Lie groups, lie algebras, and representations: an elementary introduction
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| Format: | Book |
| Language: | English |
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Cham ; Heidelberg ; New York ; Dordrecht ; London
Springer
[2015]
|
| Edition: | Second edition |
| Series: | Graduate texts in mathematics
222 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiii, 451 Seiten Illustrationen |
| ISBN: | 9783319134666 9783319374338 |
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Record in the Search Index
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|---|---|
| adam_text | Titel: Lie groups, Lie algebras, and representations
Autor: Hall, Brian C
Jahr: 2015
Contents
Part I General Theory
1 Matrix Lie Groups....................................................................................................................3
1.1 Definitions........................................................................................................................3
1.2 Examples ..........................................................................................................................5
1.3 Topological Properties..............................................................................................16
1.4 Homomorphisms..........................................................................................................21
1.5 Lie Groups........................................................................................................................25
1.6 Exercises............................................................................................................................26
2 The Matrix Exponential......................................................................................................31
2.1 The Exponential of a Matrix..................................................................................31
2.2 Computing the Exponential....................................................................................34
2.3 The Matrix Logarithm..............................................................................................36
2.4 Further Properties of the Exponential..............................................................40
2.5 The Polar Decomposition........................................................................................42
2.6 Exercises............................................................................................................................46
3 Lie Algebras..................................................................................................................................49
3.1 Definitions and First Examples............................................................................49
3.2 Simple, Solvable, and Nilpotent Lie Algebras............................................53
3.3 The Lie Algebra of a Matrix Lie Group..........................................................55
3.4 Examples..........................................................................................................................57
3.5 Lie Group and Lie Algebra Homomorphisms............................................60
3.6 The Complexification of a Real Lie Algebra..............................................65
3.7 The Exponential Map................................................................................................67
3.8 Consequences of Theorem 3.42..........................................................................70
3.9 Exercises............................................................................................................................73
4 Basic Representation Theory............................................................................................77
4.1 Representations..............................................................................................................77
4.2 Examples of Representations................................................................................81
4.3 New Representations from Old............................................................................84
vii
viii
Contents
4.4 Complete Reducibility..............................................................................................90
4.5 Schur s Lemma..............................................................................................................94
4.6 Representations of sl(2; C)....................................................................................96
4.7 Group Versus Lie Algebra Representations..................................................101
4.8 A Nonmatrix Lie Group............................................................................................103
4.9 Exercises............................................................................................................................105
5 The Baker-Campbell-Hausdorff Formula and Its Consequences.... 109
5.1 The Hard Questions................................................................................................109
5.2 An Illustrative Example............................................................................................110
5.3 The Baker-Campbell-Hausdorff Formula....................................................113
5.4 The Derivative of the Exponential Map..........................................................114
5.5 Proof of the BCH Formula......................................................................................117
5.6 The Series Form of the BCH Formula............................................................118
5.7 Group Versus Lie Algebra Homomorphisms..............................................119
5.8 Universal Covers..........................................................................................................126
5.9 Subgroups and Subalgebras....................................................................................128
5.10 Lie s Third Theorem..................................................................................................135
5.11 Exercises............................................................................................................................135
Part II Semisimple Lie Algebras
6 The Representations of si (3; C)......................................................................................141
6.1 Preliminaries....................................................................................................................141
6.2 Weights and Roots ......................................................................................................142
6.3 The Theorem of the Highest Weight................................................................146
6.4 Proof of the Theorem................................................................................................148
6.5 An Example: Highest Weight (1,1)..................................................................153
6.6 The Weyl Group............................................................................................................154
6.7 Weight Diagrams..........................................................................................................158
6.8 Further Properties of the Representations......................................................159
6.9 Exercises............................................................................................................................165
7 Semisimple Lie Algebras......................................................................................................169
7.1 Semisimple and Reductive Lie Algebras........................................................169
7.2 Cartan Subalgebras......................................................................................................174
7.3 Roots and Root Spaces..............................................................................................176
7.4 The Weyl Group............................................................................................................182
7.5 Root Systems..................................................................................................................183
7.6 Simple Lie Algebras ..................................................................................................185
7.7 The Root Systems of the Classical Lie Algebras......................................188
7.8 Exercises............................................................................................................................193
8 Root Systems................................................................................................................................197
8.1 Abstract Root Systems..............................................................................................197
8.2 Examples in Rank Two..............................................................................................201
8.3 Duality................................................................................................................................204
Contents ix
8.4 Bases and Weyl Chambers......................................................................................206
8.5 Weyl Chambers and the Weyl Group................................................................212
8.6 Dynkin Diagrams..........................................................................................................216
8.7 Integral and Dominant Integral Elements......................................................218
8.8 The Partial Ordering ..................................................................................................221
8.9 Examples in Rank Three..........................................................................................228
8.10 The Classical Root Systems..................................................................................232
8.11 The Classification........................................................................................................236
8.12 Exercises............................................................................................................................238
9 Representations of Semisimple Lie Algebras........................................................241
