Lie Algebras and Algebraic Groups.pdf - Index of

649 Pages·2005·5.52 MB·English

Springer Monographs in Mathematics
Patrice Tauvel
Rupert W.T. Yu
Lie Algebras
and Algebraic Groups
With44Figures
123
PatriceTauvel
RupertW.T.Yu
DépartementdeMathématiques
UniversitédePoitiers
BoulevardMarieetPierreCurie,
Téléport2–BP30179
86962FuturoscopeChasseneuilcedex,France
e-mail:[email protected]
[email protected]
LibraryofCongressControlNumber:2005922400
MathematicsSubjectClassification(2000):17-01,17-02,17Bxx,20Gxx
ISSN1439-7382
ISBN-10 3-540-24170-1 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-24170-6 SpringerBerlinHeidelbergNewYork
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Preface
The theory of groups and Lie algebras is interesting for many reasons. In
the mathematical viewpoint, it employs at the same time algebra, analysis
and geometry. On the other hand, it intervenes in other areas of science, in
particularindifferentbranchesofphysicsandchemistry.Itisanactivedomain
of current research.
Oneofthedifficultiesthatgraduatestudentsormathematiciansinterested
inthetheorycomeacross,isthefactthatthetheoryhasverymuchadvanced,
andconsequently,theyneedtoreadavastamountofbooksandarticlesbefore
they could tackle interesting problems.
One of the goals we wish to achieve with this book is to assemble in
a single volume the basis of the algebraic aspects of the theory of groups
and Lie algebras. More precisely, we have presented the foundation of the
study of finite-dimensional Lie algebras over an algebraically closed field of
characteristic zero.
Here,thegeometricalaspectisfundamental,andconsequently,weneedto
use the notion of algebraic groups. One of the main differences between this
book and many other books on the subject is that we give complete proofs
for the relationships between algebraic groups and Lie algebras, instead of
admitting them.
We have also given the proofs of certain results on commutative alge-
bra and algebraic geometry that we needed so as to make this book as self-
contained as possible. We believe that in this way, the book can be useful for
both graduate students and mathematicians working in this area.
Let us give a brief description of the material treated in this book.
As we have stated earlier, our goal is to study Lie algebras over an alge-
braicallyclosedfieldofcharacteristiczero.Thisallowsustoavoid,inconsider-
ingquestionsconcerningalgebraicgeometry,thenotionofseparability,which
simplifies considerably our presentation. In fact, under certain conditions of
separability, the correspondence between Lie algebras and algebraic groups
described in chapter 24 has a very nice generalization when the algebraically
closed base field has prime characteristic.
VI Preface
Chapters 1 to 9 treat basic results on topology, commutative algebra and
sheaves of functions that are required in the rest of the book.
In chapter 10, we recall some standard results on Jordan decompositions
and the theory of abstract groups and group actions. Here, the base field is
assumedtobealgebraicallyclosedinordertoobtainaJordandecomposition.
Chapters11to17giveanintroductiontothetheoryofalgebraicgeometry
which we shall encounter continually in the chapters which follow. We have
selected only the notions that we require in this book. The reader should by
no means consider these chapters as a thorough introduction to the theory of
algebraic geometry.
Chapters 18 and 36 are dedicated to root systems which are fundamental
to the study of semisimple Lie algebras.
We introduce Lie algebras in chapter 19. In this chapter, we prove impor-
tant results on the structure of Lie algebras such as Engel’s theorem, Lie’s
theorem and Cartan’s criterion on solvability.
In chapter 20, we define the notions of semisimple and reductive Lie al-
gebras. In addition to characterizing these Lie algebras, we discover in this
chapter how the structure of semisimple Lie algebras can be related to root
systems.
Thegeneraltheoryofalgebraicgroupsisstudiedinchapters21to28.The
relations between Lie algebras and algebraic groups, which are fundamental
to us, are established in chapters 23 and 24. Chapter 29 presents applications
of these relations to tackle the systematic study of Lie algebras. The reader
willobservethatthegeometricalaspectshaveanimportantpartinthisstudy.
