The Lie theory of connected pro-Lie groups: a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups
Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them. If a complete topological group
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Zürich
Europ. Math. Soc.
2007
|
| Series: | EMS tracts in mathematics
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them. If a complete topological group |
| Physical Description: | XV, 678 S. graph. Darst. |
| ISBN: | 9783037190326 |
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| 245 | 1 | 0 | |a The Lie theory of connected pro-Lie groups |b a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups |c Karl H. Hofmann ; Sidney A. Morris |
| 264 | 1 | |a Zürich |b Europ. Math. Soc. |c 2007 | |
| 300 | |a XV, 678 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a EMS tracts in mathematics |v 2 | |
| 520 | 3 | |a Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them. If a complete topological group | |
| 650 | 4 | |a Algèbres de Lie | |
| 650 | 4 | |a Groupes de Lie | |
| 650 | 4 | |a Groupes localement compacts | |
| 650 | 4 | |a Lie algebras | |
| 650 | 4 | |a Lie groups | |
| 650 | 4 | |a Locally compact groups | |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Morris, Sidney A. |d 1947- |e Verfasser |0 (DE-588)132019515 |4 aut | |
| 830 | 0 | |a EMS tracts in mathematics |v 2 |w (DE-604)BV022480257 |9 2 | |
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Record in the Search Index
| _version_ | 1804136457780068352 |
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| adam_text | Contents
Preface v
Panoramic Overview 1
Part 1. The Base Theory of Pro Lie Groups 6
Part 2. The Algebra of Pro Lie Algebras 19
Part 3. The Fine Lie Theory of Pro Lie Groups 28
Part 4. Global Structure Theory of Connected Pro Lie Groups 35
Part 5. The Role of Compactness on the Pro Lie Algebra Level 44
Part 6. The Role of Compact Subgroups of Pro Lie Groups 52
Part 7. Local Splitting According to Iwasawa 61
1 Limits of Topological Groups 63
Limits 63
The External Approach to Projective Limits 77
Projective Limits and Local Compactness 82
The Fundamental Theorem on Projective Limits 88
The Internal Approach to Projective Limits 91
Projective Limits and Completeness 93
The Closed Subgroup Theorem 96
The Role of Local Compactness 100
The Role of Closed Full Subcategories in Complete Categories 102
Postscript 104
2 Lie Groups and the Lie Theory of Topological Groups 107
The General Definition of a Lie Group 107
The Exponential Function of Topological Groups 110
The Lie Algebra of a Topological Group 115
The Category of Topological Groups with Lie Algebras 119
The Lie Algebra Functor Has a Left Adjoint Functor 126
Sophus Lie s Third Fundamental Theorem 130
The Adjoint Representation of a Topological Group with a Lie Algebra ... 131
Postscript 133
3 Pro Lie Groups 135
Projective Limits of Lie Groups 135
The Lie Algebras of Projective Limits of Lie Groups 137
Pro Lie Algebras 138
Weakly Complete Topological Vector Spaces and Lie Algebras 143
Pro Lie Groups 148
xii Contents
Some Examples 160
An Overview of the Definitions of a Pro Lie Group 160
Postscript 164
4 Quotients of Pro Lie Groups 168
Quotient Objects in Categories 169
Quotient Groups of Pro Lie Groups 170
The Exponential Function of Compact Abelian Groups and Quotient
Morphisms 173
The One Parameter Subgroup Lifting Theorem 182
Sufficient Conditions for Quotients to be Complete 195
Quotients and Quotient Maps between Pro Lie Groups 208
Postscript 210
5 Abelian Pro Lie Groups 212
Examples of Abelian Pro Lie Groups 212
Weil s Lemma 215
Vector Group Splitting Theorems . . 219
Compactly Generated Abelian Pro Lie Groups 233
Weakly Complete Topological Vector Spaces Revisited 235
The Duality Theory of Abelian Pro Lie Groups 237
The Toral Homomorphic Images of an Abelian Pro Lie Group 241
Postscript 246
6 Lie s Third Fundamental Theorem 249
Lie s Third Fundamental Theorem for Pro Lie Groups 249
Semidirect Products 264
Postscript 266
7 Profinite Dimensional Modules and Lie Algebras 269
Modules over a Lie Algebra 269
Duality of Modules 272
Semisimple and Reductive Modules 277
Reductive Pro Lie Algebras 281
Transfinitely Solvable Lie Algebras 285
