The Lie Theory of Connected Pro-Lie Groups 3037190329, 9783037190326

Table of contents :
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

Citation preview

hofmann_morris_titelei

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EMS Tracts in Mathematics 2

hofmann_morris_titelei

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EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, Strasbourg, France) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions

hofmann_morris_titelei

30.4.2007

8:12 Uhr

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Karl H. Hofmann Sidney A. Morris

The Lie Theory of Connected Pro-Lie Groups A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact Groups

M

M

S E M E S

S E M E S

European Mathematical Society

hofmann_morris_titelei

30.4.2007

8:12 Uhr

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Authors: Karl Heinrich Hofmann Fachbereich Mathematik, AG 5 Technische Universität Darmstadt Schloßgartenstraße 7 64289 Darmstadt Germany E-Mail: hofmann @mathematik.tu-darmstadt.de

Sidney A. Morris School of Information Technology and Mathematical Sciences University of Ballarat P.O. Box 663 Ballarat, Victoria 3353 Australia E-mail: S.Morris @ballarat.edu.au

2000 Mathematics Subject Classification: Primary 22-02; secondary 17B65, 22D05, 22E20, 22E65, 44A13, 44M40, 58B25.

Key words: pro-Lie groups, pro-Lie algebras, Lie theory of connected pro-Lie groups, exponential function, structure theory of locally compact groups, completeness, quotient groups, open mapping theorem, Levi–Malcev splitting, local Iwasawa splitting.

ISBN 978-3-03719-032-6 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Preface

Sophus Marius Lie (1842–1899) laid the foundation of the theory named Lie theory in honor of its creator. Several mathematicians, likewise prominent in the history of modern mathematics, contributed to its inception in the decades following 1873, which was the year in which Lie started to occupy himself intensively in the study of what he called continuous groups, notably: Friedrich Engel, Wilhelm Killing, Élie Cartan, Henri Poincaré, and Hermann Weyl. From the beginning, however, the advance of Lie theory bifurcated into two separate major highways, which is the reason why the words Lie theory mean different things to different people. Lie himself aimed at accomplishing for the solution of differential equations (in the widest sense) what Évariste Galois and Lie’s countryman Niels Henrik Abel achieved for the solution of algebraic equations: A profound understanding and, to the best extent possible, a classification in terms of groups. Even though Lie considered himself a “geometer,” he created a territory of analysis that is called “Lie theory” by those working in it, and that is represented by the well-known text by Peter J. Olver entitled “Applications of Lie Groups to Differential Equations” [Springer-Verlag, Berlin, New York, etc., 1986]. We should say in the beginning, that the project of Lie theory which we shall discuss in this book, in philosophy and thrust, does not belong to this line. A second highway was taken by Killing and Cartan. It led to a study of what soon became known as Lie algebras, of the group and structure theory of Lie groups, and to the geometry of homogeneous spaces. The latter notably yielded the classification of symmetric spaces by Élie Cartan. At long last it merged into the encyclopedic attempt by Nicolas Bourbaki of the nineteen hundred sixties and seventies, to summarize what had been achieved, and to the emergence of an immense collection of textbooks at all levels. In 1973 Jean Dieudonné quipped “Les groupes de Lie sont devenus le centre des Mathématiques; on ne peut rien faire de sérieux sans eux.” (Lie groups have moved to the center of mathematics. One cannot seriously undertake anything without them. Gazette des Mathématiciens, Société Mathématique de France, Octobre 1974, p. 77.) By and large, in this line of “Lie theory” the words meant the structure theory of Lie algebras and Lie groups, and in particular how the latter is based on the former. The term ‘Lie group’ originally meant ‘finite-dimensional Lie group’ and most people understand the words in this sense today. However even Sophus Lie spoke of “unendliche Gruppen” by which he meant something like infinite-dimensional Lie groups. But reasonable concepts of dimension were not yet available in the 19th century before topology was on its way. And indeed Lie’s attempts in this direction did not appear to have gotten off the ground. The significance of Lie’s discoveries was emphasized by David Hilbert by raising the question in 1900 whether (in later terminology) a locally euclidean topological group is in fact an analytic group in the sense of Lie. This was the fifth of his famous 23 problems which foreshadowed so much of the mathematical creativity

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of the 20th century. It required half a century of effort on the part of several generations of eminent mathematicians until it was settled in the affirmative. Partial solutions came along as the structure of topological groups was understood better and better: Hermann Weyl and his student F. Peter in 1923 laid the foundations of the representation and structure theory of compact groups, and a positive answer to Hilbert’s Fifth Problem for compact groups was a consequence, drawn by John von Neumann in 1932. Lev Semyonovich Pontryagin and Egbert Rudolf van Kampen developed in 1932, respectively, 1936, the duality theory of locally compact abelian groups laying the foundations for an abstract harmonic analysis flourishing throughout the second half of the 20th century and providing the central method for attacking the structure theory of compact abelian groups via duality. Again a positive response to Hilbert’s question for locally euclidean abelian groups followed in the wash. One of the most significant and seminal papers in topological group theory was published in 1949 by Kenkichi Iwasawa, some three years before Hilbert’s Problem was finally settled by the concerted contribution of Andrew Mattei Gleason, Dean Montgomery, Leon Zippin, and Hidehiko Yamabe. It was Iwasawa who clearly recognized for the first time that the structure theory of locally compact groups reduced to that of compact groups and finite-dimensional Lie groups provided one knew that they happen to be approximated by finite-dimensional Lie groups in the sense of projective limits, in other words, if they were pro-Lie groups in our parlance. And this is what Yamabe established in 1953 for all locally compact groups which have a compact factor group modulo their identity component – almost connected locally compact groups as we shall say. The most influential monograph collecting these results was the book by Montgomery and Zippin of 1955 with the title “Topological Transformation Groups”. The theories of compact groups and of abelian locally compact groups had introduced in the first half of the century classes of groups with an explicit structure theory without the restriction of finite-dimensionality, and in the middle of the century these results opened up an explicit development for numerous results on the structure theory of locally compact groups. What are the coordinates of our book in this historical thread? It was recognized in 1957 by Richard Kenneth Lashof that any locally compact group G has a Lie algebra g. If g is appropriately defined, then the exponential function exp : g → G is supplied along with the definition. Yet the fact that these observations are the nucleus of a complete and rich, although infinite-dimensional Lie theory was never exploited. The present book is devoted to the foundations, and the exploitation of such a Lie theory. At a point in the overall historical development where infinite-dimensional Lie theories gain increasing acceptance and attract much interest, this appears to be timely. The Lie theory we unfold is based on projective limits, both on the group level and on the Lie algebra level. We shall find it very helpful that category theory, as a tool for the “working mathematician” as Saunders Mac Lane formulated it, is so well developed that we see immediately what we need, and we shall exploit it. In our case, we need the theory of limits in a complete category, that is, in a category in which all limits exist, and we need the theory of pairs of adjoint functors, which is closely linked with limits.

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The machinery of projective limits is familiar to mathematicians dealing with profinite groups in their work on Galois theory and arithmetics, quite generally. But the apparatus of projective limits is also familiar to mathematicians dealing with compact groups, their representation theory and abstract harmonic analysis. Indeed all group theoreticians working on the structure theory of locally compact groups encounter projective limits sooner or later. In this book we shall call projective limits of projective systems (or, as some authors say, inverse systems) of finite-dimensional Lie groups pro-Lie groups. That is, pro-Lie groups relate to finite-dimensional Lie groups exactly as profinite groups relate to finite groups. However, in the theory of locally compact groups, one encounters a special kind of projective limit, namely, limit situations where limit maps and bonding maps are proper, that is, are closed continuous homomorphisms between locally compact groups having compact kernels. Some authors call such maps perfect. This type of projective limit has a significant element of compactness already built into its definition, and it is this type of limit that has shaped the intuitions of group theoreticians for fifty years or more. From the vantage point of category theory, however, such a restriction is entirely unnatural, as is indeed the entire focus on locally compact pro-Lie groups: The class of locally compact groups is not even closed under the formation of products – as the example of the groups RN or ZN shows immediately. Mathematicians will be naturally attracted to the problem of eliminating the focus on locally compact groups. As one proceeds in the direction of pro-Lie groups in general, however, one comes to realize that the restriction to locally compact groups is unnatural also for reasons that are entirely interior to the mathematics of topological groups and Lie groups. For several years we have been engaged in the laying of the foundations of a general theory of the category of pro-Lie groups. The results are presented in this book. On the first 60 pages, the reader will find a panoramic overview of what is contained in its 14 chapters, and the user of the book should get a more compact overview by perusing its table of contents. The Lie theory of finite-dimensional Lie groups works because for a connected Lie group G, its Lie algebra g and its exponential function exp : g → G largely determine the structure of G. We hasten to add that, except for the case that G is simply connected, they do not do so completely. As the title of our book indicates, we focus on a Lie theory for connected pro-Lie groups. As a consequence, our structure theory is one that is mainly concerned with connected pro-Lie groups, sometimes going a bit further, but rarely much beyond almost connected groups. In view of Yamabe’s Theorem, the structure theory of connected or almost connected pro-Lie groups applies at once to connected or even almost connected locally compact groups. There are several key elements to the structure theory of pro-Lie groups. Firstly, a thorough understanding of the working of projective limits is needed without the crutch of thinking in terms of proper maps all the time. Chapter 1 deals with many facets of this issue. But only after Chapter 3 will we have understood all aspects of what this means for the very definition of pro-Lie groups itself.

