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)+
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
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