The Radical and Levi Mal cev: Existence 291
Transfinitely Nilpotent Lie Algebras 296
The Nilpotent Radicals 300
Special Endomorphisms of Pro Lie Algebras 305
Levi Mal cev: Uniqueness 309
Direct and Semidirect Sums Revisited 313
Cartan Subalgebras of Pro Lie Algebras 315
Theorem of Ado 331
Postscript 332
Contents xiii
8 The Structure of Simply Connected Pro Lie Groups 335
The Adjoint Action 335
Simply Connected Pronilpotent Pro Lie Groups 336
The Topological Splitting Technique 340
Simple Connectivity 343
Universal Morphism versus Universal Covering Morphism 352
Postscript 354
9 Analytic Subgroups and the Lie Theory of Pro Lie Groups 356
The Exponential Function on the Inner Derivation Algebra 356
Analytic Subgroups 360
Automorphisms and Invariant Analytic Subgroups 369
Centralizers 370
Normalizers 373
Subalgebras and Subgroups 374
The Center 376
The Commutator Subgroup 376
Finite Dimensional Connected Pro Lie Groups 385
Compact Central Subgroups 402
Divisibility of Groups and Connected Pro Lie Groups 404
The Open Mapping Theorem 409
Completing Proto Lie Groups 413
Unitary Representations 415
Postscript 416
10 The Global Structure of Connected Pro Lie Groups 419
Solvability of Pro Lie Groups 420
The Radical 431
Semisimple and Reductive Groups 434
Nilpotency of Pro Lie Groups 443
The Nilradical and the Coreductive Radical 447
The Structure of Reductive Pro Lie Groups 452
Postscript 458
11 Splitting Theorems for Pro Lie Groups 461
Splitting Reductive Groups Semidirectly 462
Vector Group Splitting in Noncommutative Groups 473
The Structure of Pronilpotent and Prosolvable Groups 478
Conjugacy Theorems 487
Postscript 490
xiv Contents
12 Compact Subgroups of Pro Lie Groups 493
Procompact Modules and Lie Algebras 493
Procompact Lie Algebras and Compactly Embedded Lie Subalgebras
of Pro Lie Algebras 500
Maximal Compactly Embedded Subalgebras of Pro Lie Algebras 504
Conjugacy of Maximal Compactly Embedded Subalgebras 507
Compact Connected Groups 517
Compact Subgroups 519
Potentially Compact Pro Lie Groups 521
The Conjugacy of Maximal Compact Connected Subgroups 524
The Analytic Subgroups Having a Full Lie Algebra 532
Maximal Compact Subgroups of Connected Pro Lie Groups 544
An Alternative Open Mapping Theorem 556
On the Center of a Connected Pro Lie Group 558
Postscript 561
13 Iwasawa s Local Splitting Theorem 566
Locally Splitting Lie Group Quotients of Pro Lie Groups 566
The Lie Algebra Theory of the Local Splitting 571
Splitting on the Group Level 579
Some Comments on Connectedness 584
Postscript 584
14 Catalog of Examples 587
Classification of the Examples in the Catalog 587
Abelian Pro Lie Groups 588
A Simple Construction 595
Pronilpotent Pro Lie Groups 598
Prosolvable Pro Lie Groups 602
Semisimple and Reductive Pro Lie Groups 608
Mixed Groups 615
Examples Concerning the Definition of Lie and Pro Lie Groups 616
Analytic Subgroups of Pro Lie Groups 620
Examples Concerning Simple Connectivity 622
Example Concerning g Module Theory 622
Postscript 623
1 Appendix 1 The Campbell Hausdorff Formalism 624
2 Appendix 2 Weakly Complete Topological Vector Spaces 629
3 Appendix 3 Various Pieces of Information on Semisimple Lie Algebras 651
Postscript 655
Contents xv
Bibliography 657
List of Symbols 667
Index 669
|
| adam_txt |
Contents
Preface v
Panoramic Overview 1
Part 1. The Base Theory of Pro Lie Groups 6
Part 2. The Algebra of Pro Lie Algebras 19
Part 3. The Fine Lie Theory of Pro Lie Groups 28
Part 4. Global Structure Theory of Connected Pro Lie Groups 35
Part 5. The Role of Compactness on the Pro Lie Algebra Level 44
Part 6. The Role of Compact Subgroups of Pro Lie Groups 52
Part 7. Local Splitting According to Iwasawa 61
1 Limits of Topological Groups 63
Limits 63
The External Approach to Projective Limits 77
Projective Limits and Local Compactness 82
The Fundamental Theorem on Projective Limits 88
The Internal Approach to Projective Limits 91
Projective Limits and Completeness 93
The Closed Subgroup Theorem 96
The Role of Local Compactness 100
The Role of Closed Full Subcategories in Complete Categories 102
Postscript 104
2 Lie Groups and the Lie Theory of Topological Groups 107
The General Definition of a Lie Group 107
The Exponential Function of Topological Groups 110
