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

About this Title

Ernst Heintze, Institut für Mathematik, Universität Augsburg, Universitätsstrasse 14, D - 86159 Augsburg, Germany and Christian Groß, Institut für Mathematik, Universität Augsburg, Universitätsstrasse 14, D - 86159 Augsburg, Germany

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 219, Number 1030
ISBNs: 978-0-8218-6918-5 (print); 978-0-8218-9112-4 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00650-2
Published electronically: February 15, 2012
Keywords: Kac-Moody algebras, loop algebras, automorphisms of finite order, involutions, real forms, Cartan decompositions
MSC: Primary 17B67, 17B40, 53C35

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Table of Contents

Chapters

• 1. Introduction
• 2. Isomorphisms between smooth loop algebras
• 3. Isomorphisms of smooth affine Kac-Moody algebras
• 4. Automorphisms of the first kind of finite order
• 5. Automorphisms of the second kind of finite order
• 6. Involutions
• 7. Real forms
• 8. The algebraic case
• A. $\pi _0 ((\mathrm {Aut}\mathfrak {g})^\varrho )$ and representatives of its conjugacy classes
• B. Conjugate linear automorphisms of $\mathfrak {g}$
• C. Curves of automorphisms of finite order

Abstract

Let $\mathfrak {g}$ be a real or complex (finite dimensional) simple Lie algebra and $\sigma \in \mathrm {Aut}\mathfrak {g}$. We 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 we 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.

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