# Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category

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*Ernst Heintze; Christian Groß*

Let \(\mathfrak{g}\) be a real or complex (finite
dimensional) simple Lie algebra and
\(\sigma\in\mathrm{Aut}\mathfrak{g}\). The authors study automorphisms
of the twisted loop algebra \(L(\mathfrak{g},\sigma)\) of smooth
\(\sigma\)-periodic maps from \(\mathbb{R}\) to
\(\mathfrak{g}\) as well as of the “smooth” affine Kac-Moody
algebra \(\hat L(\mathfrak{g},\sigma)\), which is a
\(2\)-dimensional extension of \(L(\mathfrak{g},\sigma)\). It
turns out that these automorphisms which either preserve or reverse the
orientation of loops, and are correspondingly called to be of first and second
kind, can be described essentially by curves of automorphisms of
\(\mathfrak{g}\). If the order of the automorphisms is finite, then the
corresponding curves in \(\mathrm{Aut}\mathfrak{g}\) allow us to define
certain invariants and these turn out to parametrize the conjugacy classes of
the automorphisms. If their order is \(2\) the authors carry this out in
detail and deduce a complete classification of involutions and real forms
(which correspond to conjugate linear involutions) of smooth affine Kac-Moody
algebras.

The resulting classification can be seen as an extension of Cartan's
classification of symmetric spaces, i.e. of involutions on
\(\mathfrak{g}\). If \(\mathfrak{g}\) is compact, then conjugate
linear extensions of involutions from \(\hat L(\mathfrak{g},\sigma)\) to
conjugate linear involutions on \(\hat
L(\mathfrak{g}_{\mathbb{C}},\sigma_{\mathbb{C}})\) yield a bijection
between their conjugacy classes and this gives existence and uniqueness of
Cartan decompositions of real forms of complex smooth affine Kac-Moody
algebras.

The authors show that their methods work equally well also in the algebraic
case where the loops are assumed to have finite Fourier expansions.

#### Table of Contents

# Table of Contents

## Finite Order Automorphisms and Real Forms of Affine Kac-Moody Algebras in the Smooth and Algebraic Category

- Chapter 1. Introduction 110 free
- Chapter 2. Isomorphisms between smooth loop algebras 716 free
- Chapter 3. Isomorphisms of smooth affine Kac-Moody algebras 1322
- Chapter 4. Automorphisms of the first kind of finite order 1928
- Chapter 5. Automorphisms of the second kind of finite order 2534
- Chapter 6. Involutions 2938
- Chapter 7. Real forms 3746
- Chapter 8. The algebraic case 4352
- Appendix A. 0 ((Autg)) and representatives of its conjugacy classes 5564
- Appendix B. Conjugate linear automorphisms of g 6170
- Appendix C. Curves of automorphisms of finite order 6372
- Bibliography 6574