Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category

TL;DR

This paper classifies finite order automorphisms and real forms of smooth and algebraic affine Kac-Moody algebras, extending Cartan's finite-dimensional involution classification to infinite-dimensional cases.

Contribution

It provides a unified parametrization of automorphisms and real forms of affine Kac-Moody algebras in both smooth and algebraic contexts, linking to classical Cartan classifications.

Findings

Automorphisms are parametrized by specific invariants.

Involutions classification aligns with Cartan's finite-dimensional results.

The approach applies equally in smooth and algebraic settings.

Abstract

Automorphisms of finite order and real forms of "smooth" affine Kac-Moody algebras are studied, i.e. of 2-dimensional extensions of the algebra of smooth loops in a simple Lie algebra. It is shown that they can be parametrized by certain invariants and that in particular the classification of involutions essentially follows from Cartan's classifications in finite dimensions. We also prove that our approach works equally well in the usual algebraic setting and leads to the same results there.