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 $G$ 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. Preface......Page 5 Contents......Page 11 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 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