Linear Algebraic Groups

301 Pages·1991·41.328 MB·English

by  Armand Borel

Graduate Texts in Mathematics 126
Editorial Board
l.H. Ewing F.W. Gehring P.R. Halmos
Graduate Texts in Mathematics
I TAKEUTljZARING. Introduction to Axiomatic SetTheory. 2nded.
2 OXTOBY. Measureand Category. 2nded.
3 SCHAEFFER.Topological VectorSpaces.
4 HILTONjSTAMMBACH. ACoursein Homological Algebra.
5 MAc LANE. Categoriesfor the Working Mathematician.
6 HUGHES/PIPER. Projective Planes.
7 SERRE. ACourse in Arithmetic.
8 TAKEUTl/ZARING. AxiomaticSetTheory.
9 HUMPHREYS. Introduction to Lie Algebrasand Representation Theory.
10 COHEN. ACourse in Simple HomotopyTheory.
II CONWAY. FunctionsofOneComplex Variable. 2nd ed.
12 BEALS. Advanced Mathematical Analysis.
13 ANDERSON/FULLER. Rings and CategoriesofModules.
14 GOLUBITSKy/GUILLEMIN. Stable Mappings and TheirSingularities.
15 BERBERIAN. Lecturesin Functional Analysisand OperatorTheory.
16 WINTER. TheStructure ofFields.
17 ROSENBLATT. Random Processes. 2nded.
18 HALMOS. MeasureTheory.
19 HALMOS. AHilbert Space Problem Book. 2nded., revised.
20 HUSEMOLLER. Fibre Bundles. 2nd ed.
21 HUMPHREYS. Linear AlgebraicGroups.
22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic.
23 GREUB. Linear Algebra. 4thed.
24 HOLMES.GeometrioFunctional Analysisand its Applications.
25 HEWITT/STROMBERG. Real and Abstract Analysis.
26 MANES. AlgebraicTheories.
27 KELLEY. General Topology.
28 ZARISKI/SAMUEL. Commutative Algebra. Vol.I.
29 ZARISKI/SAMUEL. Commutative Algebra. Vol.II.
30 JACOBSON. Lecturesin Abstract Algebra I: BasicConcepts.
31 JACOBSON. Lectures in Abstract Algebra II: LinearAlgebra.
32 JACOBSON. Lectures in Abstract Algebra Ill: Theory ofFieldsand GaloisTheory.
33 HIRSCH. DifferentialTopology.
34 SPITZER. PrinciplesofRandom Walk. 2nd ed.
35 WERMER. Banach Algebrasand SeveralComplex Variables. 2nded.
36 KELLEy/NAMIOKA etal. LinearTopological Spaces.
37 MONK. Mathematical Logic.
38 GRAUERT/FRITZSCHE. Several Complex Variables.
39 ARVESON. An Invitation to C*-Algebras.
40 KEMENy/SNELL/KNAPP. Denumerable MarkovChains. 2nd ed.
41 ApOSTOL. Modular Functionsand DirichletSeriesin Number Theory. 2nded.
42 SERRE. Linear Representations ofFiniteGroups.
43 GILLMAN/JERISON. RingsofContinuous Functions.
44 KENDIG. ElemenIary AlgebraicGeometry.
45 LoEVE. ProbabilityTheory I.4thed.
46 LOEVE. Probability Theory II. 4thed.
47 MOISE. GeometricTopology in Dimensions 2and 3.
continuedafterIndex
Armand Borel
Linear Algebraic Groups
Second Enlarged Edition
Springer Science+Business Media, LLC
Annand Borel
School of Mathematics
Institute for Advanced Study
Princeton, New Jersey 08450 USA
Editorial Board
lH. Ewing F.W. Gehring P.R. Halmos
Department of Department of Department of
Mathematics Mathematics Mathematics
Indiana University University of Michigan Santa Clara University
Bloomington, IN 47401 Ann Arbor, MI 48019 Santa Clara, CA 95093
USA USA USA
First edition published by W.A. Benjamin, Inc., 1969
Library of Congress Cataloging-in-Publication Data
Borel, Armand.
Linear algebraic groups ! Armand Borel.-2nd enl. ed.
p. cm.-(Graduate texts in mathematics; 126)
lncludes bibliographical references and indexes.
ISBN 978-1-4612-6954-0 ISBN 978-1-4612-0941-6 (eBook)
DOI 10.1007/978-1-4612-0941-6
1. Linear algebraic groups. 1. Title. II. Series.
QA564.B58 1991
512'.5-dc20 90·19774
Printed on acid-free paper.