The Lie Algebra of a Topological Group 115
The Category of Topological Groups with Lie Algebras 119
The Lie Algebra Functor Has a Left Adjoint Functor 126
Sophus Lie's Third Fundamental Theorem 130
The Adjoint Representation of a Topological Group with a Lie Algebra . 131
Postscript 133
3 Pro Lie Groups 135
Projective Limits of Lie Groups 135
The Lie Algebras of Projective Limits of Lie Groups 137
Pro Lie Algebras 138
Weakly Complete Topological Vector Spaces and Lie Algebras 143
Pro Lie Groups 148
xii Contents
Some Examples 160
An Overview of the Definitions of a Pro Lie Group 160
Postscript 164
4 Quotients of Pro Lie Groups 168
Quotient Objects in Categories 169
Quotient Groups of Pro Lie Groups 170
The Exponential Function of Compact Abelian Groups and Quotient
Morphisms 173
The One Parameter Subgroup Lifting Theorem 182
Sufficient Conditions for Quotients to be Complete 195
Quotients and Quotient Maps between Pro Lie Groups 208
Postscript 210
5 Abelian Pro Lie Groups 212
Examples of Abelian Pro Lie Groups 212
Weil's Lemma 215
Vector Group Splitting Theorems . . 219
Compactly Generated Abelian Pro Lie Groups 233
Weakly Complete Topological Vector Spaces Revisited 235
The Duality Theory of Abelian Pro Lie Groups 237
The Toral Homomorphic Images of an Abelian Pro Lie Group 241
Postscript 246
6 Lie's Third Fundamental Theorem 249
Lie's Third Fundamental Theorem for Pro Lie Groups 249
Semidirect Products 264
Postscript 266
7 Profinite Dimensional Modules and Lie Algebras 269
Modules over a Lie Algebra 269
Duality of Modules 272
Semisimple and Reductive Modules 277
Reductive Pro Lie Algebras 281
Transfinitely Solvable Lie Algebras 285
The Radical and Levi Mal'cev: Existence 291
Transfinitely Nilpotent Lie Algebras 296
The Nilpotent Radicals 300
Special Endomorphisms of Pro Lie Algebras 305
Levi Mal'cev: Uniqueness 309
Direct and Semidirect Sums Revisited 313
Cartan Subalgebras of Pro Lie Algebras 315
Theorem of Ado 331
Postscript 332
Contents xiii
8 The Structure of Simply Connected Pro Lie Groups 335
The Adjoint Action 335
Simply Connected Pronilpotent Pro Lie Groups 336
The Topological Splitting Technique 340
Simple Connectivity 343
Universal Morphism versus Universal Covering Morphism 352
Postscript 354
9 Analytic Subgroups and the Lie Theory of Pro Lie Groups 356
The Exponential Function on the Inner Derivation Algebra 356
Analytic Subgroups 360
Automorphisms and Invariant Analytic Subgroups 369
Centralizers 370
Normalizers 373
Subalgebras and Subgroups 374
The Center 376
The Commutator Subgroup 376
Finite Dimensional Connected Pro Lie Groups 385
Compact Central Subgroups 402
Divisibility of Groups and Connected Pro Lie Groups 404
The Open Mapping Theorem 409
Completing Proto Lie Groups 413
Unitary Representations 415
Postscript 416
10 The Global Structure of Connected Pro Lie Groups 419
Solvability of Pro Lie Groups 420
The Radical 431
Semisimple and Reductive Groups 434
Nilpotency of Pro Lie Groups 443
The Nilradical and the Coreductive Radical 447
The Structure of Reductive Pro Lie Groups 452
Postscript 458
11 Splitting Theorems for Pro Lie Groups 461
Splitting Reductive Groups Semidirectly 462
Vector Group Splitting in Noncommutative Groups 473
The Structure of Pronilpotent and Prosolvable Groups 478
Conjugacy Theorems 487
Postscript 490
xiv Contents
12 Compact Subgroups of Pro Lie Groups 493
Procompact Modules and Lie Algebras 493
Procompact Lie Algebras and Compactly Embedded Lie Subalgebras
of Pro Lie Algebras 500
Maximal Compactly Embedded Subalgebras of Pro Lie Algebras 504
Conjugacy of Maximal Compactly Embedded Subalgebras 507
Compact Connected Groups 517
Compact Subgroups 519
Potentially Compact Pro Lie Groups 521
The Conjugacy of Maximal Compact Connected Subgroups 524
The Analytic Subgroups Having a Full Lie Algebra 532
Maximal Compact Subgroups of Connected Pro Lie Groups 544
An Alternative Open Mapping Theorem 556
On the Center of a Connected Pro Lie Group 558
Postscript 561
13 Iwasawa's Local Splitting Theorem 566
Locally Splitting Lie Group Quotients of Pro Lie Groups 566
The Lie Algebra Theory of the Local Splitting 571
Splitting on the Group Level 579
Some Comments on Connectedness 584
Postscript 584
14 Catalog of Examples 587
