Finite order automorphisms and real forms of affine KacMoody algebras in the smooth and algebraic category
 Author/Creator
 Heintze, Ernst.
 Language
 English.
 Imprint
 Providence, R.I. : American Mathematical Society, 2012.
 Physical description
 vii, 66 p. : ill ; 26 cm.
 Series
 Memoirs of the American Mathematical Society ; no. 1030.
Access
Available online

Stacks

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QA3 .A57 NO.1030

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QA3 .A57 NO.1030
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Contributors
 Contributor
 Gross, Christian, 1964
Contents/Summary
 Bibliography
 Includes bibliographical references.
 Publisher's Summary
 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 KacMoody 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 KacMoody 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 KacMoody 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.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2012
 Responsibility
 Ernst Heintze, Christian Groß.
 Series
 Memoirs of the American Mathematical Society, 00659266 ; no. 1030
 Note
 "September 2012, volume 219, number 1030 (third of 5 numbers)."
 ISBN
 9780821869185
 0821869183