Affine Kac-Moody groups and Lie algebras in the language of SGA3

TL;DR

This paper describes affine Kac-Moody groups and Lie algebras using the language of SGA3, providing a natural framework for understanding these infinite-dimensional structures through finite type group schemes and Lie algebras.

Contribution

It introduces a novel description of affine Kac-Moody groups and Lie algebras within the SGA3 framework, bridging infinite-dimensional theory with finite type algebraic geometry.

Findings

Provides a natural SGA3-based description of affine Kac-Moody objects

Connects infinite-dimensional Lie theory with finite type algebraic structures

Facilitates new approaches to studying Kac-Moody groups and algebras

Abstract

In infinite-dimensional Lie theory, the affine Kac-Moody Lie algebras and groups play a distinguished role due to their many applications to various areas of mathematics and physics. Underlying these infinite-dimensional objects there are closely related group schemes and Lie algebras of finite type over Laurent polynomial rings. The language of SGA3 is perfectly suited to describe such objects. The purpose of this short article is to provide a natural description of the affine Kac-Moody groups and Lie algebras using this language.