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Secondly, the entire theory depends on our accepting that pro-Lie groups, even though not being Lie groups, nevertheless have a working Lie theory, complete with the appropriate Lie algebras which we shall call pro-Lie algebras and working exponential functions that mediate between pro-Lie groups and their Lie algebras. Indeed we must become aware at an early stage that there is a good Lie algebra functor from the category of pro-Lie groups to the category of pro-Lie algebras. One of the very positive side effects of facing wider categories than the conventional ones in developing a Lie theory is that this enlargement of scope forces us to realize in great clarity that the Lie algebra functor is opposed by a Lie group functor that encapsulates lucidly the contents of Lie’s Third Fundamental Theorem. This applies to the classical situation as well, but it is not recognized there because the theory of universal covering Lie groups, while providing topologically satisfying results in general, tends to obscure the precise functorial set-up. Since for pro-Lie groups a classical covering theory is impossible as one knows from the theory of compact connected abelian groups, it is mandatory that one understands the functorial background of a more general universal covering theory. We shall discuss this in Chapters 2, 4, 6 and 8. Thirdly, the success of the structure theory of pro-Lie groups depends in a large measure on our success in dealing with the structure theory of pro-Lie algebras. This pervades the whole book, but most of this is done in our rather long Chapter 7. The point is that the topological vector spaces underlying pro-Lie algebras are what we call weakly complete topological vector spaces, because they are exactly the duals of real vector spaces given the weak ∗-topology, that is, the topology of pointwise convergence of linear functionals. Since the vector space duality is crucial for this class of topological vector spaces and hence for the structure theory of pro-Lie algebras we present the essential features of the linear algebra of weakly complete topological vector spaces in an appendix, namely, Appendix 2. The relevance of weakly complete topological vector spaces in the structure theory of pro-Lie groups themselves is evidenced in that chapter in which we discuss the structure of commutative pro-Lie groups, and that is Chapter 5. With all of these foundations done, the Lie and structure theory of pro-Lie groups can proceed, as it does in Chapters 9 through 13. This preface is not the place to go into the details, but we shall present to our readers in the beginning of the book, in our panoramic overview, the results which we obtain. One of the lead motives of our structure theory is to reduce the structure of connected pro-Lie groups in the optimal extent possible to the structure theory of compact connected groups, weakly complete topological vector spaces, and finite-dimensional Lie groups. We will prove some major structure theorems which expose that we, in essence, achieve this goal. Acknowledgements. Our mathematical collaboration in the area of compact and locally compact groups originated in the late seventies. We record with deep gratitude the steady support of our families that we enjoyed throughout this time, particularly on the part of our wives Isolde Hofmann and Elizabeth Morris. Our work on the structure of pro-Lie groups began in the fall of the year 2000; we are grateful to our academic home institutions, notably to the University of Ballarat in Victoria, Australia, the Darmstadt

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University of Technology in Germany, the University of South Australia in Adelaide, Australia, and Tulane University in New Orleans, USA. We thank Dr. Manfred Karbe who accompanied the project from its beginning with great interest and finally published it at the European Mathematical Society Publishing House. Our special thanks go to Dr. Irene Zimmermann who acted as our copy editor and typsetter. Only a mathematician such as Dr. Zimmermann could have identified errors and inconsistencies as she did. On the top of this she polished rough spots in the text and asked questions which resulted in improving the presentation. In short we have been blessed to have such a professional handling of our TEX script. Darmstadt and Ballarat, March 2007

Karl. H. Hofmann Sidney A. Morris

Contents

Preface Panoramic Overview Part 1. The Base Theory of Pro-Lie Groups . . . . . . . . . . Part 2. The Algebra of Pro-Lie Algebras . . . . . . . . . . . . Part 3. The Fine Lie Theory of Pro-Lie Groups . . . . . . . . Part 4. Global Structure Theory of Connected Pro-Lie Groups Part 5. The Role of Compactness on the Pro-Lie Algebra Level Part 6. The Role of Compact Subgroups of Pro-Lie Groups . . Part 7. Local Splitting According to Iwasawa . . . . . . . . . 1

2

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Limits of Topological Groups Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The External Approach to Projective Limits . . . . . . . . . . Projective Limits and Local Compactness . . . . . . . . . . . The Fundamental Theorem on Projective Limits . . . . . . . . The Internal Approach to Projective Limits . . . . . . . . . . . Projective Limits and Completeness . . . . . . . . . . . . . . The Closed Subgroup Theorem . . . . . . . . . . . . . . . . . The Role of Local Compactness . . . . . . . . . . . . . . . . The Role of Closed Full Subcategories in Complete Categories Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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63 63 77 82 88 90 93 96 100 102 104

Lie Groups and the Lie Theory of Topological Groups The General Definition of a Lie Group . . . . . . . . . . . . . . . . . The Exponential Function of Topological Groups . . . . . . . . . . . The Lie Algebra of a Topological Group . . . . . . . . . . . . . . . . The Category of Topological Groups with Lie Algebras . . . . . . . . The Lie Algebra Functor Has a Left Adjoint Functor . . . . . . . . . . Sophus Lie’s Third Fundamental Theorem . . . . . . . . . . . . . . . The Adjoint Representation of a Topological Group with a Lie Algebra Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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107 107 110 114 119 126 130 131 133

Pro-Lie Groups Projective Limits of Lie Groups . . . . . . . . . . . . . . . . . The Lie Algebras of Projective Limits of Lie Groups . . . . . . Pro-Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . Weakly Complete Topological Vector Spaces and Lie Algebras Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . .

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Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 An Overview of the Definitions of a Pro-Lie Group . . . . . . . . . . . . . 160 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4

Quotients of Pro-Lie Groups Quotient Objects in Categories . . . . . . . . . . . . . Quotient Groups of Pro-Lie Groups . . . . . . . . . . The Exponential Function of Compact Abelian Groups phisms . . . . . . . . . . . . . . . . . . . . . . The One Parameter Subgroup Lifting Theorem . . . . Sufficient Conditions for Quotients to be Complete . . Quotients and Quotient Maps between Pro-Lie Groups Postscript . . . . . . . . . . . . . . . . . . . . . . . .

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5 Abelian Pro-Lie Groups Examples of Abelian Pro-Lie Groups . . . . . . . . . . . . . . Weil’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Group Splitting Theorems . . . . . . . . . . . . . . . . Compactly Generated Abelian Pro-Lie Groups . . . . . . . . . Weakly Complete Topological Vector Spaces Revisited . . . . The Duality Theory of Abelian Pro-Lie Groups . . . . . . . . The Toral Homomorphic Images of an Abelian Pro-Lie Group Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168 169 170 173 182 194 208 210

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212 212 215 219 233 235 237 241 246

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Lie’s Third Fundamental Theorem Lie’s Third Fundamental Theorem for Pro-Lie Groups . . . . . . . . . . . . Semidirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

249 249 264 266

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Profinite-Dimensional Modules and Lie Algebras Modules over a Lie Algebra . . . . . . . . . . . . . Duality of Modules . . . . . . . . . . . . . . . . . Semisimple and Reductive Modules . . . . . . . . Reductive Pro-Lie Algebras . . . . . . . . . . . . . Transfinitely Solvable Lie Algebras . . . . . . . . . The Radical and Levi–Mal’cev: Existence . . . . . Transfinitely Nilpotent Lie Algebras . . . . . . . . The Nilpotent Radicals . . . . . . . . . . . . . . . Special Endomorphisms of Pro-Lie Algebras . . . . Levi–Mal’cev: Uniqueness . . . . . . . . . . . . . Direct and Semidirect Sums Revisited . . . . . . . Cartan Subalgebras of Pro-Lie Algebras . . . . . . Theorem of Ado . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . .

269 269 272 277 281 284 291 296 300 305 309 313 315 330 332

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8 The Structure of Simply Connected Pro-Lie Groups The Adjoint Action . . . . . . . . . . . . . . . . . . . . . Simply Connected Pronilpotent Pro-Lie Groups . . . . . . The Topological Splitting Technique . . . . . . . . . . . . Simple Connectivity . . . . . . . . . . . . . . . . . . . . . Universal Morphism versus Universal Covering Morphism Postscript . . . . . . . . . . . . . . . . . . . . . . . . . .

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335 335 336 339 342 352 354

9 Analytic Subgroups and the Lie Theory of Pro-Lie Groups The Exponential Function on the Inner Derivation Algebra . . Analytic Subgroups . . . . . . . . . . . . . . . . . . . . . . . Automorphisms and Invariant Analytic Subgroups . . . . . . . Centralizers . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . Subalgebras and Subgroups . . . . . . . . . . . . . . . . . . . The Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Commutator Subgroup . . . . . . . . . . . . . . . . . . . Finite-Dimensional Connected Pro-Lie Groups . . . . . . . . Compact Central Subgroups . . . . . . . . . . . . . . . . . . Divisibility of Groups and Connected Pro-Lie Groups . . . . . The Open Mapping Theorem . . . . . . . . . . . . . . . . . . Completing Proto-Lie Groups . . . . . . . . . . . . . . . . . Unitary Representations . . . . . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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356 356 360 369 370 372 374 376 376 385 402 404 409 413 415 416

10 The Global Structure of Connected Pro-Lie Groups Solvability of Pro-Lie Groups . . . . . . . . . . . . . . The Radical . . . . . . . . . . . . . . . . . . . . . . . Semisimple and Reductive Groups . . . . . . . . . . . Nilpotency of Pro-Lie Groups . . . . . . . . . . . . . The Nilradical and the Coreductive Radical . . . . . . The Structure of Reductive Pro-Lie Groups . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . .

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419 420 430 434 443 447 451 458

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461 461 473 478 487 490

11 Splitting Theorems for Pro-Lie Groups Splitting Reductive Groups Semidirectly . . . . . . . Vector Group Splitting in Noncommutative Groups . The Structure of Pronilpotent and Prosolvable Groups Conjugacy Theorems . . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . .

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xiv

Contents

12 Compact Subgroups of Pro-Lie Groups Procompact Modules and Lie Algebras . . . . . . . . . . . . . . . . . . . . Procompact Lie Algebras and Compactly Embedded Lie Subalgebras of ProLie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximal Compactly Embedded Subalgebras of Pro-Lie Algebras . . . . . . Conjugacy of Maximal Compactly Embedded Subalgebras . . . . . . . . . Compact Connected Groups . . . . . . . . . . . . . . . . . . . . . . . . . Compact Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potentially Compact Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . The Conjugacy of Maximal Compact Connected Subgroups . . . . . . . . . The Analytic Subgroups Having a Full Lie Algebra . . . . . . . . . . . . . Maximal Compact Subgroups of Connected Pro-Lie Groups . . . . . . . . An Alternative Open Mapping Theorem . . . . . . . . . . . . . . . . . . . On the Center of a Connected Pro-Lie Group . . . . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

500 504 507 516 519 521 524 532 544 556 558 561

13 Iwasawa’s Local Splitting Theorem Locally Splitting Lie Group Quotients of Pro-Lie Groups The Lie Algebra Theory of the Local Splitting . . . . . . Splitting on the Group Level . . . . . . . . . . . . . . . Some Comments on Connectedness . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . .

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566 566 571 578 584 584

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587 587 587 595 598 602 608 615 616 620 622 622 623

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14 Catalog of Examples Classification of the Examples in the Catalog . . . . . . . . . . Abelian Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . . . A Simple Construction . . . . . . . . . . . . . . . . . . . . . . Pronilpotent Pro-Lie Groups . . . . . . . . . . . . . . . . . . . Prosolvable Pro-Lie Groups . . . . . . . . . . . . . . . . . . . . Semisimple and Reductive Pro-Lie Groups . . . . . . . . . . . . Mixed Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples Concerning the Definition of Lie and Pro-Lie Groups . Analytic Subgroups of Pro-Lie Groups . . . . . . . . . . . . . . Examples Concerning Simple Connectivity . . . . . . . . . . . Example Concerning g-Module Theory . . . . . . . . . . . . . Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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

Panoramic Overview

Compactness is a core concept in general topology, because it introduces finiteness in otherwise infinite geometric objects. When we combine compactness with group theory and its enormous background we can expect a theory rich in results, varied in direction, and fertile in applications. And we get it as is evidenced through a sizeable body of monographs and texts having come about in the second half of last century. Naturally, we like to cite our book on compact groups [102] that appeared in 1998 and that experienced a second revised and augmented edition in 2006. The standard examples are linear groups such as the orthogonal and unitary groups, or the additive groups of p-adic integers, and this confirms that the concept of a compact group is natural. The class of compact groups is closed in the class of all topological groups under the formation of arbitrary products and the passage to closed subgroups. This makes it a closed category in its own right, and that in itself is a fact from which many desirable properties of this category follow. But there are before our eyes just as natural examples of groups that illustrate that there are many topological groups basic to analysis, geometry and algebra which are not compact; easily perceived examples are the additive groups of Rn or, more generally, finite-dimensional vector spaces over locally compact fields, and linear groups like the full linear groups Gl(n, R) and their closed subgroups. All of these groups, however, are locally compact. The most important locally compact groups are real Lie groups which are connected or have, at most, finitely many connected components. One definition of a Lie group is that it is a real analytic manifold with a group structure such that multiplication and inversion are analytic functions. A topological group which is isomorphic to a closed subgroup of the topological group Gl(n, R) is a Lie group, and we shall call such a Lie group a linear Lie group. Let us emphasize at this point that here, and in the following, when we speak about two topological groups as being isomorphic, we mean them to be isomorphic as topological groups; some writers like to stress this by saying that they are “isomorphic algebraically and topologically”. We give a definition of a general Lie group in Appendix 1 to this book which allows a quicker access to the group theoretical aspects of Lie group theory than one involving analytical manifolds. In our book [102] on compact groups we devote a whole chapter to an introduction to linear Lie groups. It is shown in that book that all compact Lie groups are matrix groups, that is, linear Lie groups. All these groups possess identity neighborhoods which are homeomorphic to Rn for a suitable dimension n: they are locally euclidean. In 1900 David Hilbert raised the question whether every locally euclidean group is a Lie group. It took half a century until this question was answered in the affirmative by the concerted joint efforts of Gleason [63], Montgomery and Zippin [144] published back to back in the Annals of Mathematics. The monograph [145] by Montgomery and Zippin appeared three years later and summarized the entire development including

2

Panoramic Overview

the important complements by Yamabe [206], [207] which followed one year later and to which we shall return presently. Montgomery and Zippin’s book became a classic which has not been replaced to this day, in spite of an excellent secondary source authored by Kaplansky [129] sixteen years later. Hilbert’s Problems numbered 23 in all; they were formulated in order to indicate the directions which research in mathematics was to take in the 20th century. The problem concerning Lie groups is number 5, and its difficulty as well as the sheer quantity of research that it fertilized was very indicative of Hilbert’s vision. So it is only natural that something even more influential came along with the affirmative solution of Hilbert’s Fifth Problem, namely, Yamabe’s Theorem. Yamabe’s Theorem tells us that every connected locally compact group G is approximated by a connected Lie group in the sense that G contains arbitrarily small normal subgroups N such that G/N is a Lie group ([206], [207]). In fact Yamabe’s Theorem applies to more than connected groups: it says that every locally compact group for which the group of connected components G/G0 is compact is approximated by Lie groups in the sense just explained. The concept of being approximated by Lie groups is so important that it certainly deserves a definition of its own. For this purpose let us first recall that a topological group is complete if every Cauchy filter (or every Cauchy net) converges; this aptly generalizes the concept of completeness of a metric space which is complete if every Cauchy sequence converges. Every locally compact group is complete and so no mention of completeness need be made when one deals with locally compact groups. Definition 1. (i) A topological group G is called a pro-Lie group if it is complete and if every identity neighborhood of G contains a normal subgroup N such that G/N is a Lie group. The category of all pro-Lie groups with continuous group morphisms between them is written proLieGr. (3.39) (ii) A topological group G is called almost connected if the factor group G/G0 of G modulo the connected component G0 of the identity is compact. Let us then reformulate Yamabe’s Theorem in this terminology: Every almost connected locally compact group is a pro-Lie group. It is a generally adopted notation that for a category A and objects A and B in A, the set of all morphisms A → B is denoted by A(A, B). For instance, if TopGr denotes the category of all topological groups and continuous group homomorphisms between them, then TopGr(G, H ) denotes the set of all continuous homomorphisms from the topological group G to the topological group H ; if G and H are pro-Lie groups, then we have proLieGr(G, H ) = TopGr(G, H ) by definition. This means that the category proLieGr is a full subcategory of the category TopGr of all topological groups. Algebraists, in particular ring theorists, are rather familiar with a concept similar to that of pro-Lie groups, namely, profinite groups. A group G is profinite if it is a complete topological group such that every identity neighborhood of G contains a normal open subgroup N such that G/N is finite. Profinite groups are compact, and

Panoramic Overview

3

they are pro-Lie groups. Profinite groups generalize finite groups in the exact same way as pro-Lie groups generalize Lie groups. Only three years before the solution of Hilbert’s Fifth Problem was found by Gleason, Montgomery and Zippin, a seminal paper by Iwasawa had appeared in the Annals of Mathematics [120]. In that paper he exposed fundamental properties of locally compact pro-Lie groups. So Yamabe’s result made all of this available for the study of the structure and the representation theory of almost connected locally compact groups. This was the culmination of half a century of research on topological groups following Hilbert’s vision in 1900. But at the same time, and certainly not less significant from the present vantage point, the work by Iwasawa, Gleason, Montgomery, Zippin and Yamabe provided motivation and incentive for another half a century’s worth of research on locally compact groups during the second half of the twentieth century. What went into this entire century of research naturally was the full body of highly developed structure and representation theory of finite-dimensional Lie groups and finite-dimensional Lie algebras. Let us briefly say what we mean by the Lie theory of a topological group on a very general level; after all, the words Lie theory appear in the title of this book. To each topological group G one can easily associate a topological space L(G), namely, the space Hom(R, G) of all continuous group homomorphisms from the additive topological group R of real numbers to the topological group G, endowed with the topology of uniform convergence on compact sets. We also have a continuous function exp : L(G) → G given by exp X = X(1) and a “scalar multiplication” (r, X) → r · X : R × L(G) → L(G) given by (r · X)(s) = X(sr). Whether these concepts are useful depends in large measure on the degree to which additional properties are satisfied. In Chapter 2 we shall elaborate on the following definition. Definition 2 (2.6ff.). A topological group G is said to have a Lie algebra, if L(G) has a continuous addition and bracket multiplication making it into a topological Lie algebra in such a fashion that   r   r  n (X + Y )(r) = lim X Y n→∞ n n and

  r   r   r −1  r −1 n2 X Y . Y X n→∞ n n n n If G has a Lie algebra, then L(G) is called the Lie algebra of G and exp : L(G) → G is called its exponential function. [X, Y ](r 2 ) = lim

Clearly a topological group G has a Lie algebra if and only if the connected component G0 of the identity has a Lie algebra G0 and L(G) = Hom(R, G) = Hom(R, G0 ) = L(G0 ). The image of the exponential function is contained in G0 . If we believe that L(G) and the exponential function encapsulate the Lie theory of G, then it is true that the identity component G0 already captures the Lie theory of G.

4

Panoramic Overview

We show in this book that every pro-Lie group G has a Lie algebra L(G) and the image exp L(G) of the exponential function algebraically generates a subgroup which is dense in the connected component G0 of the identity. We shall have much more to say about the topological Lie algebra L(G) that can arise in this fashion. But for the moment we observe this: In every totally disconnected locally compact group, the open (hence closed) subgroups form a basis of the neighborhood filter of the identity element. If G is a locally compact group, then G/G0 is a locally compact totally disconnected group, and so there is an open subgroup U of G containing G0 such that U/G0 is compact. Then U is almost connected and thus, by Yamabe’s Theorem, is a pro-Lie group. Therefore every locally compact group has an open subgroup which is a pro-Lie group and which captures the Lie theory of G. Apart from individual studies such as [134], [64], [103], [104], [106], the Lie theory of locally compact groups has never been systematically considered or exploited, although a start was made in [102] for the purpose of a structure theory of compact groups. One of the thrusts of this book is to change this situation with determination. In addition to the successful resolution of Hilbert’s Fifth Problem there is yet a second prime reason for the success of the structure and representation theory of locally compact groups: The 1932 proof by A. Haar of the existence and uniqueness of left invariant integration on a locally compact group G. Its full power for abstract harmonic analysis was recognized by A. Weil in his influential monograph [198] of 1941. Haar measure is the key to the representation theory of compact and locally compact groups on Hilbert space, and the wide field of abstract harmonic analysis with ever so many ramifications (including e.g. abstract probability theory on locally compact groups). A theorem due to A. Weil shows that, conversely, a complete topological group with a left- (or right-) invariant σ -finite measure is locally compact (see e.g. [76], [198]). Thus the category of locally compact groups is that which is exactly suited for real analysis resting on the existence of an invariant integral based on σ -additive measures. One cannot expect to extend this aspect of locally compact groups to larger classes without abandoning σ -additivity. (Bourbaki indicates in Chapter 9 of his “Intégration” [23], pp. 50–55, 70ff., how such an extension may be handled; however we shall not consider this aspect in this book.) In quiet moments of introspection one might even admire the small miracle inherent in the fact that measure theory carries as far as locally compact groups go. The proper domain for an invariant measure theory again appears to be the category of compact groups, where one has a unique invariant two sided invariant measure P with respect to which G is measurable with measure P (G) = 1. That is, P is a veritable probability measure that allows averaging over G as a remarkably simple but effective device ([102]). Yet there it is, Haar measure of locally compact groups, infinite but eminently useful making locally compact groups the analysts’ delight. However, from each of a group theoretical, of a Lie theoretical, and of a category theoretical point of view, the class of a locally compact groups has serious defects which go rather deep.

