An Introduction to Kac–Moody Groups over Fields

345 Pages·2018·2.981 MB·English

by  Marquis, Timothée

Textbooks in Mathematics
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Timothée Marquis is
An Introduction to
An Introduction
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Kac–Moody Groups over Fields
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The interest for Kac–Moody algebras and groups has grown exponentially in the past u
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decades, both in the mathematical and physics communities, and with it also the need t
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for an introductory textbook on the topic. o
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The aims of this book are twofold:
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algebras and groups; –
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- to clean the foundations and to provide a unified treatment of the theory.
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The book starts with an outline of the classical Lie theory, used to set the scene. Part II y

provides a self-contained introduction to Kac–Moody algebras. The heart of the book G
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is Part III, which develops an intuitive approach to the construction and fundamental o
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properties of Kac–Moody groups. It is complemented by two appendices, respectively p
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offering introductions to affine group schemes and to the theory of buildings. Many
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exercises are included, accompanying the readers throughout their journey. v
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The book assumes only a minimal background in linear algebra and basic topology, i
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and is addressed to anyone interested in learning about Kac–Moody algebras and/or ld
groups, from graduate (master) students to specialists. s
ISBN 978-3-03719-187-3
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Timothée Marquis
An Introduction
to Kac–Moody
Groups over Fields
Author:
Timothée Marquis
IRMP
Université Catholique de Louvain
Chemin du Cyclotron 2
1348 Louvain-la-Neuve
Belgium
E-mail: [email protected]
2010 Mathematics Subject Classification: 20G44, 20E42, 17B67
Key words: Kac–Moody groups, Kac–Moody algebras, infinite-dimensional Lie theory, highest-
weight modules, semisimple algebraic groups, loop groups, affine group schemes, Coxeter groups,
buildings, BN pairs, Tits systems, root group data
ISBN 978-3-03719-187-3
The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,
and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
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permission of the copyright owner must be obtained.
© 2018 European Mathematical Society
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∞ Printed on acid free paper
9 8 7 6 5 4 3 2 1
ToOliver

