Algebraic Groups and Arithmetic Groups

219 Pages·2006·1.49 MB·English

Algebraic Groups and Arithmetic Groups
J.S. Milne
Version1.01
June4,2006
These notes provide an introductory overview of the theory of algebraic groups, Lie
algebras,Liegroups,andarithmeticgroups. Theyarearevisionofthosepostedduringthe
teachingofacourseatCMS,ZhejiangUniversity,HangzhouinSpring,2005.
v0.00 (February28–May7,2005). Aspostedduringthecourse.
v1.00 May22,2005. Minorcorrectionsandrevisions;addedtableofcontentsandindex.
v1.01 June4,2006. Fixedproblemwiththediagrams.
[email protected]
Availableathttp://www.jmilne.org/math/
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Contents
Introduction 1
Notations2;Prerequisites2;References3
1 Overviewandexamples 4
Thebuildingblocks4;Semisimplegroups5;Extensions6;Summary7;Exercises8
2 Definitionofanaffinealgebraicgroup 10
Principle of permanence of identities 10; Affine algebraic groups 10; Homomorphisms of al-
gebraic groups 13; The Yoneda lemma 13; The coordinate ring of an algebraic group 14; Very
brief review of tensor products. 14; Products of algebraic groups 15; Fibred products of al-
gebraic groups 15; Extension of the base field (extension of scalars) 15; Algebraic groups and
bi-algebras16;Homogeneity18;Reducedalgebrasandtheirtensorproducts19;Reducedalge-
braic groups and smooth algebraic groups 20; Smooth algebraic groups and group varieties 20;
Algebraicgroupsincharacteristiczeroaresmooth22;Cartierduality23;Exercises24
3 Linearrepresentations 25
Linearrepresentationsandcomodules25;Stabilizersofsubspaces30
4 MatrixGroups 32
Anelementaryresult32;Howtogetbialgebrasfromgroups32;Alittlealgebraicgeometry33;
Variant34;ClosedsubgroupsofGL andalgebraicsubgroups35
n
5 Example: thespingroup 36
Quadratic spaces 36; The orthogonal group 40; Super algebras 40; Brief review of the tensor
algebra41;TheCliffordalgebra42;TheSpingroup46;TheCliffordgroup47;ActionofO.q/
onSpin.q/48;Restatementintermsofalgebraicgroups48
6 GroupTheory 49
Reviewofgrouptheory49;Reviewofflatness49;Thefaithfulflatnessofbialgebras51;Definitions;
factorizationtheorem51;Embeddings;subgroups. 52;Kernels52;Quotientmaps54;Existence
ofquotients55;Theisomorphismtheorem56
7 Finite(e´tale)algebraicgroups 58
Separable k-algebras 58; Classification of separable k-algebras 59; Etale algebraic groups 60;
Examples60
8 Theconnectedcomponentsofanalgebraicgroup 62
Somealgebraicgeometry62;Separablesubalgebras64;Thegroupofconnectedcomponentsof
an algebraic group 65; Connected algebraic groups 66; Exact sequences and connectedness 68;
Whereweare69
9 Diagonalizablegroups;tori 70
A remark about homomorphisms 70; Group-like elements in a bialgebra 70; The characters of
an algebraic group 70; The algebraic group D.M/ 71; Characterizing the groups D.M/ 72;
Diagonalizablegroups73;Diagonalizablegroupsarediagonalizable74;Splittoriandtheirrep-
resentations75;Rigidity76;Groupsofmultiplicativetype76
10 Jordandecompositions 78
Jordannormalforms78;JordandecompositioninGL .V/(k Dk)79;Jordandecompositionin
n
GL.V/,k perfect80;Infinite-dimensionalvectorspaces81;Theregularrepresentationcontains
all81;TheJordandecompositionintheregularrepresentation82
11 Solvablealgebraicgroups 85
Brief review of solvable groups (in the usual sense) 85; Remarks on algebraic subgroups 85;
Commutativegroupsaretriangulizable86;Decompositionofacommutativealgebraicgroup87;
Thederivedgroupofalgebraicgroup88;Definitionofasolvablealgebraicgroup89;Independence