© 1991 Springer Science+Business Media New York
Originally published by Springer-Verlag New York Inc. in 1991
Softcover reprint of the hardcover 2nd edition 1991
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ISBN 978-1-4612-6954-0
Introduction to the First Edition
TheseNotesaimatprovidinganintroductiontothetheoryoflinearalgebraic
groupsoverfields. Theirmain objectivesare to givesome basicmaterialover
arbitrary fields (Chap. I, II), and to discuss the structure ofsolvable and of
reductivegroups overalgebraicallyclosed fields (Chap. III, IV). To complete
the picture, they also include some rationalityproperties(§§15, 18)and some
results on groups over finite fields (§16) and over fields of characteristic
zero (§7).
Apartfrom someknowledgeofLiealgebras,themain prerequisiteforthese
Notesissomefamiliarity withalgebraicgeometry.Infact, comparativelylittle
isactuallyneeded.Mostofthenotionsandresultsfrequently usedintheNotes
are summarized, a few with proofs, in a preliminary Chapter AG. As a basic
reference,wetake Mumford's Notes [14],and have tried to betosomeextent
self-contained from there. A few further results from algebraic geometry
neededonsomespecificoccasionswillberecalled(withreferences)whereused.
Thepointofviewadoptedhereisessentiallythesettheoreticone:varietiesare
identified with their setofpoints overan algebraicclosure ofthe groundfield
(endowedwiththeZariski-topology),howeverwithsometracesofthescheme
point ofview here and there.
TheseNotesare based onacoursegivenatColumbiaUniversityinSpring,
1968,* at the suggestion of Hyman Bass. Except for Chap. V, added later,
NoteswerewrittenupbyH. Bass,withsomehelpfrom MichaelStein,andare
reproduced here with few changes or additions. He did this with marvelous
efficiency, often expanding or improving the oral presentation. In particular,
the emphasis on dual numbers in§3 in his, and he wrote up Chapter AG, of
whichonlya verybriefsurveyhad beengivenin the course. Itisapleasureto
thank him most warmly for his contributions, without which these Notes
wouldhardlyhavecomeintobeingatthistime.IwouldalsoliketothankMiss
P. Murray for her careful and fast typing of the manuscript, and lE.
Humphreys, J.S. Joel for their help in checking and proofreading it.
A. Borel
Princeton, February, 1969
*Lecturesfrom May 7th on qualified as liberatedclass, under thesponsorship ofthe Students
Strike Committee. Space was generously made available on one occasion by the Union
TheologicalSeminary.
Introduction to the Second Edition
This is a revised and enlarged edition ofthe set ofNotes: "Linear algebraic
groups" published by Benjaminin 1969.The added material pertainsmainly
torationalityquestionsoverarbitraryfields with,asamaingoal,propertiesof
the rational points of isotropic reductive groups. Besides, a number of
corrections, additions and changes to the original text have been made. In
particular:
§3 on Lie algebras has been revised.
§6 on quotient spaces contains a briefdiscussion ofcategorical quotients.
The existence ofa quotient by finite groups has been added to §6, that ofa
categorical quotient under the action ofa torus to §8.
In §11, the original proof of Chevalley's normalizer theorem has been
replaced by an argument I found in 1973, (and is used in the books of
Humphreys and Springer).
In §14, some material on parabolic subgroups has been added.
§15, on split solvable groups now contains a proof of the existence of a
rationalpointonany homogeneousspaceofasplitsolvablegroup,a theorem
ofRosenlicht's proved in the first edition only for GL and Ga.
I
§§19to24arenew.Thefirstoneshowsthatinaconnectedsolvablek-group,
all Cartan k-subgroups are conjugate under G(k), a result also due to M.
Rosenlicht. §§20, 21 aredevoted to the so-called relative theory for isotropic
reductivegroupsoverafield k: Conjugacy theoremsfor minimal parabolick
subgroups,maximal k-split tori,existenceofaTitssystemon G(k), rationality
ofthequotientofGbyaparabolick-subgroupanddescriptionoftheclosureof
a Bruhat cell. As a necessary complement, §22 discusses central isogenies.
§23 isdevoted toexamplesanddescribes theTitssystemsofmanyclassical
groups. Finally, §24 surveys without proofs some main results on classific
ations and linear representations of semi-simple groups and, assuming Lie
theory, relates the Tits system on the real points ofa reductive group to the
similar notions introduced much earlier by E. Cartan in a Lie theoretic
framework.