Inparticular,theorbitsofpointsundertheactionofanalgebraicgroupplays
a central role.
Chapter 30 gives a short introduction of the theory of representations of
semisimple Lie algebras which we need in order to prove Chevalley’s theorem
on invariants in chapter 31.
We define in chapter 32 S-triples which are essential to the study of semi-
simple Lie algebras. Another fundamental notion, treated in chapters 33 to
35, is the notion of nilpotent orbits in semisimple Lie algebras.
We introduce symmetric Lie algebras in chapter 37, and semisimple sym-
metric Lie algebras in chapter 38. In these chapters, we give generalizations
of certain results of chapters 32 to 35.
In addition to presenting the essential classical results of the theory, some
of the results we have included in the final chapters are recent, and some are
yet to be published.
Attheendofeachchapter,thereadermayfindalistofrelevantreferences,
and in some cases, remarks concerning the contents of the chapter.
There are many approaches to reading this book. We need not read this
book linearly. A reader familiar with the theory of commutative algebra may
skip chapters 2 to 8, and consider these chapters for references only. Let us
also point out that chapters 18, 19 and 20 constitute a short introduction to
Preface VII
the theory of finite-dimensional Lie algebras and the structure of semisimple
Lie algebras.
WewishtothankourcolleaguesA.BouazizandH.Sabourinoftheuniver-
sity of Poitiers with whom we had many useful discussion during the prepa-
ration of this book.
Poitiers, Patrice Tauvel
January 2005 Rupert W.T. Yu
Contents
1 Results on topological spaces .............................. 1
1.1 Irreducible sets and spaces ............................... 1
1.2 Dimension ............................................. 4
1.3 Noetherian spaces....................................... 5
1.4 Constructible sets....................................... 6
1.5 Gluing topological spaces ................................ 8
2 Rings and modules ........................................ 11
2.1 Ideals ................................................. 11
2.2 Prime and maximal ideals................................ 12
2.3 Rings of fractions and localization......................... 13
2.4 Localizations of modules................................. 17
2.5 Radical of an ideal ...................................... 18
2.6 Local rings............................................. 19
2.7 Noetherian rings and modules ............................ 21
2.8 Derivations ............................................ 24
2.9 Module of differentials................................... 25
3 Integral extensions ........................................ 31
3.1 Integral dependence ..................................... 31
3.2 Integrally closed domains ................................ 33
3.3 Extensions of prime ideals ............................... 35
4 Factorial rings ............................................. 39
4.1 Generalities ............................................ 39
4.2 Unique factorization..................................... 41
4.3 Principal ideal domains and Euclidean domains............. 43
4.4 Polynomials and factorial rings ........................... 45
4.5 Symmetric polynomials .................................. 48
4.6 Resultant and discriminant............................... 50
X Contents
5 Field extensions ........................................... 55
5.1 Extensions ............................................. 55
5.2 Algebraic and transcendental elements..................... 56
5.3 Algebraic extensions .................................... 56
5.4 Transcendence basis..................................... 58
5.5 Norm and trace......................................... 60
5.6 Theorem of the primitive element ......................... 62
5.7 Going Down Theorem ................................... 64
5.8 Fields and derivations ................................... 67
5.9 Conductor ............................................. 70
6 Finitely generated algebras ................................ 75
6.1 Dimension ............................................. 75
6.2 Noether’s Normalization Theorem......................... 76
6.3 Krull’s Principal Ideal Theorem........................... 81
6.4 Maximal ideals ......................................... 82
6.5 Zariski topology ........................................ 84
7 Gradings and filtrations ................................... 87
7.1 Graded rings and graded modules......................... 87
7.2 Graded submodules ..................................... 88
7.3 Applications ........................................... 90
7.4 Filtrations ............................................. 91
7.5 Grading associated to a filtration ......................... 92
8 Inductive limits............................................ 95
8.1 Generalities ............................................ 95
8.2 Inductive systems of maps ............................... 96
8.3 Inductive systems of magmas, groups and rings ............. 98
8.4 An example ............................................100
8.5 Inductive systems of algebras.............................100
9 Sheaves of functions .......................................103
9.1 Sheaves................................................103
9.2 Morphisms.............................................104
9.3 Sheaf associated to a presheaf ............................106
9.4 Gluing ................................................109
9.5 Ringed space ...........................................110
10 Jordan decomposition and some basic results on groups ...113