ofcharacters90;TheLie-Kolchintheorem91;Unipotentgroups92;Structureofsolvablegroups
93;Toriinsolvablegroups93;Theradicalofanalgebraicgroup94;Structureofageneral(affine)
algebraicgroup94;Exercises95
12 TheLiealgebraofanalgebraicgroup: basics 96
Lie algebras: basic definitions 96; The Lie algebra of an algebraic group 97; The functor Lie
98; Examples 98; Extension of the base field 101; Definition of the bracket 101; Alternative
constructionofthebracket. 102;Theunitarygroup103; Liepreservesfibredproducts104
13 TheLiealgebraofanalgebraicgroup 106
Somealgebraicgeometry106;Applications107;Stabilizers108;Isotropygroups109;Normalizers
andcentralizers110;Anastyexample111
14 SemisimplealgebraicgroupsandLiealgebras 112
Semisimple Lie algebras 112; Semisimple Lie algebras and algebraic groups 112; The map ad
113; The Lie algebra of Aut .C/ 113; The map Ad 114; Interlude on semisimple Lie algebras
k
115;Semisimplealgebraicgroups119
15 Reductivealgebraicgroups 121
Structure of reductive groups 121; Representations of reductive groups 122; A criterion to be
reductive124
16 Splitreductivegroups: theprogram 126
Splittori126;Splitreductivegroups127;Program129
17 Therootdatumofasplitreductivegroup 130
Roots130;Example: GL 130;Example: SL 130;Example: PGL 131;Example: GL 131;
2 2 2 n
Definitionofarootdatum132;Firstexamplesofrootdata132;Semisimplegroupsofrank0or1
134;Centralizersandnormalizers134;Definitionofthecoroots135;Computingthecentre137;
Semisimpleandtoralrootdata137;Themaintheorems. 138;Examples138
18 Generalitiesonrootdata 142
Definition142
19 Classificationofsemisimplerootdata 146
Generalities on symmetries 146; Generalities on lattices 147; Root systems 147; Root systems
andsemisimplerootdata148;Thebigpicture149;Classificationofthereducedrootsystem149;
TheCoxetergraph153
20 Theconstructionofallsplitreductivegroups 155
Preliminariesonrootdata/systems155;Briefreviewofdiagonalizablegroups156;Construction
ofallalmost-simplesplitsemisimplegroups157;Splitsemisimplegroups. 157;Splitreductive
groups157;Exercise157
21 Borelfixedpointtheoremandapplications 158
Briefreviewofalgebraicgeometry158;TheBorelfixedpointtheorem159;Quotients159;Borel
subgroups160;Parabolicsubgroups162;ExamplesofBorelandparabolicsubgroups162
22 Parabolicsubgroupsandroots 164
Liealgebras165;Algebraicgroups166
23 Representationsofsplitreductivegroups 167
The dominant weights of a root datum 167; The dominant weights of a semisimple root datum
167;Theclassificationofrepresentations167;Example:168;Example:GL 168;Example:SL
n n
169
24 Tannakaduality 170
Recoveringagroupfromitsrepresentations170;PropertiesofG versusthoseofRep .G/170;
k
(Neutralized)Tannakiancategories171;Applications172
25 AlgebraicgroupsoverRandC;relationtoLiegroups 174
The Lie group attached to an algebraic group 174; Negative results 174; Complex groups 175;
Realgroups176
26 Thecohomologyofalgebraicgroups;applications 177
Introduction177;Non-commutativecohomology. 177;Applications180;Classifyingtheforms
ofanalgebraicgroup181;InfiniteGaloisgroups182;Exactsequences183;Examples183;(Weil)
restriction of the base field 184; Reductive algebraic groups 184; Simply connected semisimple
groups184;Absolutelyalmost-simplesimply-connectedsemisimplegroups185;Themaintheo-
remsonthecohomologyofgroups186
27 Classicalgroupsandalgebraswithinvolution 188
The forms of M .k/ 188; The inner forms of SL 189; Involutions of k-algebras 190; All the
