CHAPTER 2
Isomorphisms between smooth loop algebras
Let g be a finite dimensional Lie algebra over F = R or C and σ Autg (not
necessarily of finite order). Then the (smooth, twisted) loop algebra
L(g,σ) := {u : R g | u(t + 2π) = σu(t) t, u
C∞}
is a Lie algebra with pointwise bracket
[u, v]0(t) := [u(t),v(t)] .
L(g) := L(g,id) is also called the untwisted algebra.
Remark 2.1. One may weaken the regularity assumption in the definition of
L(g,σ) and consider e.g.
Lk(g,σ) := {u : R g | u(t + 2π) = σu(t), u locally of Sobolev class
Hk}
for any k 1. Then the results of this paper go through also for Lk(g,σ) without
difficulties. In fact, the proof that two automorphisms are conjugate (the main
problem) becomes in our approach the less difficult the weaker the regularity is
assumed. The strongest regularity class is the class of algebraic loops which are
given by finite Laurent series, assuming σ to be of finite order. The corresponding
algebra Lalg(g,σ) and its automorphisms will be studied in Chapter 8.
A homomorphism ϕ : L(g,σ)
L(˜,
g ˜) σ between two loop algebras is only
supposed to be F-linear and to preserve brackets, no continuity assumptions are
made. Simple examples of homomorphisms are mappings ϕ : L(g,σ)
L(˜,
g ˜) σ
with (ϕu)(t) = ϕt(u(λ(t))) where ϕt : g
˜
g are homomorphisms and λ : R R is
a function such that t ϕt and λ are smooth (=
C∞).
Definition 2.2. A homomorphism ϕ : L(g,σ) L(˜, g ˜) σ is called standard if
it is of the above form
(ϕu)(t) = ϕt(u(λ(t))) .
The main goal of this chapter is to show that all isomorphisms L(g,σ)
L(˜,
g ˜) σ
are standard if
˜
g is simple.
The following result is obvious.
Lemma 2.3. Let t0 R and I be an open interval around t0. Then there
exists for each smooth function u : I g an ˜ u L(g,σ) with ˜(t) u = u(t) in a
neighborhood of t0. In particular the evaluation map L(g,σ) g, u u(t0), is
surjective.
The assumption that ϕt and λt are smooth in the definition of a standard
homomorphism can be almost deleted.
7
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