9.1 Weights of Representations....................................................................................241
9.2 Introduction to Verma Modules............................................................................244
9.3 Universal Enveloping Algebras............................................................................246
9.4 Proof of the PBW Theorem....................................................................................250
9.5 Construction of Verma Modules..........................................................................254
9.6 Irreducible Quotient Modules ..............................................................................257
9.7 Finite-Dimensional Quotient Modules............................................................260
9.8 Exercises............................................................................................................................263
10 Further Properties of the Representations..............................................................265
10.1 The Structure of the Weights ................................................................................265
10.2 The Casimir Element..................................................................................................269
10.3 Complete Reducibility..............................................................................................273
10.4 The Weyl Character Formula................................................................................275
10.5 The Weyl Dimension Formula..............................................................................281
10.6 The Kostant Multiplicity Formula......................................................................287
10.7 The Character Formula for Verma Modules................................................294
10.8 Proof of the Character Formula............................................................................295
10.9 Exercises............................................................................................................................303
Part III Compact Lie Groups
11 Compact Lie Groups and Maximal Tori..................................................................307
11.1 Tori........................................................................................................................................308
11.2 Maximal Tori and the Weyl Group....................................................................312
11.3 Mapping Degrees..........................................................................................................315
11.4 Quotient Manifolds......................................................................................................321
11.5 Proof of the Toms Theorem ..................................................................................326
11.6 The Weyl Integral Formula....................................................................................330
11.7 Roots and the Stmcture of the Weyl Group..................................................333
11.8 Exercises............................................................................................................................339
12 The Compact Group Approach to Representation Theory........................343
12.1 Representations..............................................................................................................343
12.2 Analytically Integral Elements ............................................................................346
12.3 Orthonormality and Completeness for Characters....................................351
X Contents
12.4 The Analytic Proof of the Weyl Character Formula................................357
12.5 Constructing the Representations........................................................................361
12.6 The Case in Which 8 is Not Analytically Integral....................................366
12.7 Exercises............................................................................................................................369
13 Fundamental Groups of Compact Lie Groups....................................................371
13.1 The Fundamental Group..........................................................................................371
13.2 Fundamental Groups of Compact Classical Groups................................373
13.3 Fundamental Groups of Noncompact Classical Groups........................377
13.4 The Fundamental Groups of K and T ............................................................377
13.5 Regular Elements..........................................................................................................383
13.6 The Stiefel Diagram....................................................................................................389
13.7 Proofs of the Main Theorems................................................................................394
13.8 The Center of K............................................................................................................399
13.9 Exercises............................................................................................................................403
A Linear Algebra Review........................................................................................................407
A.l Eigenvectors and Eigenvalues..............................................................................407
A.2 Diagonalization..............................................................................................................408
A.3 Generalized Eigenvectors and the SN Decomposition..........................409
A.4 The Jordan Canonical Form..................................................................................411
A.5 The Trace..........................................................................................................................411
A.6 Inner Products................................................................................................................412
A. 7 Dual Spaces......................................................................................................................414
A. 8 Simultaneous Diagonalization..............................................................................415
B Differential Forms....................................................................................................................419
C Clebsch-Gordan Theory and the Wigner-Eckart Theorem......................425
C. 1 Tensor Products of sl(2; C) Representations................................................425
C.2 The Wigner-Eckart Theorem................................................................................428
C.3 More on Vector Operators......................................................................................432
D Completeness of Characters..............................................................................................435
References..................................................................................................................................................443
Index..............................................................................................................................................................445
|
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| dewey-raw | 512.55 |
| dewey-search | 512.55 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
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| isbn | 9783319134666 9783319374338 |
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| spelling | Hall, Brian C. Verfasser (DE-588)1038321050 aut Lie groups, lie algebras, and representations an elementary introduction Brian C. Hall Second edition Cham ; Heidelberg ; New York ; Dordrecht ; London Springer [2015] © 2015 xiii, 451 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 222 Lie-Gruppe - Lie-Algebra - Darstellungstheorie Lie algebras Lie groups Representations of algebras Representations of groups Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Lie-Algebra (DE-588)4130355-6 gnd rswk-swf Darstellungstheorie (DE-588)4148816-7 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 s Lie-Algebra (DE-588)4130355-6 s Darstellungstheorie (DE-588)4148816-7 s DE-604 Erscheint auch als Online-Ausgabe 978-3-319-13467-3 Graduate texts in mathematics 222 (DE-604)BV000000067 222 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028052749&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hall, Brian C. Lie groups, lie algebras, and representations an elementary introduction Graduate texts in mathematics Lie-Gruppe - Lie-Algebra - Darstellungstheorie Lie algebras Lie groups Representations of algebras Representations of groups Lie-Gruppe (DE-588)4035695-4 gnd Lie-Algebra (DE-588)4130355-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| subject_GND | (DE-588)4035695-4 (DE-588)4130355-6 (DE-588)4148816-7 |
| title | Lie groups, lie algebras, and representations an elementary introduction |
| title_auth | Lie groups, lie algebras, and representations an elementary introduction |
| title_exact_search | Lie groups, lie algebras, and representations an elementary introduction |
| title_full | Lie groups, lie algebras, and representations an elementary introduction Brian C. Hall |
| title_fullStr | Lie groups, lie algebras, and representations an elementary introduction Brian C. Hall |
| title_full_unstemmed | Lie groups, lie algebras, and representations an elementary introduction Brian C. Hall |
| title_short | Lie groups, lie algebras, and representations |
| title_sort | lie groups lie algebras and representations an elementary introduction |
| title_sub | an elementary introduction |
| topic | Lie-Gruppe - Lie-Algebra - Darstellungstheorie Lie algebras Lie groups Representations of algebras Representations of groups Lie-Gruppe (DE-588)4035695-4 gnd Lie-Algebra (DE-588)4130355-6 gnd Darstellungstheorie (DE-588)4148816-7 gnd |
| topic_facet | Lie-Gruppe - Lie-Algebra - Darstellungstheorie Lie algebras Lie groups Representations of algebras Representations of groups Lie-Gruppe Lie-Algebra Darstellungstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028052749&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT hallbrianc liegroupsliealgebrasandrepresentationsanelementaryintroduction |