Compact Lie Groups

207 Pages·2006·2.11 MB·English

by  Mark R. Sepanski

235
Graduate Texts in Mathematics
Editorial Board
S.Axler
K.A.Ribet
Graduate Texts in Mathematics
1 TAKEUTI/ZARING.Introduction to 34 SPITZER.Principles ofRandom Walk.
Axiomatic Set Theory.2nd ed. 2nd ed.
2 OXTOBY.Measure and Category.2nd ed. 35 ALEXANDER/WERMER.Several Complex
3 SCHAEFER.Topological Vector Spaces. Variables and Banach Algebras.3rd ed.
2nd ed. 36 KELLEY/NAMIOKAet al.Linear
4 HILTON/STAMMBACH.A Course in Topological Spaces.
Homological Algebra.2nd ed. 37 MONK.Mathematical Logic.
5 MACLANE.Categories for the Working 38 GRAUERT/FRITZSCHE.Several Complex
Mathematician.2nd ed. Variables.
6 HUGHES/PIPER.Projective Planes. 39 ARVESON.An Invitation to C*-Algebras.
7 J.-P.SERRE.A Course in Arithmetic. 40 KEMENY/SNELL/KNAPP.Denumerable
8 TAKEUTI/ZARING.Axiomatic Set Theory. Markov Chains.2nd ed.
9 HUMPHREYS.Introduction to Lie 41 APOSTOL.Modular Functions and
Algebras and Representation Theory. Dirichlet Series in Number Theory.2nd ed.
10 COHEN.A Course in Simple Homotopy 42 J.-P.SERRE.Linear Representations of
Theory. Finite Groups.
11 CONWAY.Functions ofOne Complex 43 GILLMAN/JERISON.Rings ofContinuous
Variable I.2nd ed. Functions.
12 BEALS.Advanced Mathematical Analysis. 44 KENDIG.Elementary Algebraic Geometry.
13 ANDERSON/FULLER.Rings and 45 LOÈVE.Probability Theory I.4th ed.
Categories ofModules.2nd ed. 46 LOÈVE.Probability Theory II.4th ed.
14 GOLUBITSKY/GUILLEMIN.Stable 47 MOISE.Geometric Topology in
Mappings and Their Singularities. Dimensions 2 and 3.
15 BERBERIAN.Lectures in Functional 48 SACHS/WU.General Relativity for
Analysis and Operator Theory. Mathematicians.
16 WINTER.The Structure ofFields. 49 GRUENBERG/WEIR.Linear Geometry.
17 ROSENBLATT.Random Processes.2nd ed. 2nd ed.
18 HALMOS.Measure Theory. 50 EDWARDS.Fermat’s Last Theorem.
19 HALMOS.A Hilbert Space Problem 51 KLINGENBERG.A Course in Differential
Book.2nd ed. Geometry.
20 HUSEMOLLER.Fibre Bundles.3rd ed. 52 HARTSHORNE.Algebraic Geometry.
21 HUMPHREYS.Linear Algebraic Groups. 53 MANIN.A Course in Mathematical
22 BARNES/MACK.An Algebraic Logic.
Introduction to Mathematical Logic. 54 GRAVER/WATKINS.Combinatories with
23 GREUB.Linear Algebra.4th ed. Emphasis on the Theory ofGraphs.
24 HOLMES.Geometric Functional Analysis 55 BROWN/PEARCY.Introduction to
and Its Applications. Operator Theory I:Elements of
25 HEWITT/STROMBERG.Real and Abstract Functional Analysis.
Analysis. 56 MASSEY.Algebraic Topology:An
26 MANES.Algebraic Theories. Introduction.
27 KELLEY.General Topology. 57 CROWELL/FOX.Introduction to Knot
28 ZARISKI/SAMUEL.Commutative Algebra. Theory.
Vol.I. 58 KOBLITZ.p-adic Numbers,p-adic
29 ZARISKI/SAMUEL.Commutative Algebra. Analysis,and Zeta-Functions.2nd ed.
Vol.II. 59 LANG.Cyclotomic Fields.
30 JACOBSON.Lectures in Abstract Algebra 60 ARNOLD.Mathematical Methods in
I.Basic Concepts. Classical Mechanics.2nd ed.
31 JACOBSON.Lectures in Abstract Algebra 61 WHITEHEAD.Elements ofHomotopy
II.Linear Algebra. Theory.