Panoramic Overview

5

Indeed, if we consider a family of Lie groups Gj , j ∈ J for an index set J , then its product j ∈J Gj is a perfectly good Hausdorff topological group with a lucid structure, but it fails to be locally compact whenever infinitely many of the Gj fail to be compact. Furthermore, while every locally compact group G does have a Lie algebra L(G), the additive group of the Lie algebra is never locally compact unless it is finitedimensional. Indeed even the additive topological group of the Lie algebra of a compact def

abelian group need not be locally compact; for example the product G = TJ of circle groups T = R/Z has a Lie algebra L(G) isomorphic to RJ and thus fails to be locally compact as soon as J is infinite, while the group TJ is comfortably within the realm of compact groups. Each Lie group G has a tangent bundle which is again a Lie group, namely, the semidirect product L(G) Ad G with G acting on its Lie algebra by adjoint action induced by inner automorphisms. Does a locally compact group have a tangent bundle? The answer is yes, it does, in fact every pro-Lie group has one (as we shall show in this book), but it is almost never a locally compact group except when the group itself is finite-dimensional. Thus the category of locally compact groups appears to have two major drawbacks: – The topological abelian group underlying the Lie algebra L(G) and the tangent bundle of a locally compact group fail to be locally compact unless L(G) is finitedimensional. In other words, the very Lie theory that makes the structure theory of locally compact groups interesting leads us outside the class. – The category of locally compact groups is not closed under the forming of products, even of copies of R; it is not closed under projective limits of projective systems of finite-dimensional Lie groups, let alone under arbitrary limits. In other words, the category of locally compact groups is badly incomplete. This book presents an argument for a shift in the vantage point of looking at locally compact groups. We plead for a structure theory of topological groups that places the focus squarely and systematically on pro-Lie groups. Recall that we denote the category of all (Hausdorff) topological groups and continuous group homomorphisms by TopGr. It turns out that the full subcategory proLieGr of TopGr consisting of all projective limits of finite-dimensional Lie groups avoids both of these difficulties. This would perhaps not yet be a sufficient reason for advocating this category if it were not for two facts: – Firstly, while not every locally compact group is a projective limit of Lie groups, every locally compact group has an open subgroup which is a projective limit of Lie groups, so that, in particular, every connected locally compact group is a pro-Lie group; also all compact groups and all locally compact abelian groups are pro-Lie groups. – Secondly, the category proLieGr is astonishingly well-behaved. Not only is it a complete category, it is closed under passing to closed subgroups and to those quotients which are complete, and it has a demonstrably good Lie theory.

6

Panoramic Overview

It is therefore indeed surprising that this class of groups has been little investigated in a systematic fashion. A serious attempt at such an investigation is made in this book where it is submitted that not only a general structure theory of locally compact groups can be based on a good understanding of the category proLieGr of pro-Lie groups, but that the category of pro-Lie groups is well worth a thorough study on its own account. In this book we will prove general structure theorems on pro-Lie groups which will include the best known general structure theorems on locally compact groups. Since the main strategy of the book is to provide a structure theory via Lie theory, en route we shall have to develop a full grown structure theory of those topological Lie algebras which occur as Lie algebras of pro-Lie groups. We shall call these pro-Lie algebras, because each of them is a complete topological Lie algebra such that every 0-neighborhood contains a closed ideal modulo which it is finite-dimensional (3.6).

Part 1. The Base Theory of Pro-Lie Groups For a description of some basic results on the theory of projective limits of Lie groups some technical background information appears inevitable even for an overview, long before we delve into the actual study of our topic.

Core Definitions and Facts on Pro-Lie Groups Definition 3. A projective system D of topological groups is a family of topological groups (Cj )j ∈J indexed by a directed set J and a family of morphisms {fj k : Ck → Cj | (j, k) ∈ J × J, j ≤ k}, such that fjj is always the identity morphism and i ≤ j ≤ k in J implies fik= fij  fj k . Then the projective limit of the system limj ∈J Cj is the subgroup of j ∈J Cj consisting of all J -tuples (xj )j ∈J for which the equation xj = fj k (xk ) holds for all j, k ∈ J such that j ≤ k. Every cartesian product of topological groups may be considered as a projective limit. Indeed, if (Gα )α∈A is an arbitrary family of topological groups indexed by an infinite set A, one obtains a projective system by considering J to be the set of finite  subsets of A directed by inclusion, by setting Cj = a∈j Ga for j ∈ J , and by letting fj k : Ck → Cj for j ≤ k in J be the projection onto the partial product. The projective  limit of this system is isomorphic to a∈A Ga . There are a few sample facts one should recall about the basic properties of projective limits (see e.g. [25], [64], [107], or this book 1.27 and 1.33): Let G = limj ∈J Gj be a projective limit of a projective system P = {fj k : Gk → Gj | (j, k) ∈ J × J, j ≤ k} of topological groups with limit morphisms fj : G → Gj , and let Uj denote the filter of identity neighborhoods of Gj , U the filter of identity neighborhoods of G,

Panoramic Overview

7

and N the set {ker fj | j ∈ J }. Then U has a basis of identity neighborhoods {fk−1 (U ) | k ∈ J, U ∈ Uk } and N is a filter basis of closed normal subgroups converging to 1. If all bonding maps fj k : Gk → Gj are quotient morphisms and all limit maps fj are surjective, then the limit maps fj : G → Gj are quotient morphisms. The limit G is complete if all Gj are complete. Definition 4 (3.25). For a topological group G let N (G) denote the set of closed normal subgroups N such that all quotient groups G/N are finite-dimensional real Lie groups. Then G ∈ N (G) and G is said to be a proto-Lie group if every identity neighborhood contains a member of N (G). By our earlier Definition 1, if in addition, G is a complete topological group, then G is a pro-Lie group. While not every topological group can be embedded as a subgroup into a complete topological group, this is the case for proto-Lie groups, indeed every proto-Lie group has a completion which is a pro-Lie group. (See 4.1.)  Every product of a family of finite-dimensional Lie groups j ∈J Gj is a pro-Lie group. In particular, RJ is a pro-Lie group for any set J which is locally compact if and only if the set J is finite. The product ZN , accordingly, is a pro-Lie group. It is well known that the space ZN is homeomorphic to the space of irrational real numbers in the natural topology. We may formulate this by saying that the space of irrational numbers supports the structure of a pro-Lie group. It is a remarkable fact (which we discuss in Chapter 4) that the free abelian group Z(N) in countably many generators carries the structure of a nondiscrete pro-Lie group. The underlying topological space cannot be a Baire space and so certainly cannot be Polish (second countable completely metrizable), nor locally compact; indeed a countable homogeneous Baire space is necessarily discrete. These examples help us to realize from the beginning, that our general intuition of the topology of pro-Lie groups cannot be based on experience gathered from locally compact groups. If {Gj : j ∈ J } is a family of finite-dimensional real Lie groups then the subgroup    (gj )j ∈J ∈ Gj : {j ∈ J : gj = 1} is finite j ∈J

 of the direct product j ∈J Gj is a proto-Lie group which is not a pro-Lie group if J is infinite and the Gj are nonsingleton. We reiterate that a topological group G is called almost connected if the factor group G/G0 modulo the connected component G0 of the identity is compact. Everything that is proved for almost connected topological groups therefore is true for all connected groups and for all compact groups. One of the very weighty reasons why this concept is relevant for the theory of topological groups is the existence of Yamabe’s crucial result:

8

Panoramic Overview

Every almost connected locally compact group is a pro-Lie group. The group PSl(2, Qp ) of projective transformations of the p-adic projective line is locally compact, but has no nontrivial normal subgroups and is therefore a locally compact group which is not a pro-Lie group in our sense, while it is, of course, a p-adic Lie group. Every pro-Lie group G gives rise to a projective system {pN M : G/M → G/N : M ⊇ N in N (G)} whose projective limit it is (up to isomorphism). The converse is a difficult issue, but it is true. Theorem 5 (3.34, 3.35 (The Closed Subgroup Theorem)). Every projective limit of pro-Lie groups is a pro-Lie group. Every closed subgroup of a pro-Lie group is a pro-Lie group. A topological group is a pro-Lie group if and only if it is isomorphic to a closed subgroup of a product of Lie groups. In fact in simple category theoretical parlance the following theorem holds. Theorem 6 (3.3, 3.36). The category proLieGr of pro-Lie groups is closed in TopGr under the formation of all limits and is therefore complete. It is the smallest full subcategory of TopGr that contains all finite-dimensional Lie groups and is closed under the formation of all limits. This shows that the category proLieGr does not have some of the shortcomings of the category of locally compact groups which is obviously incomplete. It remains yet to be seen how good the Lie theory of the category proLieGr is and we shall say good things about it shortly. However, one must, at an early stage, admit that the category of pro-Lie groups has certain problems which are invisible as long as one stays inside the subcategory of locally compact pro-Lie groups. Indeed, every quotient group of a locally compact group is locally compact (which is a consequence of the fairly elementary observation that in any topological group, the product H K of a closed subset H and a compact subset K is closed, and the application of this fact to the case that H is a closed (normal) subgroup and K a compact identity neighborhood of G). It is one of the less elementary facts of Lie group theory that a quotient of a Lie group is a Lie group. Indeed the quotient of a linear Lie group need not be a linear Lie group, but is a Lie group nevertheless. The simplest example is the group of upper triangular matrices ⎫ ⎧⎛ ⎞ ⎬ ⎨ 1 x z def ⎝ 0 1 y ⎠ : x, y, z ∈ R G = ⎭ ⎩ 0 0 1 and the discrete central subgroup ⎫ ⎧⎛ ⎞ ⎬ ⎨ 1 0 n def Z = ⎝0 1 0 ⎠ : n ∈ Z ; ⎭ ⎩ 0 0 1

Panoramic Overview

9

here G is clearly a linear Lie group but the factor group G/Z, which is even locally isomorphic to G is not a linear Lie group. This was proved by Garret Birkhoff in 1936 [9] by astute but elementary linear algebra. (See also [102], p. 169ff.) It is a much more debilitating fact for the study of pro-Lie groups that quotient groups of pro-Lie groups need not be pro-Lie groups. (Corollary 4.11) Still, every quotient group of a pro-Lie group is a proto-Lie group and has a completion which is a pro-Lie group. (4.1) So the defect here arises from a phenomenon that is well observed and studied, that quotients of complete topological groups may fail to be complete. (See [176].) We shall explain in Chapter 4 that the additive group of the topological vector space R[0,1] has a nondiscrete closed subgroup K algebraically isomorphic to the free abelian group Z(N) in countably many generators such that R[0,1] /K is an abelian proto-Lie group which is dense in a compact connected and locally connected group (Corollary 4.11). We use this example in various places in the book to construct counterexamples. In this sense, this example is very helpful to build up our intuition on certain aspects of pro-Lie group theory that are invisible as long as we stay in the locally compact domain. Curiously, the counterexample itself arises from the theory of compact abelian groups, and it was discovered not so long ago ([106]). The defect of proLieGr of not being closed under the passing to quotients is, as we have said, debilitating, because passing to quotient groups is an extremely helpful device of reduction to simple situations in many proofs; therefore it is a handicap not having this tool available at all times inside proLieGr. Fortunately, we shall see that, even regarding quotients, the category proLieGr has its redeeming features. Theorem 7 (The Quotient Theorem; 4.28). Let G be an almost connected pro-Lie group and N a closed normal subgroup. Then G/N is a pro-Lie group provided at least one of the following conditions are satisfied by N : (i) N is almost connected. (ii) N is the kernel of a morphism from G onto some pro-Lie group. (iii) N is locally compact or Polish. Part (iii) of this theorem arises from general topological group theory, and we refer to sources like the book [176] of Dierolf and Roelke for such pieces of information. Parts (i) and (ii) belong to the proper substance of this book, and neither of the two is a trivial matter (See Theorem 4.28 and Corollary 9.58.) In fact, Part (ii) is a consequence of another core result concerning pro-Lie groups, namely, the Open Mapping Theorem that is well known to functional analysts as applying to a variety of operators between suitable topological vector spaces, and that is equally well known to people working with locally compact or Polish topological groups. If conditions are right, then the surjectivity of a continuous group homomorphism f : G → H from a topological

