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 $G$ can be approximated by Lie groups in the sense that every identity neighborhood $U$ of $G$ contains a normal subgroup $N$ such that $G/N$ is a Lie group, then it is called a pro-Lie group. Every locally compact connected topological group and every compact group is a pro-Lie group. While the class of locally compact groups is not closed under the formation of arbitrary products, the class of pro-Lie groups is. For half a century, locally compact pro-Lie groups have drifted through the literature, yet this is the first book which systematically treats the Lie and structure theory of pro-Lie groups irrespective of local compactness. This study fits very well into the current trend which addresses infinite-dimensional Lie groups. The results of this text are based on a theory of pro-Lie algebras which parallels the structure theory of finite-dimensional real Lie algebras to an astonishing degree, even though it has had to overcome greater technical obstacles. This book exposes a Lie theory of connected pro-Lie groups (and hence of connected locally compact groups) and illuminates the manifold ways in which their structure theory reduces to that of compact groups on the one hand and of finite-dimensional Lie groups on the other. It is a continuation of the authors' fundamental monograph on the structure of compact groups (1998, 2006) and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis, and representation theory. It is written to be accessible to advanced graduate students wishing to study this fascinating and important area of current research, which has so many fruitful interactions with other fields of mathematics.
Author(s): Karl H. Hofmann and Sidney A. Morris
Series: EMS Tracts in Mathematics
Publisher: European Mathematical Society
Year: 2007
Language: English
Pages: 694
Preface......Page 6
Contents......Page 12
Panoramic Overview......Page 17
Part 1. The Base Theory of Pro-Lie Groups......Page 22
Part 2. The Algebra of Pro-Lie Algebras......Page 35
Part 3. The Fine Lie Theory of Pro-Lie Groups......Page 43
Part 4. Global Structure Theory of Connected Pro-Lie Groups......Page 51
Part 5. The Role of Compactness on the Pro-Lie Algebra Level......Page 60
Part 6. The Role of Compact Subgroups of Pro-Lie Groups......Page 68
Part 7. Local Splitting According to Iwasawa......Page 76
Limits......Page 79
Nilpotency of Pro-Lie Groups......Page 93
Projective Limits and Local Compactness......Page 98
The Fundamental Theorem on Projective Limits......Page 104
The Internal Approach to Projective Limits......Page 105
Projective Limits and Completeness......Page 109
The Closed Subgroup Theorem......Page 112
The Role of Local Compactness......Page 116
The Role of Closed Full Subcategories in Complete Categories......Page 118
Postscript......Page 120
The General Definition of a Lie Group......Page 123
The Exponential Function of Topological Groups......Page 126
The Lie Algebra of a Topological Group......Page 130
The Category of Topological Groups with Lie Algebras......Page 91
The Lie Algebra Functor Has a Left Adjoint Functor......Page 142
Sophus Lie's Third Fundamental Theorem......Page 143
The Adjoint Representation of a Topological Group with a Lie Algebra......Page 147
Postscript......Page 149
Projective Limits of Lie Groups......Page 151
The Lie Algebras of Projective Limits of Lie Groups......Page 82
Pro-Lie Algebras......Page 153
Weakly Complete Topological Vector Spaces and Lie Algebras......Page 158
Pro-Lie Groups......Page 163
An Overview of the Definitions of a Pro-Lie Group......Page 176
Postscript......Page 180
4 Quotients of Pro-Lie Groups......Page 184
Quotient Groups of Pro-Lie Groups......Page 185
The Exponential Function of Compact Abelian Groups and Quotient Morphisms......Page 186
Normalizers......Page 81
Sufficient Conditions for Quotients to be Complete......Page 210
Quotients and Quotient Maps between Pro-Lie Groups......Page 224
Postscript......Page 226
Examples of Abelian Pro-Lie Groups......Page 228
Weil's Lemma......Page 230