32 JACOBSON.Lectures in Abstract Algebra 62 KARGAPOLOV/MERLZIAKOV.
III.Theory ofFields and Galois Theory. Fundamentals ofthe Theory ofGroups.
33 HIRSCH.Differential Topology. 63 BOLLOBAS.Graph Theory.
(continued after index)
Mark R. Sepanski
Compact Lie Groups
Mark R.Sepanski
Department ofMathematics
Baylor University
Waco,TX 76798-7328
USA
[email protected]
Editorial Board
S.Axler K.A.Ribet
Mathematics Department Mathematics Department
San Francisco State University University ofCalifornia,Berkeley
San Francisco,CA 94132 Berkeley,CA 94720-3840
USA USA
[email protected] [email protected]
Mathematics Subject Classification (2000):22E15,17B20
Library ofCongress Control Number:2006937104
ISBN-10:0-387-30263-8 e-ISBN-10:0-387-49158-9
ISBN-13:978-0-387-30263-8 e-ISBN-13:978-0-387-49158-5
Printed on acid-free paper.
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Contents
Preface.......................................................... xi
1 CompactLieGroups ............................................ 1
1.1 BasicNotions 1
1.1.1 Manifolds 1
1.1.2 LieGroups 2
1.1.3 LieSubgroupsandHomomorphisms 2
1.1.4 CompactClassicalLieGroups 4
1.1.5 Exercises 6
1.2 BasicTopology 8
1.2.1 Connectedness 8
1.2.2 SimplyConnectedCover 10
1.2.3 Exercises 12
1.3 TheDoubleCoverofSO(n) 13
1.3.1 CliffordAlgebras 14
1.3.2 Spin (R)andPin (R) 16
n n
1.3.3 Exercises 18
1.4 Integration 19
1.4.1 VolumeForms 19
1.4.2 InvariantIntegration 20
1.4.3 Fubini’sTheorem 22
1.4.4 Exercises 23
2 Representations ................................................ 27
2.1 BasicNotions 27
2.1.1 Definitions 27
2.1.2 Examples 28
2.1.3 Exercises 32
2.2 OperationsonRepresentations 34
2.2.1 ConstructingNewRepresentations 34
viii Contents
2.2.2 IrreducibilityandSchur’sLemma 36
2.2.3 Unitarity 37
2.2.4 CanonicalDecomposition 39
2.2.5 Exercises 40
2.3 ExamplesofIrreducibility 41
2.3.1 SU(2)andV (C2) 41
n
2.3.2 SO(n)andHarmonicPolynomials 42
2.3.3 SpinandHalf-SpinRepresentations 44
2.3.4 Exercises 45
3 HarmonicAnalysis ............................................. 47
3.1 MatrixCoefficients 47
3.1.1 SchurOrthogonality 47
3.1.2 Characters 49
3.1.3 Exercises 52
3.2 Infinite-DimensionalRepresentations 54
3.2.1 BasicDefinitionsandSchur’sLemma 54
3.2.2 G-FiniteVectors 56
3.2.3 CanonicalDecomposition 58
3.2.4 Exercises 59
3.3 ThePeter–WeylTheorem 60
3.3.1 TheLeftandRightRegularRepresentation 60
3.3.2 MainResult 64
3.3.3 Applications 66
3.3.4 Exercises 69
3.4 FourierTheory 70
3.4.1 Convolution 71
3.4.2 PlancherelTheorem 72
3.4.3 ProjectionOperatorsandMoreGeneralSpaces 77
3.4.4 Exercises 79
4 LieAlgebras ................................................... 81
4.1 BasicDefinitions 81
4.1.1 LieAlgebrasofLinearLieGroups 81
4.1.2 ExponentialMap 83
4.1.3 LieAlgebrasfortheCompactClassicalLieGroups 84
4.1.4 Exercises 86
4.2 FurtherConstructions 88
4.2.1 LieAlgebraHomomorphisms 88
4.2.2 LieSubgroupsandSubalgebras 91
4.2.3 CoveringHomomorphisms 92
4.2.4 Exercises 93
Contents ix
5 AbelianLieSubgroupsandStructure.............................. 97
5.1 AbelianSubgroupsandSubalgebras 97
5.1.1 MaximalToriandCartanSubalgebras 97
5.1.2 Examples 98
5.1.3 ConjugacyofCartanSubalgebras 100
5.1.4 MaximalTorusTheorem 102
5.1.5 Exercises 103
5.2 Structure 105
5.2.1 ExponentialMapRevisited 105
5.2.2 LieAlgebraStructure 108
5.2.3 CommutatorTheorem 109
5.2.4 CompactLieGroupStructure 110
5.2.5 Exercises 111
6 RootsandAssociatedStructures .................................. 113
6.1 RootTheory 113
6.1.1 RepresentationsofLieAlgebras 113
6.1.2 ComplexificationofLieAlgebras 115