n n
forms of SL 190; Forms of Sp 191; The forms of Spin.(cid:30)/ 192; Algebras admitting an in-
n 2n
volution192;Theinvolutionsonanalgebra193;Hermitianandskew-hermitianforms194;The
groupsattachedtoalgebraswithinvolution194;Conclusion. 195
28 Arithmeticsubgroups 196
Commensurable groups 196; Definitions and examples 196; Questions 197; Independence of (cid:26)
and L. 197; Behaviour with respect to homomorphisms 198; Ade`lic description of congruence
subgroups199;Applicationstomanifolds200;Torsion-freearithmeticgroups200;Afundamen-
tal domain for SL 201; Application to quadratric forms 202; “Large” discrete subgroups 203;
2
Reduction theory 204; Presentations 206; The congruence subgroup problem 207; The theorem
ofMargulis208;Shimuravarieties209
Indexofdefinitions 211

1
Introduction
For one who attempts to unravel the story, the
problems are as perplexing as a mass of hemp
withathousandlooseends.
DreamoftheRedChamber,TsaoHsueh-Chin.
Algebraic groups are groups of matrices determined by polynomial conditions. For
example, the group of matrices of determinant 1 and the orthogonal group of a symmetric
bilinearformarebothalgebraicgroups. Theelucidationofthestructureofalgebraicgroups
and the classification of them were among the great achievements of twentieth century
mathematics (Borel, Chevalley, Tits and others, building on the work of the pioneers on
Lie groups). Algebraic groups are used in most branches of mathematics, and since the
famous work of Hermann Weyl in the 1920s they have also playeda vital role in quantum
mechanicsandotherbranchesofphysics(usuallyasLiegroups).
Arithmetic groups are groups of matrices with integer entries. They are an important
sourceofdiscretegroupsactingonmanifolds,andrecentlytheyhaveappearedasthesym-
metrygroupsofseveralstringtheoriesinphysics.
These are the notes for a 40 hour course that I gave at CMS, Zhejiang University,
Hangzhou, in the spring of 2005. My goal was to give an introductory overview of al-
gebraic groups, Lie algebras, Lie groups, and arithmetic groups. However, to adequately
cover this topic would take twice as long and twice as many pages (but not more!). Thus,
the treatment is very sketchy in places, and some important topics (for example, the cru-
cial real case) are barely mentioned. Nevertheless, I hope that the notes may be useful for
someone looking for a rapid introduction to the subject. Sometime I plan to produce an
expandedversion.
The approach to algebraic groups taken in these notes In most of the expository lit-
erature, the theory of algebraic groups is based (in spirit if not in fact) on the algebraic
geometry of Weil’s Foundations.1 Thus coordinate rings are not allowed to have nonzero
nilpotents, which means, for example, that the centre of SL in characteristic p is vis-
p
ible only through its Lie algebra. Moreover, the isomorphism theorem in group theory,
HN=N ' H=N \H,fails,andsotheintuitionprovidedbygrouptheoryisunavailable. It
istruethatincharacteristiczero,allalgebraicgroupsarereduced,butthisisatheoremthat
canonlybestatedwhennilpotentsareallowed. Anotherproblemisthatanalgebraicgroup
overafieldk isdefinedtobeanalgebraicgroupoversomelargealgebraicallyclosedfield
together with a k-structure. This leads to a confusing terminology in conflict with that of
today’s algebraic geometry and prevents, for example, the theory of split reductive groups
tobedevelopedintrinsicallyoverthebasefield.
