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

81 Pages·2011·0.716 MB·English

by  Gross, Christian, Heintze, Ernst

MEMOIRS
of the
American Mathematical Society
Number 1030
Finite Order Automorphisms
and Real Forms of Affine
Kac-Moody Algebras in the
Smooth and Algebraic Category
Ernst Heintze
Christian Groß
September 2012 • Volume 219 • Number 1030 (third of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 1030
Finite Order Automorphisms
and Real Forms of Affine
Kac-Moody Algebras in the
Smooth and Algebraic Category
Ernst Heintze
Christian Groß
September2012 • Volume219 • Number1030(thirdof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Heintze,Ernst.
Finite order automorphismsand realformsofAffine Kac-Moodyalgebrasin the smoothand
algebraiccategory/ErnstHeintze,ChristianGross.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 1030)
“September2012,volume219,number1030(thirdof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-6918-5(alk. paper)
1. Kac-Moody algebras. 2. Automorphisms. 3. Symmetric spaces. I. Gross, Christian,
1964- II.Title.
QA252.3.H45 2011
512(cid:2).482—dc23 2012015560
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Dedicated to Richard Palais on the occasion of his eightieth birthday

Contents
Chapter 1. Introduction 1
Chapter 2. Isomorphisms between smooth loop algebras 7
Chapter 3. Isomorphisms of smooth affine Kac-Moody algebras 13
Chapter 4. Automorphisms of the first kind of finite order 19
Chapter 5. Automorphisms of the second kind of finite order 25
Chapter 6. Involutions 29
6.1. Involutions of the first kind 29
6.2. Involutions of the second kind 33
Chapter 7. Real forms 37
Chapter 8. The algebraic case 43
8.1. Preliminaries 43
8.2. Isomorphisms between loop algebras 44
8.3. Isomorphisms between affine Kac-Moody algebras 46
8.4. Automorphisms of finite order 48
8.5. Injectivity of AutqL (g,σ)/Aut L (g,σ)→Jq(g,σ) 50
i alg 1 alg i
8.6. Real forms and Cartan decompositions 54
Appendix A. π ((Autg)(cid:2)) and representatives
0
of its conjugacy classes 55
Appendix B. Conjugate linear automorphisms of g 61
Appendix C. Curves of automorphisms of finite order 63
Bibliography 65
v

Abstract
Let g be a real or complex (finite dimensional) simple Lie algebra and σ ∈
Autg. We study automorphisms of the twisted loop algebra L(g,σ) of smooth σ-
periodic maps from R to g as well as of the “smooth” affine Kac-Moody algebra
Lˆ(g,σ), which is a 2-dimensional extension of L(g,σ). 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 g. If the order of the automorphisms is finite, then
the corresponding curves in Autg 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’sclassificationofsymmetricspaces, i.e.ofinvolutionsong. Ifgiscompact,
then conjugate linear extensions of involutions from Lˆ(g,σ) to conjugate linear
involutions on Lˆ(gC,σC) 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.
ReceivedbytheeditorDecember23,2009and,inrevisedform,March29,2011.
ArticleelectronicallypublishedonFebruary15,2012;S0065-9266(2012)00650-2.
2010 MathematicsSubjectClassification. Primary17B67,17B40,53C35.
Key words and phrases. Kac-Moodyalgebras,loopalgebras,automorphismsoffiniteorder,
involutions,realforms,Cartandecompositions.
(cid:2)c2012 American Mathematical Society
vii

CHAPTER 1
Introduction
WestudyautomorphismsoffiniteorderandrealformsofaffineKac-Moodyal-
gebras,thatisofcertainextensionsofthealgebraoftwistedloops(foramoreprecise
definition see below). These automorphisms and real forms have already been con-
sideredextensivelyinthealgebraiccategorywheretheloopsareassumedtohavefi-
niteFourierexpansion([BP],[Kob],[Lev],[Bau],[BR],[Rou1],[Rou2],[Rou3],
[Cor1], [Cor2], [Cor3], [And], [B R], [KW], [Bat], [JZ], [BMR], [BMR(cid:2)]). In
3
particular involutions (i.e. automorphisms of order 2) and real forms have been
classified in the algebraic case ([B R] and [BMR]).
3
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
immediatelytothefinitedimensionalcase. Wemainlyworkinthesmoothcategory
where the loops are assumed to be smooth. But with some modifications and a
resultofLevsteinour methodsalsocarryovertothealgebraic settingandnotonly
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.
Todescribeourapproachandresultsinmoredetail,letgbeafinitedimensional
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∞}
atwistedloopalgebraandLˆ(g,σ):=L(g,σ)+Fc+Fd asmoothaffineKac-Moody
algebraorjust affine Kac-Moodyalgebrainthefollowing. Herecliesinthecenter,
dactsontheloopsasderivationandthebracketbetweentwoloopsisthepointwise
bracket plus a certain multiple of c (cf. Chapter 3).
An isomorphism ϕˆ : Lˆ(g,σ) → Lˆ(˜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
asϕˆarecalledstandardifϕu(t)=ϕ (u(λ(t)))whereλ:R→Risadiffeomorphism
t
and ϕ : g → ˜g is a smooth curve of isomorphisms. Our first main result is the
t
following.
Theorem A. Any isomorphism ϕ : L(g,σ) → L(g,σ˜) is standard. More pre-
cisely,thereexists(cid:6)∈{±1},adiffeomorphismλ:R→Rwithλ(t+2π)=λ(t)+(cid:6)2π
for all t ∈ R and a smooth curve t (cid:5)→ ϕ ∈ Autg of automorphisms with ϕ =
t t+2π
σ˜ϕ σ−(cid:4) such that ϕu(t)=ϕ (u(λ(t))).
t t
1