Lie Algebras, Algebraic Groups, and Lie Groups

186 Pages·2013·1.34 MB·German

Lie Algebras, Algebraic Groups,
and
Lie Groups
J.S. Milne
Version2.00
May5,2013
These notes are an introduction to Lie algebras, algebraic groups, and Lie groups in
characteristiczero,emphasizingtherelationshipsbetweentheseobjectsvisibleintheircat-
egoriesofrepresentations. Eventuallythesenoteswillconsistofthreechapters,eachabout
100pageslong,andashortappendix.
BibTeXinformation:
@misc{milneLAG,
author={Milne, James S.},
title={Lie Algebras, Algebraic Groups, and Lie Groups},
year={2013},
note={Available at www.jmilne.org/math/}
}
v1.00 March11,2012;142pages.
V2.00 May1,2013;186pages.
Pleasesendcommentsandcorrectionstomeattheaddressonmywebsite
http://www.jmilne.org/math/.
ThephotoisofagrottoonThePeakThatFlewHere,Hangzhou,Zhejiang,China.
Copyright(cid:13)c 2012,2013J.S.Milne.
Singlepapercopiesfornoncommercialpersonalusemaybemadewithoutexplicitpermis-
sionfromthecopyrightholder.
Table of Contents
TableofContents 3
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I LieAlgebras 11
1 Definitionsandbasicproperties. . . . . . . . . . . . . . . . . . . . . . . . 11
2 NilpotentLiealgebras: Engel’stheorem . . . . . . . . . . . . . . . . . . . 26
3 SolvableLiealgebras: Lie’stheorem . . . . . . . . . . . . . . . . . . . . . 33
4 SemisimpleLiealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 RepresentationsofLiealgebras: Weyl’stheorem . . . . . . . . . . . . . . 46
6 ReductiveLiealgebras;Levisubalgebras;Ado’stheorem . . . . . . . . . . 56
7 Rootsystemsandtheirclassification . . . . . . . . . . . . . . . . . . . . . 66
8 SplitsemisimpleLiealgebras . . . . . . . . . . . . . . . . . . . . . . . . . 77
9 RepresentationsofsplitsemisimpleLiealgebras . . . . . . . . . . . . . . . 100
10 RealLiealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11 ClassicalLiealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
II AlgebraicGroups 105
1 Algebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
2 Representationsofalgebraicgroups;tensorcategories . . . . . . . . . . . . 110
3 TheLiealgebraofanalgebraicgroup . . . . . . . . . . . . . . . . . . . . 118
4 Semisimplealgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . 134
5 Reductivegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Algebraicgroupswithunipotentcentre . . . . . . . . . . . . . . . . . . . . 151
7 Realalgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8 Classicalalgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 155
III Liegroups 157
1 Liegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
2 Liegroupsandalgebraicgroups . . . . . . . . . . . . . . . . . . . . . . . 158
3 Compacttopologicalgroups . . . . . . . . . . . . . . . . . . . . . . . . . 161
A ArithmeticSubgroups 163
1 Commensurablegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
2 Definitionsandexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
3 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4 Independenceof(cid:26)andL. . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5 Behaviourwithrespecttohomomorphisms . . . . . . . . . . . . . . . . . 166
6 Ade`licdescriptionofcongruencesubgroups . . . . . . . . . . . . . . . . . 167
3
7 Applicationstomanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8 Torsion-freearithmeticgroups . . . . . . . . . . . . . . . . . . . . . . . . 169
9 AfundamentaldomainforSL . . . . . . . . . . . . . . . . . . . . . . . . 169
2
10 Applicationtoquadraticforms . . . . . . . . . . . . . . . . . . . . . . . . 170
11 “Large”discretesubgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 171
12 Reductiontheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
13 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
14 Thecongruencesubgroupproblem . . . . . . . . . . . . . . . . . . . . . . 175
15 ThetheoremofMargulis . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
16 Shimuravarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
Bibliography 181
Index 185
4
Preface
[Lie] did not follow the accepted paths...I
would compare him rather to a pathfinder in
a primal forest who always knows how to find
the way, whereas others thrash around in the
thicket...moreover, his pathway always leads
pastthebestvistas,overunknownmountainsand
valleys.
FriedrichEngel.