Classification of the Examples in the Catalog 587
Abelian Pro Lie Groups 588
A Simple Construction 595
Pronilpotent Pro Lie Groups 598
Prosolvable Pro Lie Groups 602
Semisimple and Reductive Pro Lie Groups 608
Mixed Groups 615
Examples Concerning the Definition of Lie and Pro Lie Groups 616
Analytic Subgroups of Pro Lie Groups 620
Examples Concerning Simple Connectivity 622
Example Concerning g Module Theory 622
Postscript 623
1 Appendix 1 The Campbell Hausdorff Formalism 624
2 Appendix 2 Weakly Complete Topological Vector Spaces 629
3 Appendix 3 Various Pieces of Information on Semisimple Lie Algebras 651
Postscript 655
Contents xv
Bibliography 657
List of Symbols 667
Index 669 |
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| id | DE-604.BV022397373 |
| illustrated | Illustrated |
| index_date | 2024-07-02T17:17:01Z |
| indexdate | 2024-07-09T20:56:43Z |
| institution | BVB |
| isbn | 9783037190326 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015606087 |
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| physical | XV, 678 S. graph. Darst. |
| publishDate | 2007 |
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| publisher | Europ. Math. Soc. |
| record_format | marc |
| series | EMS tracts in mathematics |
| series2 | EMS tracts in mathematics |
| spelling | Hofmann, Karl H. 1932- Verfasser (DE-588)115780734 aut The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups Karl H. Hofmann ; Sidney A. Morris Zürich Europ. Math. Soc. 2007 XV, 678 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier EMS tracts in mathematics 2 Lie groups were introduced in 1870 by the Norwegian mathematician Sophus Lie. A century later Jean Dieudonné quipped that Lie groups had moved to the center of mathematics and that one cannot undertake anything without them. If a complete topological group Algèbres de Lie Groupes de Lie Groupes localement compacts Lie algebras Lie groups Locally compact groups Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 s DE-604 Morris, Sidney A. 1947- Verfasser (DE-588)132019515 aut EMS tracts in mathematics 2 (DE-604)BV022480257 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015606087&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hofmann, Karl H. 1932- Morris, Sidney A. 1947- The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups EMS tracts in mathematics Algèbres de Lie Groupes de Lie Groupes localement compacts Lie algebras Lie groups Locally compact groups Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4035695-4 |
| title | The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups |
| title_auth | The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups |
| title_exact_search | The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups |
| title_exact_search_txtP | The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups |
| title_full | The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups Karl H. Hofmann ; Sidney A. Morris |
| title_fullStr | The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups Karl H. Hofmann ; Sidney A. Morris |
| title_full_unstemmed | The Lie theory of connected pro-Lie groups a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups Karl H. Hofmann ; Sidney A. Morris |
| title_short | The Lie theory of connected pro-Lie groups |
| title_sort | the lie theory of connected pro lie groups a structure theory for pro lie algebras pro lie groups and connected locally compact groups |
| title_sub | a structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups |
| topic | Algèbres de Lie Groupes de Lie Groupes localement compacts Lie algebras Lie groups Locally compact groups Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Algèbres de Lie Groupes de Lie Groupes localement compacts Lie algebras Lie groups Locally compact groups Lie-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015606087&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV022480257 |
| work_keys_str_mv | AT hofmannkarlh thelietheoryofconnectedproliegroupsastructuretheoryforproliealgebrasproliegroupsandconnectedlocallycompactgroups AT morrissidneya thelietheoryofconnectedproliegroupsastructuretheoryforproliealgebrasproliegroupsandconnectedlocallycompactgroups |