Preface
TheinterestforKac–Moodyalgebrasandgroupshasgrownexponentiallyinthe
pastdecades,bothinthemathematicalandphysicscommunities. Inphysics,this
interest has essentially been focused on affine Kac–Moody algebras and groups
(seee.g.[Kac90]),untiltherecentdevelopmentofM-theory,whichalsobrought
into the game certain Kac–Moody algebras and groups of indefinite type (see
e.g., [DHN02], [DN05], [FGKP18]). Within mathematics, Kac–Moody groups
have been studied from a wide variety of perspectives, reflecting the variety of
flavours in which they appear: as for the group functor SL , which associates
n
to each field K the groupSL .K/ D fA 2 Mat .K/ j detA D 1g, Kac–Moody
n n
groupscanbeconstructedoveranyfieldK. Inaddition,Kac–Moodygroupscome
in two versions (minimal and maximal). To a given Kac–Moody algebra is thus
in factattacheda familyofgroups,whosenaturecangreatlyvary. Justto givea
glimpseofthis variety, here is a (neitherexhaustivenorevenrepresentative,and
possiblyrandom)listofrecentresearchdirections.
Note first thatany Kac–Moodygroup naturally acts (in a nice way) on some
geometric object, called a building. Buildings have an extensive theory of their
own(see[AB08])andadmitseveralmetricrealisations,amongstwhichCAT(0)-
realisations. Inturn,CAT(0)-spaceshavebeenextensivelystudied(see[BH99]).
This already provides powerful machineries to study Kac–Moody groups, and
connectsKac–Moodytheorytomanytopicsofgeometricgrouptheory.
Over K D R or K D C, minimal Kac–Moody groups G are connected
Hausdorfftopologicalgroups. In[FHHK17],symmetricspaces(intheaxiomatic
sense of Loos) associated to G are defined and studied. In [Kit14], cohomo-
logical properties of the unitary form K of G (i.e. the analogue of a maximal
compact subgroup in SL .C/) are investigated. Maximal Kac–Moody groups
n
over K D C, on the other hand, have a rich algebraic-geometric structure (see
e.g.,[Mat88b],[Kum02],[Pez17]).
Overanon-ArchimedeanlocalfieldK,theauthorsof[GR14]associatespher-
ical Hecke algebras to Kac–Moody groups of arbitrary type, using a variant of
buildings,calledhovels.
When K is a finite field, minimal Kac–Moodygroupsprovidea class of dis-
cretegroupsthatcombinevariouspropertiesinaverysingularway. Forinstance,
they share many properties with arithmetic groups (see [Re´m09]), but are typi-
cally simple; they in fact provide the first infinite finitely presented examples of
discretegroupsthatarebothsimpleandKazhdan(see[CR09]). Theyalsohelped
constructGolod–ShafarevichgroupsthatdisprovedaconjecturebyE.Zelmanov
(see[Ers08]).
viii Preface
MaximalKac–Moodygroupsoverfinite fields,ontheotherhand,providean
important family of simple (non-discrete) totally disconnected locally compact
groups(see[Re´m12],[CRW17],andalso(cid:2)9.4).
Despite themanifold attractionsofgeneralKac–Moodygroups, the vastma-
jorityoftheworksinKac–MoodytheorystillfocusonaffineKac–Moodygroups.
Westronglybelievethatthisisinpartduetotheabsenceofanintroductorytext-
book on the subject (apart from Kumar’s book [Kum02] which, however, only
covers the case K D C), which can make learning about general Kac–Moody
groupsa longand difficultjourney. The presentbookwas bornoutofthe desire
to fill this gap in the literature, and to provide an accessible, intuitive, reader-
friendly, self-contained and yet concise introduction to Kac–Moody groups. It
also aims at “cleaning” the foundations and providing a unified treatment of the
theory. The targetedaudienceincludesanyoneinterestedin learningaboutKac–
Moodyalgebrasand/orgroups(withaminimalbackgroundinlinearalgebraand
basictopology—thisbookactuallygrewoutoflecturenotesforaMastercourse
on Kac–Moody algebras and groups), as well as more seasoned researchers and
expertsin Kac–Moodytheory,who mayfindin this booksomeclarificationsfor
themanyroughspotsofthecurrentliteratureonKac–Moodygroups. Adescrip-
tion ofthe structureofthebook,aswellas aguidetothe reader,are providedat
theendoftheintroduction.
Toconclude,someacknowledgementsareinorder. Iamverymuchindebted
to Guy Rousseau, first for his paper [Rou16] which made it possible for me to
writeChapter8ofthisbook,andsecondforhisthoroughcommentsonanearlier
versionofthatchapter. IamalsoindebtedtoPierre-EmmanuelCaprace,forintro-
ducingme to the world ofKac–Moodygroupsin the first place, andforhis pre-
ciouscommentsonanearlierversionofChapter7. Finally,Iextendmywarmest
thankstoRalfKo¨hlandanonymousreviewersfortheirpreciouscommentsonan
earlier version of the book. Needless to say, all remaining mistakes are entirely
mine.
Brussels,December2017
Timothe´eMarquis1
1F.R.S.-F.N.R.SResearchFellow
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I A few words onthe classicalLietheory . . . . . . . . . . . . . 13
1 FromLiegroupstoLiealgebras . . . . . . . . . . . . . . . . . . . . 15
2 Finite-dimensional(realorcomplex)Liealgebras . . . . . . . . . . 19
2.1 Afewdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Levidecomposition . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 SemisimpleLiealgebras . . . . . . . . . . . . . . . . . . . . . . 22
2.4 ClassificationofsimpleLiealgebras . . . . . . . . . . . . . . . . 25
II Kac–Moodyalgebras . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Basicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Preliminaries: presentationsandenvelopingalgebra
ofaLiealgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 TowardsKac–Moodyalgebras . . . . . . . . . . . . . . . . . . . 39
3.3 GeneralisedCartanmatrices . . . . . . . . . . . . . . . . . . . . 40
3.4 Gradations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 RealisationsofGCMandKac–Moodyalgebras . . . . . . . . . . 41
3.6 SimplicityofKac–Moodyalgebras . . . . . . . . . . . . . . . . . 49
3.7 Theinvariantbilinearform . . . . . . . . . . . . . . . . . . . . . 51
4 TheWeylgroupofaKac–Moodyalgebra . . . . . . . . . . . . . . . 57
4.1 Integrablemodules . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2 TheWeylgroupofg.A/. . . . . . . . . . . . . . . . . . . . . . . 63
4.3 GeometryoftheWeylgroup . . . . . . . . . . . . . . . . . . . . 72
5 Kac–Moodyalgebrasoffiniteandaffinetype . . . . . . . . . . . . . 75
5.1 TypesofgeneralisedCartanmatrices . . . . . . . . . . . . . . . . 75
5.2 Kac–Moodyalgebrasoffinitetype . . . . . . . . . . . . . . . . . 78
5.3 Kac–Moodyalgebrasofaffinetype* . . . . . . . . . . . . . . . . 79
6 Realandimaginaryroots . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Realroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Imaginaryroots . . . . . . . . . . . . . . . . . . . . . . . . . . . 90