Lie algebras are an essential tool in studying both algebraic groups and Lie groups.
ChapterIdevelopsthebasictheoryofLiealgebras,includingthefundamentaltheoremsof
Engel,Lie,Cartan,Weyl,Ado,andPoincare´-Birkhoff-Witt. Theclassificationofsemisim-
pleLiealgebrasintermsoftheDynkindiagramsisexplained,andthestructureofsemisim-
pleLiealgebrasandtheirrepresentationsdescribed.
In Chapter II we apply the theory of Lie algebras to the study of algebraic groups in
characteristiczero. AsCartier(1956)noted,therelationbetweenLiealgebrasandalgebraic
groupsincharacteristiczeroisbestunderstoodthroughtheircategoriesofrepresentations.
For example, when g is a semisimple Lie algebra, the representations of g form a tan-
nakiancategoryRep.g/whoseassociatedaffinegroupG isthesimplyconnectedsemisim-
plealgebraicgroupG withLiealgebrag. Inotherwords,
Rep.G/DRep.g/ (1)
withG asimplyconnectedsemisimplealgebraicgrouphavingLiealgebrag. Itispossible
tocomputethecentreofG fromRep.g/, andtoidentifythesubcategoryofRep.g/corre-
spondingtoeachquotientofG byafinitesubgroup. Thismakesitpossibletoreadoffthe
entiretheoryofsemisimplealgebraicgroupsandtheirrepresentationsfromthe(apparently
simpler)theoryofsemisimpleLiealgebras.
ForageneralLiealgebrag,weconsiderthecategoryRepnil.g/ofrepresentationsofg
such that the elements in the largest nilpotent ideal of g act as nilpotent endomorphisms.
Ado’s theorem assures us that g has a faithful such representation, and from this we are
abletodeduceacorrespondencebetweenalgebraicLiealgebrasandalgebraicgroupswith
unipotentcentre.
LetG beareductivealgebraicgroupwithasplitmaximaltorusT. TheactionofT on
theLiealgebragofG inducesadecomposition
M
gDh˚ g˛; hDLie.T/,
˛2R
of g into eigenspaces g˛ indexed by certain characters ˛ of T, called the roots. A root ˛
determinesacopys ofsl ing. Fromthecompositeoftheexacttensorfunctors
˛ 2
Rep.G/!Rep.g/!Rep.s /(D1)Rep.S /,
˛ ˛
weobtainahomomorphismfromacopyS ofSL intoG. Regard˛ asarootofS ;then
˛ 2 ˛
its coroot ˛_ can be regarded as an element of X(cid:3).T/. The system .X(cid:3).T/;R;˛ 7!˛_/
is a root datum. From this, and the Borel fixed point theorem, the entire theory of split
reductivegroupsoverfieldsofcharacteristiczerofollowseasily.
5
Although there are many books on algebraic groups, and even more on Lie groups,
there are few that treat both. In fact it is not easy to discover in the expository literature
what the precise relation between the two is. In Chapter III we show that all connected
complexsemisimpleLiegroupsarealgebraicgroups,andthatallconnectedrealsemisimple
Liegroupsariseascoveringgroupsofalgebraicgroups. Thusreaderswhounderstandthe
theoryofalgebraicgroupsandtheirrepresentationswillfindthattheyalsounderstandmuch
ofthetheoryofLiegroups. Again,thekeytoolistannakianduality.
Realizing a Lie group as an algebraic group is the first step towards understanding the
discretesubgroupsoftheLiegroup. Wediscussthediscretegroupsthatariseinthiswayin
anappendix.
At present, only the split case is covered in Chapter I, only the semisimple case is
coveredindetailinChapterII,andonlyapartialsummaryofChapterIIIisavailable.
Notations; terminology
We use the standard (Bourbaki) notations: N D f0;1;2;:::g; Z D ring of integers; Q D
fieldofrationalnumbers;RDfieldofrealnumbers;CDfieldofcomplexnumbers;F D
p
Z=pZD field with p elements, p a prime number. For integers m and n, mjn means that
mdividesn,i.e.,n2mZ. Throughoutthenotes,p isaprimenumber,i.e.,pD2;3;5;:::.