10

Panoramic Overview

group onto another implies already that f is an open function, or, in equivalent terms that the canonical decomposition G ⏐ ⏐ quot G/ ker f

f

−−−−−→ H  ⏐id ⏐ H −−−−−→ H F

produces an isomorphism of topological groups F : G/ ker f → H . If we let G be the additive group of real numbers Rd with the discrete topology, H the same group R but considered with its natural topology, then the identity function f : G → H is a bijective morphism between locally compact metric groups that is not open. We mentioned earlier that we shall expose a nondiscrete pro-Lie structure on the countable free abelian group H with infinitely many generators. So the identity morphism f from the discrete countable (hence locally compact Polish) group G = Z(N) to H is a continuous morphism between pro-Lie groups which is not open. These examples show that the following theorem is not likely to be either obvious or trivial: Theorem 8 (Open Mapping Theorem for Pro-Lie Groups; 9.60). Let G be an almost connected pro-Lie group and f : G → H a continuous group homomorphism onto a pro-Lie group. Then f is an open mapping. That is, under these circumstances, f is equivalent to a quotient homomorphism. One of the major impediments in the group theory of topological groups is the unavailability of the Second Isomorphism Theorem. The so called First Isomorphism Theorem says that if G is a topological group and M ⊆ N are normal subgroups of G then the morphism gM  → gN : G/M → G/N factors through an isomorphism of topological groups (G/M)/(N/M) → G/N. This is a very robust theorem belonging to universal algebra. The environment of the so-called Second Isomorphism Theorem is as follows: Assume that G is a topological group with a closed normal subgroup N and a closed subgroup H such that G = H N = N H . Then the surjective morphism h  → hN : H → G/N factors through a bijective continuous group homomorphism H /(H ∩ N) → H N/N . This may fail to be open even if H ∩ N = {1}. In [108] there is an example of a topological abelian group G and two (isomorphic) closed subgroups H and N such that G is algebraically the direct sum of H and N and G/H and G/N are (isomorphic) compact groups, while G blatantly fails to be compact. However, if G is a pro-Lie group, then a closed subgroup H is a pro-Lie group by the Closed Subgroup Theorem. If N is an almost connected closed normal subgroup of G and G is almost connected, then G/N is a pro-Lie group by the Quotient Theorem. Therefore, from the Open Mapping Theorem we get the next theorem. Theorem 9 (The Second Isomorphism Theorem for Pro-Lie Groups; 9.62). Let N be an almost connected normal subgroup and H an almost connected subgroup of a topological group G and assume that H , N , and H N are pro-Lie groups. Then N/(H ∩ N) and H N/N are naturally isomorphic.

Panoramic Overview

11

The Coarse Lie Theory of Pro-Lie Groups Let us consider a topological Lie algebra g and on it the filter basis of closed ideals j such that dim g/j < ∞; we shall denote it by  (g). Definition 10 (3.6). A topological Lie algebra g is called a pro-Lie algebra (short for profinite-dimensional Lie algebra) if  (g) converges to 0 and if g is a complete topological vector space. Under these circumstances, g ∼ = limj∈ (g) g/j, and the underlying vector space is a weakly complete topological vector space, that is, it is the algebraic dual of a real vector space with the weak ∗-topology. We give a systematic treatment of the duality of vector spaces and weakly complete topological vector spaces in Appendix 2 of this book. The category of pro-Lie algebras and continuous Lie algebra morphisms is denoted proLieAlg. Proposition 11 (3.3, 3.36). The category proLieAlg of pro-Lie algebras is closed in the category of topological Lie algebras and continuous Lie algebra morphisms under the formation of all limits and is therefore complete. It is the smallest category that contains all finite-dimensional Lie algebras and is closed under the formation of all limits. See also [104]. Our demonstration that Lie theory is applicable to pro-Lie groups begins with our showing results like the following: Theorem 12 (3.12, 2.25). Every pro-Lie group G has a pro-Lie algebra g as Liealgebra, and the assignment L which associates with a pro-Lie group G its pro-Lie algebra is a limit preserving functor. These matters will be shown in Chapters 2 and 3. In fact, a portion of this set-up allows a considerable improvement which we summarize in the next section.

The Category Theoretical Version of Lie’s Third Theorem Theorem 13 (Lie’s Third Theorem for Pro-Lie groups; 6.5, 6.6, 8.15). The Lie algebra functor L : proLieGr → proLieAlg has a left adjoint . It associates with every pro-Lie algebra g a unique simply connected pro-Lie group (g) and a natural isomorphism ηg : g → L((g)) such that for every morphism ϕ : g → L(G) there is a unique morphism ϕ  : (g) → G such that ϕ = L(ϕ  )  ηg . A good portion of this theorem we shall prove in Chapter 6, but we find it necessary to introduce a preliminary concept of simple connectivity. Indeed we shall call a proLie group prosimply connected if every member of N (G) contains a member N of N (G) such that G/N is a simply connected Lie group. This turns out to be, for a while, a very useful concept of simple connectivity for pro-Lie groups in all respects, and it

12

Panoramic Overview

reduces correctly to simple connectivity in the case of finite-dimensional Lie groups. Once we have developed enough structure theory we will be able in Chapter 8 to show that a pro-Lie group is prosimply connected if and only if it is simply connected. (See Theorem 8.15.) Indeed, for each pro-Lie algebra g, the group (g) is a projective limit of a projective system of simply connected Lie groups. The fact that L is a right adjoint confirms its property of preserving all limits. There is more to Theorem 13 than meets the eye, and we should alert the reader to these circumstances because they shed new light on the situation even when everything is restricted to the classical situation of finite-dimensional Lie groups. The adjointness of the two functors L and  may be expressed in terms of universal properties as follows. There is a natural isomorphism ηg : g → L((g)) such that for any morphism f : g → L(H ) of topological Lie algebras there is a unique morphism f  : (g) → H such that f = L(f  )  ηg . In diagrams: proLieAlg

proLieGr

ηg

g ⏐ ⏐ ∀f 

−−−−−→ L((g)) ⏐ ⏐  L(f )

L(H ) −−−−−→ idL(H )

(g) ⏐ ⏐  ∃!f

L(H )

H.

In fact, the natural isomorphism really allows us to identify g with the Lie algebra of (g). Sophus Lie’s Third Fundamental Theorem (in his own terminology) says that for every finite-dimensional Lie algebra there is a Lie group having as Lie algebra the given one. So this theorem persists for pro-Lie groups. The natural morphism η is what category theoreticians call the front adjunction or the unit of the adjunction. But any adjoint situation between two functors also has a back adjunction or counit with an appropriate version of the universal property. In the case of the present adjoint situation between L and , the back adjunction set-up is as follows. There is a natural morphism πG : (L(G)) → G of pro-Lie groups with the following universal property: Given a pro-Lie group G and any morphism f : (h) → G for some pro-Lie algebra h, there is a unique morphism f  : h → L(G) of pro-Lie algebras such that f = πG  (f  ). proLieAlg

L(G)  ⏐ ∃!f  ⏐ h

proLieGr πG

(L(G)) −−−−−→  ⏐  ⏐(f ) (h)

G  ⏐∀f ⏐

−−−−−→ (h) id(h)

Panoramic Overview

13

 and call the morphism πG : G  → G the We shall abbreviate (L(G)) by G, universal morphism of G. If G happens to be a pro-Lie group which has a universal covering group in the topological sense (in particular, if G is a finite dimensional Lie  → G is the universal covering morphism (8.21). In general the group), then πG : G universal morphism is neither surjective nor a local isomorphism. This is best realized at an early stage by considering any connected compact abelian group G together with  R), its exponential function expG : L(G) → G, L(G) = Hom(R, G) ∼ = Hom(G,  where G = Hom(G, T), T = R/Z is the discrete character group of G. These things  may be equated are explained in great detail in [102], Chapters 7 and 8. In this case G with the additive group of L(G) and πG with expG : L(G) → G. The world of compact abelian groups of course is full of examples for which the exponential function fails to be surjective, beginning with the one-dimensional examples that are different from the circle group, that is, the solenoids, the character groups of which are the noncyclic infinite subgroups of Q. Let us consider within the complete category proLieGr the full subcategory proSimpConLieGr of all simply connected pro-Lie groups. Then we have the following corollary. Corollary 14 (6.6(vi)). The restrictions and corestrictions of the functors L and  implement an equivalence of categories proLieAlg

L

−−−−−→ ←−−−−−

proSimpConLieGr.



Therefore, the category of pro-Lie algebras has a faithful copy inside the category of all pro-Lie groups, namely, the full subcategory of all simply connected pro-Lie groups.  → G is a group theoretical substitute for In this light, the universal morphism πG : G the exponential function expG : g → G; indeed for abelian pro-Lie groups the two functions agree for all practical intents and purposes. These matters are discussed in Chapter 6 but thereafter will pervade the whole book. Considering the problems we have encountered with quotients in the category of pro-Lie groups, it is remarkable that the functor L behaves well with regard to quotient morphisms. Indeed we see next that L not only preserves all limits, but some colimits as well.

Conservation Laws for L and  Theorem 15 (4.20). The functor L preserves quotients. Specifically, assume that G is a pro-Lie group and N a closed normal subgroup and denote by q : G → G/N the quotient morphism. Then G/N is a proto-Lie group whose Lie algebra L(G/N ) is a pro-Lie algebra and the induced morphism of pro-Lie algebras L(q) : L(G) → L(G/N) is a quotient morphism. The exact sequence 0 → L(N ) → L(G) → L(G/N ) → 0

14

Panoramic Overview

induces an isomorphism X + L(N ) → L(f )(X) : L(G)/L(N ) → L(G/N ). The core of Theorem 15 is proved by showing that for every quotient morphism f : G → H of topological groups, where G is a pro-Lie group, every one parameter subgroup Y : R → H lifts to one of G, that is, there is a one parameter subgroup σ of G such that Y = f  σ . (See 4.19, 4.20.) This requires the Axiom of Choice. It should be emphasized that, according to Theorem 15, a quotient group of a pro-Lie group always has a complete Lie algebra even if it is itself incomplete. Therefore, a proto-Lie group with an incomplete Lie algebra such as R(N) cannot be a quotient of a pro-Lie group. Corollary 16 (4.21). Let G be a pro-Lie group. Then {L(N ) | N ∈ N (G)} converges to zero and every closed ideal i of L(G) such that L(G)/i is finite-dimensional contains an L(N) for some N ∈ N (G). Furthermore, L(G) is the projective limit limN ∈N (G) L(G)/L(N ) of a projective system of bonding morphisms and limit maps all of which are quotient morphisms, and there is a commutative diagram L(γG ) G ∼ L(G) −−−−−→ L(limN ∈N ⏐ ⏐ (G) N ) = limn∈N (G) ⏐ ⏐ expG  L(limN∈N (G) expG/N ) G −−−−−→ limN ∈N (G) G/N.

