Lie groups, lie algebras, and representations: an elementary introduction
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cham ; Heidelberg ; New York ; Dordrecht ; London
Springer
[2015]
|
| Ausgabe: | Second edition |
| Schriftenreihe: | Graduate texts in mathematics
222 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xiii, 451 Seiten Illustrationen |
| ISBN: | 9783319134666 9783319374338 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV042620021 | ||
| 003 | DE-604 | ||
| 005 | 20220525 | ||
| 007 | t | ||
| 008 | 150616s2015 xxua||| |||| 00||| eng d | ||
| 020 | |a 9783319134666 |c hardcover |9 978-3-319-13466-6 | ||
| 020 | |a 9783319374338 |c softcover |9 978-3-319-37433-8 | ||
| 035 | |a (OCoLC)910905786 | ||
| 035 | |a (DE-599)BVBBV042620021 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 044 | |a xxu |c US | ||
| 049 | |a DE-11 |a DE-19 |a DE-29T |a DE-355 |a DE-188 |a DE-83 | ||
| 050 | 0 | |a QA387 | |
| 082 | 0 | |a 512.55 | |
| 084 | |a SK 340 |0 (DE-625)143232: |2 rvk | ||
| 084 | |a MAT 173f |2 stub | ||
| 084 | |a MAT 225f |2 stub | ||
| 100 | 1 | |a Hall, Brian C. |e Verfasser |0 (DE-588)1038321050 |4 aut | |
| 245 | 1 | 0 | |a Lie groups, lie algebras, and representations |b an elementary introduction |c Brian C. Hall |
| 250 | |a Second edition | ||
| 264 | 1 | |a Cham ; Heidelberg ; New York ; Dordrecht ; London |b Springer |c [2015] | |
| 264 | 4 | |c © 2015 | |
| 300 | |a xiii, 451 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate texts in mathematics |v 222 | |
| 650 | 4 | |a Lie-Gruppe - Lie-Algebra - Darstellungstheorie | |
| 650 | 4 | |a Lie algebras | |
| 650 | 4 | |a Lie groups | |
| 650 | 4 | |a Representations of algebras | |
| 650 | 4 | |a Representations of groups | |
| 650 | 0 | 7 | |a Lie-Gruppe |0 (DE-588)4035695-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lie-Algebra |0 (DE-588)4130355-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Darstellungstheorie |0 (DE-588)4148816-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Lie-Gruppe |0 (DE-588)4035695-4 |D s |
| 689 | 0 | 1 | |a Lie-Algebra |0 (DE-588)4130355-6 |D s |
| 689 | 0 | 2 | |a Darstellungstheorie |0 (DE-588)4148816-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-319-13467-3 |
| 830 | 0 | |a Graduate texts in mathematics |v 222 |w (DE-604)BV000000067 |9 222 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u 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 |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-028052749 | ||
Datensatz im Suchindex
| _version_ | 1804174794701144064 |
|---|---|
| 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
|
| any_adam_object | 1 |
| author | Hall, Brian C. |
| author_GND | (DE-588)1038321050 |
| author_facet | Hall, Brian C. |
| author_role | aut |
| author_sort | Hall, Brian C. |
| author_variant | b c h bc bch |
| building | Verbundindex |
| bvnumber | BV042620021 |
| callnumber-first | Q - Science |
| callnumber-label | QA387 |
| callnumber-raw | QA387 |
| callnumber-search | QA387 |
| callnumber-sort | QA 3387 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 340 |
| classification_tum | MAT 173f MAT 225f |
| ctrlnum | (OCoLC)910905786 (DE-599)BVBBV042620021 |
| dewey-full | 512.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.55 |
| dewey-search | 512.55 |
| dewey-sort | 3512.55 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02171nam a2200553 cb4500</leader><controlfield tag="001">BV042620021</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20220525 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">150616s2015 xxua||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783319134666</subfield><subfield code="c">hardcover</subfield><subfield code="9">978-3-319-13466-6</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783319374338</subfield><subfield code="c">softcover</subfield><subfield code="9">978-3-319-37433-8</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)910905786</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV042620021</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">xxu</subfield><subfield code="c">US</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-11</subfield><subfield code="a">DE-19</subfield><subfield code="a">DE-29T</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-188</subfield><subfield code="a">DE-83</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA387</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">512.55</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 340</subfield><subfield code="0">(DE-625)143232:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 173f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 225f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Hall, Brian C.</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)1038321050</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Lie groups, lie algebras, and representations</subfield><subfield code="b">an elementary introduction</subfield><subfield code="c">Brian C. Hall</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">Second edition</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Cham ; Heidelberg ; New York ; Dordrecht ; London</subfield><subfield code="b">Springer</subfield><subfield code="c">[2015]</subfield></datafield><datafield tag="264" ind1=" " ind2="4"><subfield code="c">© 2015</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">xiii, 451 Seiten</subfield><subfield code="b">Illustrationen</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Graduate texts in mathematics</subfield><subfield code="v">222</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Lie-Gruppe - Lie-Algebra - Darstellungstheorie</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Lie algebras</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Lie groups</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Representations of algebras</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Representations of groups</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Lie-Gruppe</subfield><subfield code="0">(DE-588)4035695-4</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Lie-Algebra</subfield><subfield code="0">(DE-588)4130355-6</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Darstellungstheorie</subfield><subfield code="0">(DE-588)4148816-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Lie-Gruppe</subfield><subfield code="0">(DE-588)4035695-4</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Lie-Algebra</subfield><subfield code="0">(DE-588)4130355-6</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="2"><subfield code="a">Darstellungstheorie</subfield><subfield code="0">(DE-588)4148816-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-3-319-13467-3</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Graduate texts in mathematics</subfield><subfield code="v">222</subfield><subfield code="w">(DE-604)BV000000067</subfield><subfield code="9">222</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">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</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-028052749</subfield></datafield></record></collection> |
| id | DE-604.BV042620021 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:06:03Z |
| institution | BVB |
| isbn | 9783319134666 9783319374338 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028052749 |
| oclc_num | 910905786 |
| open_access_boolean | |
| owner | DE-11 DE-19 DE-BY-UBM DE-29T DE-355 DE-BY-UBR DE-188 DE-83 |
| owner_facet | DE-11 DE-19 DE-BY-UBM DE-29T DE-355 DE-BY-UBR DE-188 DE-83 |
| physical | xiii, 451 Seiten Illustrationen |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| 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 |