Throughout k is the ground field, usually of characteristic zero, and R always denotes
a commutative k-algebra. A k-algebra A is a k-module equipped with a k-bilinear (mul-
tiplication) map A(cid:2)A ! k. Associative k-algebras are required to have an element 1,
and fc1jc 2kg is contained in the centre of the algebra. Unadorned tensor products are
over k. Notations from commutative algebra are as in my primer. When k is a field, ksep
denotes a separable algebraic closure of k and kal an algebraic closure of k. The dual
_
Hom .V;k/ of a k-module V is denoted by V . The transpose of a matrix M is de-
k-linear
noted by Mt. We define the eigenvalues of an endomorphism of a vector space to be the
rootsofitscharacteristicpolynomial.
We use the terms “morphism of functors” and “natural transformation of functors” in-
0
terchangeably. WhenF andF arefunctorsfromacategory,wesaythat“ahomomorphism
F.a/!F0.a/isnaturalina”whenwehaveafamilyofsuchmaps,indexedbytheobjects
a of the category, forming a natural transformation F !F0. For a natural transformation
˛WF !F0,weoftenwrite˛ forthemorphism˛.R/WF.R/!F0.R/. Whenitsactionon
R
morphismsisobvious, weusuallydescribeafunctorF bygivingitsactionR(cid:32)F.R/on
objects. Categories are required to be locally small (i.e., the morphisms between any two
objectsformaset),exceptforthecategoryA_offunctorsA!Set. AdiagramA!B⇒C
is said to be exact if the first arrow is the equalizer of the pair of arrows; in particular, this
meansthatA!B isamonomorphism.
Thesymbol(cid:16)denotesasurjectivemap,and,!aninjectivemap.
Weusethefollowingconventions:
X (cid:26)Y X isasubsetofY (notnecessarilyproper);
X DdefY X isdefinedtobeY,orequalsY bydefinition;
X (cid:25)Y X isisomorphictoY;
X 'Y X andY arecanonicallyisomorphic(orthereisagivenoruniqueisomorphism);
Passages designed to prevent the reader from falling into a possibly fatal error are sig-
nalledbyputtingthesymbolAinthemargin.
ASIDESmaybeskipped; NOTESareoftenreminderstotheauthor.
6
Prerequisites
The only prerequisite for Chapter I (Lie algebras) is the algebra normally taught in first-
year graduate courses and in some advanced undergraduate courses. Chapter II (algebraic
groups)makesuseofsomealgebraicgeometryfromthefirst11chaptersofmynotesAG,
andChapterIII(Liegroups)assumessomefamiliaritywithmanifolds.
References
Inadditiontothereferenceslistedattheend(andinfootnotes),Ishallrefertothefollowing
ofmynotes(availableonmywebsite):
GT GroupTheory(v3.13,2013).
CA APrimerofCommutativeAlgebra(v2.23,2013).
AG AlgebraicGeometry(v5.22,2012).
AGS BasicTheoryofAffineGroupSchemes(v1.00,2012).
The links to GT, CA, AG, and AGS in the pdf file will work if the files are placed in the
samedirectory.
Also,Iusethefollowingabbreviations:
BourbakiA Bourbaki,Alge`bre.
BourbakiLIE Bourbaki,GroupesetAlge`bresdeLie(I1972;II–III1972;IV–VI1981).
DG DemazureandGabriel,GroupesAlge´briques,TomeI,1970.
SophusLie Se´minaire“SophusLie”,Paris,1954–56.
monnnn http://mathoverflow.net/questions/nnnn/
TheworksofCasselmancitedcanbefoundonhishomepageunder“Essaysonrepre-
sentationsandautomorphicforms”.
Acknowledgements
I thank the following for providing comments and corrections for earlier versions of these
notes: LyoshaBeshenov;RolandLoetscher;BhupendraNathTiwari,andothers.
7
DRAMATIS PERSONÆ
JACOBI(1804–1851). Inhisworkonpartialdifferentialequations,hediscoveredtheJacobi
identity. Jacobi’sworkhelpedLietodevelopananalyticframeworkforhisgeometricideas.
RIEMANN (1826–1866). Definedthespacesthatbearhisname. Thestudyofthesespaces
ledtotheintroductionoflocalLiegroupsandLiealgebras.