L(G) L(N )

γG

Theorem 15 expresses a version of exactness of L. But there is also an exactness theorem for , the left adjoint of L. Theorem 17 (6.7, 6.8, 6.9). If h is a closed ideal of a pro-Lie algebra g, then the exact sequence q i 0 → h −−−→ g −−−→ g/h → 0 induces an exact sequence (j )

(q)

1 → (h) −−−→ (g) −−−→ (g/h) → 1, in which (j ) is an algebraic and topological embedding and (q) is a quotient morphism. There are some other noteworthy consequences of Theorem 15. Proposition 18 (4.22 (iv)). Any quotient morphism f : G → H of pro-Lie groups onto a finite-dimensional Lie group is a locally trivial fibration. Proposition 19 (4.22 (i)). For a pro-Lie group G, the subgroup exp g generated by the image of the exponential function, is dense in G0 , that is, exp g = G0 . In particular, a connected nonsingleton pro-Lie group has nontrivial one parameter subgroups. This may be viewed as an Existence Theorem for one parameter subgroups in proLie groups, indeed of an abundance of them – unless of course, the group in question is totally disconnected. So, as an illuminating consequence we get the following characterisation of a pro-Lie group to be totally disconnected.

Panoramic Overview

15

Corollary 20 (4.23). For a pro-Lie group G the following statements are equivalent: (a) G is totally disconnected. (b) L(G) = {0}. (c) The filter basis of open normal subgroups of G converges to 1. In this book we shall call topological groups satisfying these equivalent conditions prodiscrete groups. So every prodiscrete group is, in particular, a pro-Lie group. As we already mentioned and will observe again later, there are locally compact totally disconnected groups which are not prodiscrete. The group ZN in the product topology is prodiscrete but not locally compact. It is, as we remarked earlier, homeomorphic to the space of irrational numbers. Semidirect products of two topological groups (and semidirect sums of topological Lie algebras) permeate the whole book, beginning from Chapter 1 where we remind the reader of its definition in Exercise E1.5 through Chapter 11, that is specifically devoted to the splitting of pro-Lie groups, that is to results that assert that, under suitable circumstances, a given pro-Lie group may be represented as a semidirect product. If π : H → Aut(N ) is a representation of a topological group in the group def

of automorphisms of a topological group N such that the function (h, n)  → h · n = π(n)(h) : H × N → N is continuous, then the semidirect product N π H of N by H is the topological product N × H endowed with the multiplication (m, h)(n, k) = (m(h · n), hk). That N π H is a topological group is straightforwardly verified. Very simple examples show that semidirect products of pro-Lie groups need not be pro-Lie groups (see Examples 4.29). We shall demonstrate in this book that every proLie group acts under what will be called the adjoint action or adjoint representation Ad : G → Gl(L(G)) on L(G) (see 2.27ff.). So we can form the semidirect product L(G) Ad G,

(X, g)(Y, h) = (X + Ad(g)Y, gh),

and obtain this result. def

Proposition 21 (4.29 (iii)). For each pro-Lie group G, the semidirect product T (G) = L(G) Ad G is a well-defined pro-Lie group.

We call T (G) the tangent bundle of G. Thus pro-Lie groups have tangent bundles that are pro-Lie groups. In particular, all (almost) connected locally compact groups have tangent bundles within the category of pro-Lie groups. However, for a locally compact group G its tangent bundle T (G) is locally compact only if G is finite-dimensional. We have seen that the category of pro-Lie groups – contains all finite-dimensional real Lie groups, – is closed in the category of topological groups under the formation of all limits and the passing to closed subgroups, – has a substantial Lie algebra functor that possesses a very reasonable left adjoint,

16

Panoramic Overview

– is closed under the passing from a group to the additive group of its Lie algebra and under the passing from a group to its tangent bundle. In other words, we have seen that the category of pro-Lie groups has none of the defects which plague the category of locally compact groups while it still contains all almost connected locally compact groups. That is, it still houses comfortably all those locally compact groups that support all the Lie theory there is for locally compact groups. But can we exhibit, one might ask, enough fine structure theory of pro-Lie algebras and pro-Lie groups so that at least the known structure theory of locally compact groups is recovered? Like with all categories of groups, the first test that a group theory has to face is how well it elucidates the structure of its abelian representatives.

Abelian Pro-Lie groups Apart from a territory far removed from the domain of connected or even almost connected commutative pro-Lie groups, the situation is very satisfactory and is, as a first coarse approximation to the general structure theory of almost connected proLie groups, rather representative and a good guide for one’s intuition. A weakly complete vector space is a real topological vector space V for which there is a real vector space E such that V ∼ = E ∗ , where E ∗ is the algebraic dual E ∗ HomR (E, R) ⊆ R endowed with the weak -topology, that is, the topology of pointwise convergence induced from RE given the product topology. (See Appendix 2, notably Theorem A2.8.) If the cardinal dim E is the linear dimension of E, that is, the cardinality of one, hence every basis of E, then E ∼ = R(dim E) and thus V ∼ = Rdim E . Therefore, an equivalent definition of a weakly complete topological vector space is the postulate that there be a set J such that V ∼ = RJ (see Corollary A2.9). If N S(V ) denotes the filter basis of all closed vector subspaces F of a locally convex Hausdorff topological vector space V such that dim V /F < ∞, then V is a weakly complete vector space if and only if the natural morphism λV : V → limF ∈N S(V ) V /F , λV (v) = (v + F )F ∈N S(V ) is an isomorphism of topological vector spaces. If an abelian topological group is isomorphic to the additive group of a weakly complete topological vector space, that is, to RJ for some set J , then we shall call it a weakly complete vector group. The abelian pro-Lie groups we know best are the compact abelian groups and the weakly complete vector groups. So it is very pleasing that we can state the following result. Lemma 22 (Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups; 5.12). Any abelian almost connected pro-Lie group is isomorphic to the direct product of a weakly complete vector group and a compact abelian group.

Panoramic Overview

17

This result is succinct and very lucid. It illustrates that abelian pro-Lie groups, at least if they are almost connected are built up from weakly complete vector groups and compact abelian groups in a certainly simple fashion. In reality, we have better and more accurate information. For the more accurate information we have to pay a price: the formulations get more complicated. First we have a clean cut intermediate result showing that weakly complete topological vector spaces and tori are injectives in the category of abelian pro-Lie groups. Theorem 23 (5.19). Assume that G is an abelian pro-Lie group with a closed subgroup G1 and assume that there are sets I and J such that G1 ∼ = RI × TJ . Then G1 is a homomorphic retract of G, that is, G1 is a direct summand algebraically and topologically. So G ∼ = G1 × G/G1 . This allows us to argue that every abelian pro-Lie G group has a weakly complete vector subgroup V such that G is isomorphic to the direct product V × (G/V ) where the factor G/V has no nontrivial vector subgroup. We call any such subgroup V a vector group complement. For a topological group G we let comp(G) denote the set of all elements which are contained in a compact subgroup. Theorem 24 (Vector Group Splitting Theorem for Abelian Pro-Lie Groups; 5.20). Let G be an abelian pro-Lie group and V a vector group complement. Then there is a closed subgroup H such that (i) (v, h) → v + h : V × H → G is an isomorphism of topological groups. (ii) H0 is compact and equals comp G0 and comp(H ) = comp(G); in particular, comp(G) ⊆ H . (iii) H /H0 ∼ = G/G0 , and this group is prodiscrete. (iv) G/ comp(G) ∼ = V × S for some prodiscrete abelian group S without nontrivial compact subgroups. (v) G has a characteristic closed subgroup G1 = G0 + comp(G) which is isomorphic to V × comp(H ) such that G/G1 is prodiscrete without nontrivial compact subgroups. (vi) The exponential function expG of G = V ⊕ H decomposes as expG = expV ⊕ expH where expV : L(V ) → V is an isomorphism of weakly complete vector groups and expH = expcomp(G0 ) : L((comp(G0 )) → comp(G0 ) is the exponential function of the unique largest compact connected subgroup; here L(comp(G0 )) = comp(L)(G) is the set of relatively compact one parameter subgroups of G. (vii) The arc component Ga of G is V ⊕Ha = V ⊕comp(G0 )a = im L(G). Moreover, if h is a closed vector subspace of L(G) such that exp h = Ga , then h = L(G). This theorem actually is the basis of a rather explicit structure theory of abelian pro-Lie groups. We recall that all locally compact abelian groups belong to this class. There are still some portions of an abelian pro-Lie group G which we do not fully control:

18

Panoramic Overview

– The factor group G/G1 ∼ = H / comp G is prodiscrete and has no compact subgroups but is otherwise uncharted. – comp(G) = comp(H ) is a pro-Lie group that is a directed union of compact subgroups. We do not know much more in the absence of local compactness. If we impose certain natural additional hypothesis that are traditionally invoked in topological group theory, the situation is at once much better. A topological group is called compactly generated if it is algebraically generated by a compact subset. Theorem 25 (The Compact Generation Theorem for Abelian Pro-Lie Groups; 5.32). (i) For a compactly generated abelian pro-Lie group G the characteristic closed subgroup comp(G) is compact and the characteristic closed subgroup G1 is locally compact. (ii) In particular, every vector group complement V is isomorphic to a euclidean group Rm for some m ∈ N0 = {0, 1, 2, . . . }. (iii) The factor group G/G1 is a compactly generated prodiscrete group without compact subgroups. If G/G1 is Polish, then G is locally compact and G∼ = Rm × comp(G) × Zn . (iv) If G is a pro-Lie group containing a finitely generated abelian dense subgroup, then comp(G) is compact and G ∼ = comp(G)× Zm . In particular, G is locally compact. (v) A finitely generated abelian pro-Lie group is discrete. The full subcategory of locally compact abelian groups in the category of abelian pro-Lie groups has a celebrated structure theory that is primarily due to the highly effective and elegant duality theory going back to L. S. Pontryagin [169] and E. R. van Kampen [125] in the early thirties of the 20th century. For any topological abelian group  = Hom(G, T) denote its dual with the compact open topology. (See e.g. G we let G   given [102, Chapter 7].) There is a natural morphism of abelian groups ηG : G → G by ηG (g)(χ) = χ (g) which may or may not be continuous; information regarding this issue is to be found for instance in [102, pp. 298ff], notably in Theorem 7.7 on p. 300.   is bijective and We shall call a topological abelian group semireflexive if ηG : G → G reflexive if ηG is an isomorphism of topological groups; in the latter case G is also said to have duality (see [102, p. 305]). In the direction of a duality theory of abelian pro-Lie groups we offer the following results. Proposition 26 (5.35). Let G be an abelian pro-Lie group and let V be a vector group complement. Then G is reflexive, respectively, semireflexive iff G/V is reflexive,  is isomorphic to a product E × A respectively, semireflexive. The character group G where E is the additive group of a real vector space with its finest locally convex topology and A is the character group of an abelian pro-Lie group whose identity component is compact. Theorem 27 (5.36). Every almost connected abelian pro-Lie group is reflexive, and its character group is a direct sum of the additive topological group of a real vector space

Panoramic Overview

19

endowed with the finest locally convex topology and a discrete abelian group. Pontryagin duality establishes a contravariant functorial bijection between the categories of almost connected abelian pro-Lie groups and the full subcategory of the category of topological abelian groups containing all direct sums of vector groups with the finest locally convex topology and discrete abelian groups.

