extended over Laurent polynomial rings. They play a crucial role in the construction of Extended Affine Lie Algebras (which are higher nullity analogues of the affine Kac-Moody Lie algebras). The torsor approach that we take draws heavily for the theory of reductive group schemes developed by M. Demazure and A. Grothendieck. It also allows us to find a bridge between multiloop algebras and the work of J. Tits on reductive groups over complete local fields.