LIE(1842–1899). Foundedanddevelopedthesubjectthatbearshisnamewiththeoriginal
intentionoffindinga“Galoistheory”forsystemsofdifferentialequations.
KILLING (1847–1923). He introduced Lie algebras independently of Lie in order to un-
derstand the different noneuclidean geometries (manifolds of constant curvature), and he
classifiedthepossibleLiealgebrasoverthecomplexnumbersintermsofrootsystems. In-
troducedCartansubalgebras,Cartanmatrices,Weylgroups,andCoxetertransformations.
MAURER (1859–1927). Histhesiswasonlinearsubstitutions(matrixgroups). Hecharac-
terizedtheLiealgebrasofalgebraicgroups,andessentiallyprovedthatgroupvarietiesare
rational(incharacteristiczero).
ENGEL (1861–1941). IncollaboratingwithLieonthethree-volumeTheoriederTransfor-
mationsgruppen and editing Lie’s collected works, he helped put Lie’s ideas into coherent
formandmakethemmoreaccessible.
E. CARTAN (1869–1951). Corrected and completed the work of Killing on the classifi-
cation of semisimple Lie algebras over C, and extended it to give a classification of their
representations. HealsoclassifiedthesemisimpleLiealgebrasoverR, andheusedthisto
classifysymmetricspaces.
WEYL (1885–1955). He was a pioneer in the application of Lie groups to physics. He
provedthatthefinite-dimensionalrepresentationsofsemisimpleLiealgebrasandsemisim-
pleLiegroupsaresemisimple(completelyreducible).
NOETHER(1882–1935).
They found a classification of semisimple algebras
HASSE(1898–1979).
over number fields, which leads to a classification of
BRAUER(1901–1977).
theclassicalalgebraicgroupsoverthesamefields.
ALBERT(1905–1972).
HOPF(1894–1971). Observedthatamultiplicationmaponamanifolddefinesacomultipli-
cation map on the cohomology ring, and exploited this to study the ring. This observation
ledtothenotionofaHopfalgebra.
VON NEUMANN (1903–1957). Proved that every closed subgroup of a real Lie group is
againaLiegroup.
WEIL(1906–1998). Foundationalworkonalgebraicgroupsoverarbitraryfields. Classified
the classical algebraic groups over arbitrary fields in terms of semisimple algebras with
involution (thereby winning the all India cocycling championship for 1960). Introduced
ade´lesintothestudyofarithmeticproblemsonalgebraicgroups.
CHEVALLEY (1909–1984). He proved the existence of the simple Lie algebras and of
their representations without using a case-by-case argument. Was the leading pioneer in
the development of the theory algebraic groups over arbitrary fields. Classified the split
semisimple algebraic groups over any field, and in the process found new classes of finite
simplegroups.
JACOBSON (1910–1999). Proved that most of the classical results on Lie algebras remain
trueoveranyfieldofcharacteristiczero(atleastforsplitalgebras).
8
KOLCHIN (1916–1991). Obtained the first significant results on matrix groups over arbi-
traryfieldsaspreparationforhisworkondifferentialalgebraicgroups.
IWASAWA (1917–1998). Found the Iwasawa decomposition, which is fundamental for the
structureofrealsemisimpleLiegroups.
HARISH-CHANDRA(1923–1983). IndependentlyofChevalley,heshowedtheexistenceof
thesimpleLiealgebrasandoftheirrepresentationswithoutusingacase-by-caseargument.
WithBorelheprovedsomebasicresultsonarithmeticgroups. Wasoneofthefoundersof
thetheoryofinfinite-dimensionalrepresentationsofLiegroups.
BOREL (1923–2003). He applied algebraic geometry to study algebraic groups, thereby
simplifyingandextendingearlierworkofChevalley,whothenadoptedthesemethodshim-
self. Borel made many fundamental contributions to the theory of algebraic groups and of
theirarithmeticsubgroups.
SATAKE (1927–). He classified reductive algebraic groups over perfect fields (indepen-
dentlyofTits).
TITS(1930–). Histheoryofbuildingsgivesageometricapproachtothestudyofalgebraic
groups, especially the exceptional simple groups. With Bruhat he used them to study the
structureofalgebraicgroupsoverdiscretevaluationrings.
MARGULIS(1946–). ProvedfundamentalresultsondiscretesubgroupsofLiegroups.
9