Part 2. The Algebra of Pro-Lie Algebras The success of the Lie theory of classical Lie groups as well as in our case the Lie theory of pro-Lie groups depends on the effectiveness of the mechanism that allows us to translate problems of the topological group structure on the group level to algebraic problems on the Lie algebra level and back. Experience demonstrates that problems are more easily attacked in a purely algebraic environment. In the present case we know, however, that the Lie algebra of a pro-Lie group is a topological algebra itself. So we hope to repeat the classical success story only to the extent to which the topological algebra and the representation theory of pro-Lie groups themselves reduce to pure algebra – more or less. We shall see that this is largely the case for pro-Lie algebras due to the fact that the underlying topological vector spaces are weakly complete vector spaces and that these have a perfect duality theory that allows us to translate their topological linear algebra to pure linear algebra upon passing to the vector space duals. (See Appendix 2.)

The Module Theory of Pro-Lie Algebras We saw that for every pro-Lie group G there exists a simply connected pro-Lie group  → G. Thus the structure of  and a natural morphism with dense image πG : G G simply connected pro-Lie groups has no small influence on the structure of pro-Lie groups in general. We further saw that the structure of simply connected pro-Lie groups, in a well-understood sense, is completely determined by the structure of their Lie algebra. The lesson learned from Lie Theory of finite-dimensional Lie groups is that one must first study the structure of Lie algebras carefully and then apply the information gathered in this fashion to the group theory of Lie groups. It is no different with pro-Lie groups even though the connection between pro-Lie algebras and pro-Lie groups is more tenuous than in the finite-dimensional case. We develop the representation theory and structure theory of pro-Lie groups simultaneously. Elementary module theory is usually preceded by a rush of simple definitions which still turn out to be very effective. We record some to the extent they are necessary for the reader to follow this overview. Let L be a Lie algebra and E a vector space. Then E is an L-module if there is a bilinear map (x, v)  → x · v : L × E → E

satisfying

[x, y] · v = x · (y · v) − y · (x · v)

20

Panoramic Overview

for all x, y ∈ L and v ∈ E. A function f : E1 → E2 between L-modules is said to be a morphism of L-modules if it is linear and satisfies (∀x ∈ L, v ∈ E1 )

f (x · v) = x · f (v).

A submodule F of an L-module E is a vector subspace such that L · F ⊆ F . An L-module E is said to be simple if {0} and E = {0} are its only submodules. An L-module E is called semisimple if every submodule is a direct module summand. If L is a topological Lie algebra, then a topological vector space V is said to be a topological L-module if (x, v)  → x · v : L × V → V is continuous in each variable separately. If the topological vector space V is weakly complete, and if the filter basis of closed submodules W such that dim V /W < ∞ converges to 0, then V is said to be a profinite-dimensional L-module. The profinite-dimensional modules have a perfect duality; indeed if E is the topological dual of a profinite-dimensional L- module, then E is an L-module with respect to the module operation defined by x · ω, v = −ω, x · v for x ∈ L, v ∈ V and ω ∈ E. Duality permits us to transfer concepts from algebraic module theory to topological module theory. For instance, let V be a profinite-dimensional topological vector space and an L-module. Then the module is said to be reductive if its dual module is semisimple. Duality then permits us to prove theorems like the following: Theorem 28 (7.18). (a) Let V be a profinite-dimensional L-module for a Lie algebra L. Then the following statements are equivalent: (i) (ii) (iii) (iv)

V is reductive. Every finite-dimensional quotient module of V is reductive. V is the projective limit of finite-dimensional reductive module quotients. V is isomorphic to a product of finite-dimensional simple modules.

(b) Every profinite-dimensional L-module has a unique smallest submodule V ss such that V /V ss is reductive. The theory and duality of L-modules are discussed in great detail, among many other things, in Chapter 7. Now these module theoretical concepts apply to the structure theory of pro-Lie algebras. The key is the following remark. If g is a pro-Lie algebra, then the underlying weakly complete topological vector space |g| is a topological L-module with respect to the module operation defined by x · v = [x, v] for x ∈ g and v ∈ |g|. This module is called the adjoint module gad . A pro-Lie algebra g is called reductive if its adjoint module gad is a reductive g-module. It is called semisimple if it is reductive and its center z(g) is zero.

Panoramic Overview

21

While the duality theory of profinite-dimensional L-modules works perfectly, the theory of pro-Lie algebras has no duality theory in the sense that a pro-Lie algebra g could have a Lie algebra as a dual object. However, its adjoint g-module gad has a dual g-module, also called its coadjoint module gcoad . This module duality, however, attaches to each pro-Lie algebra g an almost purely algebraic object, the coadjoint module gcoad , and that is extremely helpful for the structure theory of pro-Lie algebras as the following results will show. Theorem 29 (The Structure Theorem of Reductive and Semisimple Pro-Lie Algebras). (a) For a pro-Lie algebra g the following conditions are equivalent. (i) g is reductive. (ii) g is the product of a family of finite-dimensional simple or one-dimensional ideals of g. (b) Let g be a reductive pro-Lie algebra. Then the commutator algebra [g, g] is closed and is a product of finite simple real Lie algebras. Further g ∼ = z(g) ⊕ [g, g] I ∼ algebraically and topologically, and z(g) = R for some set I . (c) A pro-Lie algebra is semisimple iff it is a product of finite simple real Lie algebras. (d) Every pro-Lie algebra has a unique smallest ideal ncored (g) such that g/ncored (g) is reductive. Consequently, for a pro-Lie algebra, the following statements are equivalent. (I) g is semisimple. (II) g is the product of a family of finite-dimensional simple ideals of g. (7.27, 7.29) In the light of the fact that pro-Lie algebras g arise as the Lie algebras of pro-Lie groups G, the very appealing duality theory of profinite-dimensional g-modules is a surprisingly effective tool for making the structure theory of pro-Lie groups algebraic.

Pro-Lie Algebras and Solvability Recalling in the structure theory of finite-dimensional Lie algebras that there is always a unique largest solvable ideal, called the radical, we cannot hope to be able to bypass the question of solvability in any structure theory of pro-Lie algebras that is deserving of this name. The fact that the underlying vector spaces of pro-Lie algebras are infinitedimensional as soon as the theory begins to be new and interesting is an ominous warning that solvability is going to be a delicate matter likely to involve set theory including well-ordering and ordinals. Firstly, on a purely algebraic basis, in any Lie algebra we must define a transfinite commutator series and use this transfinite series to define a general concept of solvability. This proceeds as follows. Let g be a Lie algebra. Set g(0) = g and define sequences of ideals g(α) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g(α) is defined for α < β.

22

Panoramic Overview

 (i) If β is a limit ordinal, set g(β) = α 1. Let 0 = g ∈ G. Since G satisfies (i), G = [Z · g] ⊕ H and [Z · g] is free. Since the subgroup H has smaller rank than G, it is not a counterexample and is therefore free. Hence G is free and thus cannot be a counterexample. Step 3. Now we prove that every finite rank pure subgroup G splits. By Step 2, N G = m=1 Z · em , where N ∈ N. By (i) there is a subgroup H1 of A such that A = Z · e1 ⊕ H1 and G = Z · e1 ⊕ (H1 ∩ G). Assume that H1 ⊇ H2 ⊇ · · · ⊇ Hn , n < N has been constructed in such a fashion that A = Z · e1 ⊕ · · · ⊕ Z · em ⊕ Hm

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and G = Z · e1 ⊕ · · · ⊕ Z · em ⊕ (Hm ∩ G), m = 1, 2, . . . , n. By the first step we may apply (i) to the pure subgroup Hn and find a subgroup Hn+1 of Hn such that Hn = Z · en+1 ⊕ Hn+1 and Hn ∩ G = Z · en+1 ∩ (Hn+1 ∩ G). This yields a descending family of subgroups Hn such that A = nm=1 Z · em ⊕ Hn and em ∈ Hn for m > n. We set H = HN . Then A = N m=1 Z · em ⊕ HN = G ⊕ H , as was to be shown. Definition 4.3. We say that an abelian group A is an S-group if it satisfies the equivalent conditions of Proposition 4.2. The S-groups have been called separable [49] which is not an advisable terminology here because we will deal with topological abelian groups for which the adjective separable refers to groups having a dense countable subset, and this is entirely different. One might have called S-groups strongly ℵ1 -free; our terminology reflects the “strongly” as well. Every free group is an S-group, since every subgroup of a free group is free, and the quotient of a free group modulo a pure subgroup is free (see e.g. [102, Proposition A1.24 (ii)]); thus every pure subgroup splits. A Whitehead group is an abelian group A such that Ext(A, Z) = {0}, that is, every extension 0→Z→G→A→0 splits. Example 4.4. The group A = ZN has the following properties: (i) A is an S-group. (ii) A is not a Whitehead group. (iii) The subgroup Z(N) of A is a countable free pure subgroup which does not split. Proof. (i), (ii) The group ZN is an ℵ1 -free group which is not a Whitehead group: see e.g. [102, Example A1.65]. We verify Condition 4.2 (ii) for A = ZN . Let P be a rank one pure subgroup of A and let k = (kn )n∈N be an element of P such that [Z · k] = P . Then the greatest common divisor of {kn : n ∈ N} is 1. The decreasing sequence gcd{k1 , k2 , . . . , kn } is eventually constant; that is there is a natural number N such that k1 , . . . , kN have the greatest common divisor 1. Then the subgroup PN generated in the finitely generated free group ZN by (k1 , . . . , kN ) is pure. By the Elementary Divisor Theorem (see e.g. [102, Theorem A1.10]) applied to ZN , after choosing a new basis, we may assume that PN = Z × {0} × · · · × {0}. It is therefore no loss of generality to assume that (kn )n∈N = (1, 0, . . . , 0, kN +1 , kN +2 , . . . ). Set G = {0} × Z × Z × · · · . Then ZN = P ⊕ G and ZN is an S-group as asserted. (iii) If m · (kn )n∈N ∈ Z(N) for some m ∈ Z then mkn = 0 for all but a finite number of n ∈ N. Then (kn )n∈N . Thus Z(N) is a pure subgroup of Z N which is obviously countable and free. The group ZN /Z(N) is a torsion-free algebraically compact group and contains a copy Zp of the p-adic integers for each prime as a direct summand. (See e.g. [58, p. 176, 42.2 and p. 169, 40.4.]) Since Zp contains countable groups which

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are not free (e.g. q1∞ Z for any prime q different from p), and since ZN is ℵ1 -free, ZN cannot contain a subgroup isomorphic to A/Z(N) . Example 4.5. There is an abelian group B with a subgroup C ∼ = Z such that B/C ∼ = ZN and that every morphism B → Z annihilates C. The group B is an ℵ1 -free group which is not an S-group. Proof. In the proof of Proposition A1.66 of [102], the following lemmas are proved: Lemma A. Let E = [0 → C → B → X → 0] be any extension of C ∼ = Z by an abelian group X. Then there is a homomorphism f : B → Z whose restriction to C is nontrivial if and only if E represents an element of finite order in Ext(X, Z). Lemma B. Ext(ZN , Z) contains 2(2

ℵ0 )

elements of infinite order.