Vector Group Splitting Theorems......Page 235
Compactly Generated Abelian Pro-Lie Groups......Page 248
Weakly Complete Topological Vector Spaces Revisited......Page 251
The Duality Theory of Abelian Pro-Lie Groups......Page 252
The Toral Homomorphic Images of an Abelian Pro-Lie Group......Page 257
Postscript......Page 262
Lie's Third Fundamental Theorem for Pro-Lie Groups......Page 265
Semidirect Products......Page 279
Postscript......Page 282
Modules over a Lie Algebra......Page 285
Duality of Modules......Page 288
Semisimple and Reductive Modules......Page 85
Reductive Pro-Lie Algebras......Page 297
Transfinitely Solvable Lie Algebras......Page 300
The Radical and Levi–Mal'cev: Existence......Page 307
Transfinitely Nilpotent Lie Algebras......Page 312
The Nilpotent Radicals......Page 96
Special Endomorphisms of Pro-Lie Algebras......Page 319
Levi–Mal'cev: Uniqueness......Page 325
Direct and Semidirect Sums Revisited......Page 329
Cartan Subalgebras of Pro-Lie Algebras......Page 331
Theorem of Ado......Page 346
Postscript......Page 348
Simply Connected Pronilpotent Pro-Lie Groups......Page 351
Simple Connectivity......Page 358
Universal Morphism versus Universal Covering Morphism......Page 368
Postscript......Page 370
The Exponential Function on the Inner Derivation Algebra......Page 372
Analytic Subgroups......Page 375
Automorphisms and Invariant Analytic Subgroups......Page 384
Centralizers......Page 386
Subalgebras and Subgroups......Page 389
The Center......Page 391
The Commutator Subgroup......Page 392
Finite-Dimensional Connected Pro-Lie Groups......Page 400
Compact Central Subgroups......Page 83
Divisibility of Groups and Connected Pro-Lie Groups......Page 420
The Open Mapping Theorem......Page 425
Completing Proto-Lie Groups......Page 429
Unitary Representations......Page 430
Postscript......Page 432
10 The Global Structure of Connected Pro-Lie Groups......Page 435
Solvability of Pro-Lie Groups......Page 436
The Radical......Page 446
Semisimple and Reductive Groups......Page 449
The Nilradical and the Coreductive Radical......Page 94
The Structure of Reductive Pro-Lie Groups......Page 467
Postscript......Page 474
Splitting Reductive Groups Semidirectly......Page 477
Vector Group Splitting in Noncommutative Groups......Page 489
The Structure of Pronilpotent and Prosolvable Groups......Page 494
Conjugacy Theorems......Page 503
Postscript......Page 506
Procompact Modules and Lie Algebras......Page 509
Procompact Lie Algebras and Compactly Embedded Lie Subalgebras of Pro-Lie Algebras......Page 515
Maximal Compactly Embedded Subalgebras of Pro-Lie Algebras......Page 519
Conjugacy of Maximal Compactly Embedded Subalgebras......Page 523
Compact Connected Groups......Page 532
Compact Subgroups......Page 535
Potentially Compact Pro-Lie Groups......Page 537
The Conjugacy of Maximal Compact Connected Subgroups......Page 540
The Analytic Subgroups Having a Full Lie Algebra......Page 548
Maximal Compact Subgroups of Connected Pro-Lie Groups......Page 560
An Alternative Open Mapping Theorem......Page 572
On the Center of a Connected Pro-Lie Group......Page 574
Postscript......Page 577
Locally Splitting Lie Group Quotients of Pro-Lie Groups......Page 582
The Lie Algebra Theory of the Local Splitting......Page 587
Splitting on the Group Level......Page 593
Postscript......Page 600
Abelian Pro-Lie Groups......Page 603
A Simple Construction......Page 610
Pronilpotent Pro-Lie Groups......Page 614
Prosolvable Pro-Lie Groups......Page 618
Semisimple and Reductive Pro-Lie Groups......Page 624
Mixed Groups......Page 631
Examples Concerning the Definition of Lie and Pro-Lie Groups......Page 632
Analytic Subgroups of Pro-Lie Groups......Page 636
Example Concerning g-Module Theory......Page 638
Postscript......Page 639
Appendix 1 The CampbellŒHausdorff Formalism......Page 640
Appendix 2 Weakly Complete Topological Vector Spaces......Page 645
Appendix 3 Various Pieces of Information on Semisimple Lie Algebras......Page 667
Postscript......Page 671
Bibliography......Page 673
List of Symbols......Page 683
Index......Page 685