10.1 Jordan decomposition ...................................113
10.2 Generalities on groups...................................117
10.3 Commutators ..........................................118
10.4 Solvable groups.........................................120
10.5 Nilpotent groups........................................121
Contents XI
10.6 Group actions ..........................................122
10.7 Generalities on representations ...........................123
10.8 Examples ..............................................126
11 Algebraic sets .............................................131
11.1 Affine algebraic sets.....................................131
11.2 Zariski topology ........................................132
11.3 Regular functions .......................................133
11.4 Morphisms.............................................134
11.5 Examples of morphisms..................................136
11.6 Abstract algebraic sets ..................................138
11.7 Principal open subsets...................................140
11.8 Products of algebraic sets ................................142
12 Prevarieties and varieties ..................................147
12.1 Structure sheaf .........................................147
12.2 Algebraic prevarieties ...................................149
12.3 Morphisms of prevarieties................................151
12.4 Products of prevarieties..................................152
12.5 Algebraic varieties ......................................155
12.6 Gluing ................................................158
12.7 Rational functions ......................................159
12.8 Local rings of a variety ..................................162
13 Projective varieties ........................................167
13.1 Projective spaces .......................................167
13.2 Projective spaces and varieties............................168
13.3 Cones and projective varieties ............................171
13.4 Complete varieties ......................................176
13.5 Products...............................................178
13.6 Grassmannian variety ...................................180
14 Dimension.................................................183
14.1 Dimension of varieties ...................................183
14.2 Dimension and the number of equations ...................185
14.3 System of parameters ...................................187
14.4 Counterexamples .......................................190
15 Morphisms and dimension .................................191
15.1 Criterion of affineness ...................................191
15.2 Affine morphisms .......................................193
15.3 Finite morphisms .......................................194
15.4 Factorization and applications ............................197
15.5 Dimension of fibres of a morphism ........................199
15.6 An example ............................................203
XII Contents
16 Tangent spaces ............................................205
16.1 A first approach ........................................205
16.2 Zariski tangent space....................................207
16.3 Differential of a morphism ...............................209
16.4 Some lemmas ..........................................213
16.5 Smooth points..........................................215
17 Normal varieties...........................................219
17.1 Normal varieties ........................................219
17.2 Normalization ..........................................221
17.3 Products of normal varieties..............................223
17.4 Properties of normal varieties ............................225
18 Root systems ..............................................233
18.1 Reflections .............................................233
18.2 Root systems...........................................235
18.3 Root systems and bilinear forms ..........................238
18.4 Passage to the field of real numbers .......................239
18.5 Relations between two roots..............................240
18.6 Examples of root systems ................................243
18.7 Base of a root system ...................................244
18.8 Weyl chambers .........................................247
18.9 Highest root ...........................................250
18.10Closed subsets of roots ..................................250
18.11Weights ...............................................253
18.12Graphs ................................................255
18.13Dynkin diagrams .......................................256
18.14Classification of root systems.............................259
19 Lie algebras ...............................................277
19.1 Generalities on Lie algebras ..............................277
19.2 Representations.........................................279
19.3 Nilpotent Lie algebras ...................................282
19.4 Solvable Lie algebras ....................................286
19.5 Radical and the largest nilpotent ideal.....................289
19.6 Nilpotent radical........................................291
19.7 Regular linear forms.....................................292
19.8 Cartan subalgebras .....................................294
20 Semisimple and reductive Lie algebras.....................299
20.1 Semisimple Lie algebras .................................299
20.2 Examples ..............................................301
20.3 Semisimplicity of representations..........................302
20.4 Semisimple and nilpotent elements ........................305
20.5 Reductive Lie algebras...................................307