6.1.3 Weights 116
6.1.4 Roots 117
6.1.5 CompactClassicalLieGroupExamples 118
6.1.6 Exercises 120
6.2 TheStandardsl(2,C)Triple 123
6.2.1 CartanInvolution 123
6.2.2 KillingForm 124
6.2.3 TheStandardsl(2,C)andsu(2)Triples 125
6.2.4 Exercises 129
6.3 Lattices 130
6.3.1 Definitions 130
6.3.2 Relations 131
6.3.3 CenterandFundamentalGroup 132
6.3.4 Exercises 134
6.4 WeylGroup 136
6.4.1 GroupPicture 136
6.4.2 ClassicalExamples 137
6.4.3 SimpleRootsandWeylChambers 139
6.4.4 TheWeylGroupasaReflectionGroup 143
6.4.5 Exercises 146
7 HighestWeightTheory .......................................... 151
7.1 HighestWeights 151
7.1.1 Exercises 154
7.2 WeylIntegrationFormula 156
7.2.1 RegularElements 156
x Contents
7.2.2 MainTheorem 159
7.2.3 Exercises 162
7.3 WeylCharacterFormula 163
7.3.1 Machinery 163
7.3.2 MainTheorem 166
7.3.3 WeylDenominatorFormula 168
7.3.4 WeylDimensionFormula 168
7.3.5 HighestWeightClassification 169
7.3.6 FundamentalGroup 170
7.3.7 Exercises 173
7.4 Borel–WeilTheorem 176
7.4.1 InducedRepresentations 176
7.4.2 ComplexStructureonG/T 178
7.4.3 HolomorphicFunctions 180
7.4.4 MainTheorem 182
7.4.5 Exercises 184
References....................................................... 187
Index ........................................................... 193
Preface
Asanundergraduate,Iwasofferedareadingcourseontherepresentationtheoryof
finite groups. When I learned this basically meant studying homomorphisms from
groups into matrices, I was not impressed. In its place I opted for a reading course
onthemuchmoreglamoroussoundingtopicofmultilinearalgebra.Ironically,when
IfinallytookacourseonrepresentationtheoryfromB.Kostantingraduateschool,I
wasimmediatelycaptivated.
Inbroadterms,representationtheoryissimplythestudyofsymmetry.Inprac-
tice, the theory often begins by classifying all the ways in which a group acts on
vectorspacesandthenmovesintoquestionsofdecomposition,unitarity,geometric
realizations,andspecialstructures.Ingeneral,eachoftheseproblemsisextremely
difficult.HoweverinthecaseofcompactLiegroups,answerstomostoftheseques-
tions are well understood. As a result, the theory of compact Lie groups is used
extensivelyasasteppingstoneinthestudyofnoncompactLiegroups.
Regarding prerequisites for this text, the reader must first be familiar with the
definitionofagroupandbasictopology.Secondly,elementaryknowledgeofdiffer-
entialgeometryisassumed.Studentslackingaformalcourseinmanifoldtheorywill
be able to follow most of this book if they are willing to take a few facts on faith.
Thismostlyconsistsofacceptingtheexistenceofaninvariantintegralin§1.4.1.Ina
bitmoredetail,thenotionofasubmanifoldisusedin§1.1.3,thetheoryofcovering
spaces is used in §1.2, §1.3, §4.2.3, and §7.3.6, integral curves are used in §4.1.2,
and Frobenius’ theorem on integral submanifolds is used in the proof of Theorem
4.14.Athirdprerequisiteiselementaryfunctionalanalysis.Again,studentslacking
formal course work in this area can follow most of the text if they are willing to
assume a few facts. In particular, the Spectral Theorem for normal bounded opera-
torsisusedintheproofofTheorem3.12,vector-valuedintegrationisintroducedin
§3.2.2, and the Spectral Theorem for compact self-adjoint operators is used in the
proofofLemma3.13.
The text assumes no prior knowledge of Lie groups or Lie algebras and so all
the necessary theory is developed here. Students already familiar with Lie groups
can quickly skim most of Chapters 1 and 4. Similarly, students familiar with Lie
algebrascanquicklyskimmostofChapter6.