Of course, the theory of algebraic groups should be based on Grothendieck’s theory
of schemes. However, the language of schemes is not entirely appropriate either, since
the nonclosed points are an unnecessary complication when working over a field and they
preventtheunderlyingspaceofanalgebraicgroupfrombeingagroup. Inthesenotes, we
usuallyregardalgebraicgroupsasfunctors(orbi-algebras),exceptthat,inordertobeable
toapplyalgebraicgeometry,wesometimesinterpretthemasalgebraicvarietiesoralgebraic
spaces(inthesenseofAG(cid:144)11).
1Weil,Andre´.Foundationsofalgebraicgeometry.AMS,1962
2
Theexpertneedonlynotethatby“algebraicgroupoverafield”wemean“affinealge-
braicgroupschemeoverafield”,andthatourringedspaceshaveonlyclosedpoints(thus,
weareusingSpmratherthanSpec).
Notations
We use the standard (Bourbaki) notations: N D f0;1;2;:::g, Z D ring of integers, R D
fieldofrealnumbers,C Dfieldofcomplexnumbers,F D Z=pZ Dfieldofpelements,p
p
aprimenumber. Givenanequivalencerelation,Œ(cid:3)(cid:141)denotestheequivalenceclasscontaining
(cid:3). AfamilyofelementsofasetAindexedbyasecondsetI,denoted.ai/i2I,isafunction
i 7! a WI ! A.
i
Throughout,k isafieldandk isanalgebraicclosureofk.
Ringswillbecommutativewith1unlessstatedotherwise,andhomomorphismsofrings
arerequiredtomap1to1. Ak-algebraisaringAtogetherwithahomomorphismk ! A.
(cid:2)
ForaringA,A isthegroupofunitsinA:
A(cid:2) D fa 2 A j thereexistsab 2 Asuchthatab D 1g:
WeuseGothic(fraktur)lettersforideals:
a b c m n p q A B C M N P Q
a b c m n p q A B C M N P Q
X Ddf Y X isdefinedtobeY,orequalsY bydefinition;
X (cid:26) Y X isasubsetofY (notnecessarilyproper,i.e.,X mayequalY);
X (cid:25) Y X andY areisomorphic;
X ' Y X andY arecanonicallyisomorphic(orthereisagivenoruniqueisomorphism).
Prerequisites
˘ Astandardcourseonalgebra,forexample,agoodknowledgeoftheArtin1991.
˘ Some knowledge of the language of algebraic geometry, for example, the first few
sectionsofAG.
Acknowledgements
I thank the Scientific Committee and Faculty of CMS (Yau Shing-Tung, Liu Kefeng, Ji
Lizhen, ...) for the invitation to lecture at CMS; Xu Hongwei and Dang Ying for helping
tomakemystayinHangzhouanenjoyableone;andthoseattendingthelectures,especially
Ding Zhiguo, Han Gang, Liu Gongxiang, Sun Shenghao, Xie Zhizhang, Yang Tian, Zhou
Yangmei,andMunirAhmed,fortheirquestionsandcomments.
3
References
BASIC ALGEBRA
Artin1991: Algebra,Prentice-Hall.
FT:Milne,J.,FieldsandGaloistheory,availableatwww.jmilne.org/math/.
GT:Milne,J.,Grouptheory,availableatwww.jmilne.org/math/.
COMMUTATIVE ALGEBRA
AtiyahandMacdonald1969: Commutativealgebra,Addison-Wesley.
ALGEBRAIC GEOMETRY
AG:Milne,J.,Algebraicgeometry,availableatwww.jmilne.org/math/.
GROUP VARIETIES
Borel1991: Linearalgebraicgroups,Springer.
Humphreys1975: Linearalgebraicgroups,Springer.
Springer1998: Linearalgebraicgroups,Birkha¨user.
GROUP SCHEMES
DemazureandGabriel,1970: Groupesalge´briques. Masson,Paris.