Many corrections have been made to the text of the first edition and
my thanks are due to J. Humphreys, F.D. Veldkamp, A.E. Zalesski and
V. Platonov who pointed out most ofthem.
Vlll Introduction to the Second Edition
Iamalsogratefulto MutsumiSaito,T. WatanabeandespeciallyG. Prasad,
who read a draft of the changes and additions and found an embarrassing
number ofmisprints and minor inaccuracies. I am also glad to acknowledge
helpreceivedintheproofreadingfrom H.P.Kraft,whoread partsoftheproofs
with great care and came up with a depressing list ofcorrections, and from
D. labon.
Thefirst edition has been out ofprintfor many yearsand the questionofa
reedition has been in the air for that much time. After Addison-Wesley had
acquired the rights to the Benjamin publications they decided not to proceed
with oneand released the publication rights to me. Iam grateful to Springer
Verlag to have offered over ten years ago to publish a reedition in which
ever form I would want it and to several technical editors (starting with
W. Kaufmann-Biihler)and scientificeditorsfor having periodically prodded
me into getting on with this project. I am solely to blame for the
procrastination.
In preparing the typescript for the second edition, use was made to the
extentpossibleofcopiesofthefirst one,whosetypography wasquitedifferent
from the one present techniques allow one to produce. The insertions of
corrections,changes and additions, whichcameinsuccessiveways, presented
seriousproblemsinharmonization,pastingandcutting.Iamgratefulto Irene
Gaskill and Elly Gustafsson for having performed them with great skill.
I would also like to express my appreciation to Springer-Verlag for their
handlingofthepublicationandtheirpatienceintakingcareofmydesiderata.
A. Borel
Contents
Introduction to the First Edition . . V
Introduction to the Second Edition. Vll
Conventions and Notation . Xl
CHAPTER AG-Background Material From Algebraic Geometry
§1. Some Topological Notions... 1
§2. Some Facts from Field Theory 3
§3. Some Commutative Algebra. . 5
§4. Sheaves . 10
§5. Affine K-Schemes, Prevarieties. 11
§6. Products; Varieties . 14
§7. Projective and Complete Varieties 17
§8. Rational Functions; Dominant Morphisms . 19
§9. Dimension . . . . . . . . . . . . . . 20
§1O. Images and Fibres ofa Morphism 20
§11. k-structures on K-Schemes. 21
§12. k-Structures on Varieties . . . . 23
§13. Separable points . 26
§14. Galois Criteria for Rationality. 29
§15. Derivations and Differentials. 32
§16. Tangent Spaces. . 36
§17. Simple Points... 40
§18. Normal Varieties. 42
References. . . . . 45
CHAPTER I-General Notions Associated With Algebraic Groups
§1. The Notion ofan Algebraic Groups . 46
§2. Group Closure; Solvable and Nilpotent Groups 56
x Contents
§3. The Lie Algebra ofan Algebraic Group. 62
§4. Jordan Decomposition . 79
CHAPTER II-Homogeneous Spaces
§5. Semi-Invariants . 89
§6. Homogeneous Spaces . 94
§7. Algebraic Groups in Characteristic Zero. 105
CHAPTER 111- Solvable Groups
§8. Diagonalizable Groups and Tori 111
§9. Conjugacy Classes and Centralizers of
Semi-Simple Elements . 127
§10. Connected Solvable Groups . 134
CHAPTER IV- Borel Subgroups; Reductive Groups
§11. Borel Subgroups . 147
§12. Cartan Subgroups; Regular Elements . 159
§13. The Borel Subgroups Containing a Given Torus. 163
§14. Root Systems and Bruhat Decomposition
in Reductive Groups . 179
CHAPTER V- Rationality Questions
§IS. Split Solvable Groups and Subgroups . 203
§16. Groups over Finite Fields. . . . . . . . 210
§17. Quotient ofa Group by a Lie Subalgebra. 213
§18. Cartan Subgroups over the Groundfield. Unirationality.
Splitting ofReductive Groups. . . . . . 218
§19. Cartan Subgroups of Solvable Groups. . . . . . . . . . 222
§20. Isotropic Reductive Groups . . . . . . . . . . . . . . . . 224
§21. Relative Root System and Bruhat Decomposition for
Isotropic Reductive Groups 229
§22. Central Isogenies. . . . . . . . . 246
§23. Examples . . . . . . . . . . . . . 253
§24. Survey ofSome Other Topics. 268
A. Classification. . . . . . . 268
B. Linear Representations . 270
C. Real Reductive Groups. 274
References for Chapters I to V 280
Index ofDefinition 282
Index of Notation . . . . . . . . . . 286