CHAPTER 1

Introduction

We study automorphisms of finite order and real forms of aﬃne Kac-Moody al-

gebras, that is of certain extensions of the algebra of twisted loops (for a more precise

definition see below). These automorphisms and real forms have already been con-

sidered extensively in the algebraic category where the loops are assumed to have fi-

nite Fourier expansion ([BP], [Kob], [Lev], [Bau], [BR], [Rou1], [Rou2], [Rou3],

[Cor1], [Cor2], [Cor3], [And], [B3R], [KW], [Bat], [JZ], [BMR], [BMR ]). In

particular involutions (i.e. automorphisms of order 2) and real forms have been

classified in the algebraic case ([B3R] and [BMR]).

Our approach is different in that it is more elementary and direct. It does not

use the structure theory of Kac-Moody algebras but rather reduces the problems

immediately to the finite dimensional case. We mainly work in the smooth category

where the loops are assumed to be smooth. But with some modifications and a

result of Levstein our methods also carry over to the algebraic setting and not only

seem to give much simpler and shorter proofs of the existing results there but also

new insights. For example, it turns out that involutions (of the second kind) and

real forms of aﬃne Kac-Moody algebras are in close connection with hyperpolar

actions on compact Lie groups. Furthermore we show that Cartan decompositions

of real forms always exist.

It might be interesting to mention that our methods also work in the Hk-case,

k ≥ 1, where the loops are assumed to be of Sobolev class Hk.

To describe our approach and results in more detail, let g be a finite dimensional

simple Lie algebra over F := R or C and σ ∈ Autg be an arbitrary automorphism,

not necessarily of finite order. We call

L(g,σ) := {u : R → g | u(t + 2π) = σu(t),u ∈

C∞}

a twisted loop algebra and

ˆ(g,σ)

L := L(g,σ)+Fc+Fd a smooth aﬃne Kac-Moody

algebra or just aﬃne Kac-Moody algebra in the following. Here c lies in the center,

d acts on the loops as derivation and the bracket between two loops is the pointwise

bracket plus a certain multiple of c (cf. Chapter 3).

An isomorphism ˆ ϕ :

ˆ(g,σ)

L →

ˆ(˜,

L g ˜) σ between two such algebras induces

an isomorphism ϕ : L(g,σ) → L(˜, g ˜) σ between the loop algebras (which are the

quotients of the derived algebras by their center Fc). The isomorphisms ϕ as well

as ˆ ϕ are called standard if ϕu(t) = ϕt(u(λ(t))) where λ : R → R is a diffeomorphism

and ϕt : g →

˜

g is a smooth curve of isomorphisms. Our first main result is the

following.

Theorem A. Any isomorphism ϕ : L(g,σ) → L(g, ˜) σ is standard. More pre-

cisely, there exists ∈ {±1}, a diffeomorphism λ : R → R with λ(t+2π) = λ(t)+ 2π

for all t ∈ R and a smooth curve t → ϕt ∈ Autg of automorphisms with ϕt+2π =

˜ σϕ

t

σ− such that ϕu(t) = ϕt(u(λ(t))).

1