SGA3: Sche´masenGroupes,SeminarorganizedbyDemazureand
Grothendieck(1963–64),availableatwww.grothendieck-circle.org.
Waterhouse1979: Introductiontoaffinegroupschemes,Springer.
LIE ALGEBRAS
Humphreys1972: IntroductiontoLiealgebrasandrepresentationtheory,Springer.
Serre1987: ComplexsemisimpleLiealgebras,Springer.
LIE GROUPS
Hall2003: Liegroups,Liealgebrasandrepresentationtheory,Springer.
ARITHMETIC OF ALGEBRAIC GROUPS
PlatonovandRapinchuk1994: Algebraicgroupsandnumbertheory,Academic.
ARITHMETIC GROUPS
Borel1969: Introductionauxgroupesarithme´tiques,Hermann.
HISTORY
Borel2001: EssaysinthehistoryofLiegroupsandalgebraicgroups,AMS.
1 OVERVIEWANDEXAMPLES 4
1 Overview and examples
Loosely speaking, an algebraic group is a group defined by polynomials. Following Mike
Artin’s dictum (Artin 1991, p xiv), I give the main examples before the precise abstract
definition.
The determinant of an n (cid:2) n matrix A D .a / is a polynomial in the entries of A,
ij
specifically,
X
det.A/ D .sgn.(cid:27)//a (cid:1)(cid:1)(cid:1)a
1(cid:27).1/ n(cid:27).n/
(cid:27)2Sn
where S is the symmetric group on n letters, and sgn.(cid:27)/ is the sign of (cid:27). Moreover, the
n
entries of the product of two matrices are polynomials in the entries of the two matrices.
Therefore, for any field k, the group SL .k/ of n (cid:2) n matrices with determinant 1 is an
n
algebraicgroup(calledthespeciallineargroup).
The group GL .k/ of n (cid:2) n matrices with nonzero determinant is also an algebraic
n
group (called the general linear group) because its elements can be identified with the
n2C1-tuples..aij/1(cid:20)i;j(cid:20)n;t/suchthat
det.a /t D 1:
ij
Moregenerally,forafinite-dimensionalvectorspaceV,wedefineGL.V/(resp. SL.V/)
tobethegroupsautomorphismsofV (resp. automorphismswithdeterminant1). Theseare
againalgebraicgroups.
Ontheotherhand,thesubgroup
f.x;ex/ j x 2 Rg
of R(cid:2)R(cid:2) is not an algebraic subgroup because any polynomial f.X;Y/ 2 RŒX;Y(cid:141) zero
onitisidenticallyzero.
AnalgebraicgroupisconnectedifithasnoquotientalgebraicgroupQsuchthatQ.k/
isfiniteand¤ 1.
Thebuildingblocks
Unipotentgroups
Recallthatanendomorphism˛ ofavectorspaceV isnilpotentif˛n D 0forsomen > 0
andthatitisunipotentif1(cid:0)˛ isnilpotent. Forexample,amatrixAoftheform(cid:16)00 (cid:3)0 (cid:3)(cid:3)(cid:17)
0 0 0
isnilpotent(A3 D 0)andsoamatrixoftheform1(cid:0)A D(cid:16)10 (cid:3)1 (cid:3)(cid:3)(cid:17)isunipotent.
0 0 1
An algebraic subgroup of GL.V/ is unipotent if there exists a basis of V relative to
whichG iscontainedinthegroupofalln(cid:2)nmatricesoftheform
01 (cid:3) (cid:1)(cid:1)(cid:1) (cid:3) (cid:3)1
B0 1 (cid:1)(cid:1)(cid:1) (cid:3) (cid:3)C
B C
BB::: ::: ::: ::: :::CC; (1)
B@0 0 (cid:1)(cid:1)(cid:1) 1 (cid:3)CA
0 0 (cid:1)(cid:1)(cid:1) 0 1
whichwedenoteitU . Thus,theelementsofaunipotentgroupareunipotent.
n