Taken together, these lemmas yield the existence of a torsion-free group B and a cyclic subgroup C such that B/C ∼ = ZN , and that every morphism B → Z vanishes on C. The subgroup C is a subgroup of rank 1 which does not split, and since B/C is torsion-free, C is a pure subgroup ' +C &of B. Thus B is not an S-group. If P is a finite rank pure subgroup of B, then P C is a finite rank pure subgroup of B/C ∼ = A and is therefore finitely generated free; its full inverse image P  in B is a finitely generated torsion-free group and is, therefore, free. Thus P as a subgroup of a free group is free. Hence B is an ℵ1 -free group. In this area of the theory of abelian groups, ZN is a universal test example. For instance, Proposition 4.2 cannot be complemented by another equivalent condition which would say: Every countable pure subgroup splits. The example shows, in particular, that the class of S-groups is properly smaller than that of ℵ1 -free groups and is not contained in the class of Whitehead groups. Every Whitehead group is an S-group (see [49, p. 226]). Thus the class of S-groups is properly bigger than the class of Whitehead groups and thus a fortiori properly bigger than the class of free groups. ℵ1 -free groups  S-groups   Whitehead  groups  free groups Strengthening local connectivity for compact abelian groups. A locally compact  = Hom(G, T). abelian group G is completely characterized by its Pontryagin dual G (See e.g. [102, Chapters 7 and 8].) A topological group G is locally connected if and only if its identity component G0 is open in G and is locally connected; a connected locally compact abelian group G contains a unique characteristic maximal compact

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subgroup C and a subgroup V ∼ = Rn such that the morphism (v, c)  → v + c : V × C → G is an isomorphism of topological groups. (See e.g. [102, Theorem 7.57].) In discussing local connectivity of a locally compact abelian group G, it is no loss of generality to assume that G is compact and connected. A locally compact abelian  is discrete and group G is compact and connected if and only if its character group G torsion-free. (See e.g. [102, Proposition 7.5 (i), and Corollary 8.5].) Local connectivity of a compact connected abelian group is characterized as follows. Proposition 4.6. For a compact connected abelian group G, the following statements are equivalent: (i) There are arbitrarily small compact connected subgroups N such that G/N is a finite-dimensional torus group.  is the directed union of pure finitely generated free sub(ii) The character group G groups.  is ℵ1 -free. (iii) G (iv) G is locally connected. Proof. For a proof see [102, Theorem 8.36]. We compare this proposition with the following Proposition 4.7. For a compact connected abelian group G, the following statements are equivalent: (i) There are arbitrarily small compact connected subgroups N for which there is a finite-dimensional torus subgroup TN of G so that (n, t)  → n+t : N ×TN → G is an isomorphism of topological groups.  is the directed union of finitely generated free split sub(ii) The character group G groups.  is an S-group. (iii) G Proof. The equivalence of (i) and (ii) follows at once from duality.  is torsion-free. Let P be a rank one pure (ii) ⇒ (iii): By (ii), the abelian group G  By (ii) there is a finitely generated  The P = [Z · a] for some a ∈ G. subgroup of G.  free split subgroup F of G containing a. Since a direct summand is a pure subgroup we have P = [Z · a] ⊆ F . As a pure subgroup of a finitely generated torsion-free  P is a split group, P is a direct summand of F , and since F is a direct summand of G,  subgroup of G.  is the directed union of all of its finite (iii) ⇒ (ii): As a torsion-free abelian group, G rank pure subgroups P ; by (iii), every such P is split and free, and thus (ii) follows. The comparison of Propositions 4.6 and 4.7 justifies the following definition. Definition 4.8. A locally compact abelian group is said to be strongly locally connected if its identity component is open and its unique maximal compact connected subgroup satisfies the equivalent conditions of Proposition 4.7.

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In particular, a compact connected abelian group is strongly locally connected if and only if its character group is an S-group. def  N . Then G is a strongly locally connected and connected Example 4.9. Let G = Z but not arcwise connected compact abelian group. There is a compact connected, locally connected, but not strongly locally connected group H of weight 2ℵ0 containing G such that H /G is a circle group. G has a metric torus group quotient which is not a homomorphic retract.

Proof. A compact connected abelian group H is arcwise connected if and only if its  is a Whitehead group. (See e.g. [102, Theorem 8.30 (iv)].) The character group H claim thus follows by duality from Examples 4.4 and 4.5. The class of connected strongly locally connected compact abelian groups is properly larger than that of torus groups and properly smaller than that of connected locally connected compact abelian groups. The exponential function of strongly locally connected groups. We shall investigate when the exponential function expG : L(G) → G of a compact abelian group G is open onto its image in the present context. For a detailed exposition of the exponential function of compact abelian groups we refer to [102], notably Chapters 7 and 8. We need to know here that L(G) = Hom(R, G) is the topological vector space of all one parameter subgroups, i.e. continuous group morphisms X : R → G, where Hom(R, G) is given the topology of uniform convergence on compact sets. The exponential function is given by evaluation via exp X = X(1). By duality, L(G) may also be viewed  R) with the topology of pointwise convergence. We as the vector space Hom(G, note that L(G) is isomorphic to a product of copies of R (see [102, Theorem 7.66 (i) and Theorem 7.30 (ii)]). For any compact abelian group G, the exponential function expG : L(G) → G (see [102, Theorem 7.66]) is a morphism of abelian topological groups. Let Ga = im expG denote the arc component of the zero element 0. (See [102, Theorem 8.30.]) We note the exact sequence: expG  Z) → 0 0 → K(G) → L(G) −−−−−→ G → Ext(G,

(exp)

where K(G) = ker expG . The corestriction expG : L(G) → Ga of the exponential function to its image is a surjective morphism of topological groups. A surjective morphism between topological groups is open if and only if it is a quotient morphism. Thus expG is open iff the induced bijective morphism of topological groups L(G)/K(G) → Ga is an isomorphism. Proposition 4.10. Let G be a connected and strongly locally connected compact abelian group. Then expG : L(G) → Ga is open.  Then P ∈ P is Proof. Let P denote the set of pure finite rank subgroups of G.  such  a finitely generated free split subgroup of G. We select a subgroup SP ⊆ G def  = P ⊕ SP . Let NP = P ⊥ denote the annihilator of P in G and TP = S ⊥ that G P

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the annihilator of SP . By duality (n, t)  → n + t : NP × TP → G is an isomorphism of topological groups, that is, G = NP ⊕ TP algebraically and topologically.  are naturally isomorphic by the Annihilator MechaThe groups TP , G/NP and P nism (see [102, Theorem 7.64]) and thus are finite-dimensional torus groups. The morphism expNP × expTP : L(NP ) × L(TP ) → NP × TP is naturally equivalent to the exponential function of NP × TP . Let UP be the set of arcwise connected open zero-neighborhoods U of L(TP ) mapped homeomorphically onto an open zero neighborhood V of TP by expTP ; such neighborhoods U and V exist as TP is a Lie group. Then (expNP × expTP )(L(NP ) × U ) = (NP )a × V = (NP × V ) ∩ (NP × TP )a . It follows that expG (L(NP ) ⊕ U ) is an identity neighborhood of Ga in the topology induced from that of G. We claim that {L(NP ) ⊕ U : P ∈ P , U ∈ UP } is a basis for the open zero-neighborhoods of L(G), where L(NP ) is naturally considered as a cofinite-dimensional vector subspace of L(G). Once this claim is established, expG : L(G) → Ga is open and the proof of the proposition is complete.   = Since G P = colimP ∈P P by duality, we have G = limP ∈P G/NP . The functor L = Hom(R, −) preserves projective limits (see e.g. [102, Proposition 7.38 (iv)]; in fact L preserves all limits). Furthermore, L preserves quotients (see [102, Theorem 7.66 (iv)]). Hence L(G) = limP ∈P L(G)/L(NP ). Let rP : L(G) → L(G)/L(NP ) denote the quotient morphism, qP : L(G)/L(NP ) → L(TP ) the natural isomorphism, and pP = qP ◦ rP : L(G) = L(NP ) ⊕ L(TP ) → L(TP ) the projection. For any zero neighborhood W of L(G), by the Fundamental Theorem on Projective Limits 1.27 (i), there is a P ∈ P and a zero neighborhood U ∈ UP in L(TP ) such that pP−1 (U ) ⊆ W . Since pP−1 (U ) = L(NP ) ⊕ U , the claim is proved. We now get the result which shows that Theorem 4.1 cannot be improved. The product RR is a pro-Lie group (see 3.4 (ii) and 3.28), and it is of the simplest kind while not being locally compact. def

Corollary 4.11. (i) The uncountable product V = RR has a closed totally disconnected algebraically free subgroup K of countable rank such that the quotient V /K is incomplete and its completion G is a compact connected and strongly locally connected abelian group of continuum weight. (ii) Let q : V → V /K denote the quotient map and γV /K : V /K → G be the completion map. There is a morphism f : V → C into a compact hence complete group whose kernel is K and which has the property that the factorisation map f  : G → C determined uniquely by f = f   γV /K  q is not injective.  N be the compact abelian group of 4.4. The rank of ZN Proof. (i): Let G = Z agrees with the cardinal of ZN and that is the cardinal 2N of the continuum. Then N L(G) = Hom(R, G) = Hom(ZN , R) ∼ = RR . Thus we take for V the ad= R2 ∼ ditive group of L(G) and K = K(G) and know that ε : V /K → Ga is an isomorphism of topological groups by Proposition 4.10. The completion of Ga is G, and w(G) = card ZN = 2ℵ0 . The kernel K(G) is algebraically isomorphic to Hom(ZN , Z) (see [102, Theorem 7.66 (ii)]). But Hom(ZN , Z) ∼ = Z(N) (see [49, p. 61, Corollary 2.5]). Thus K is free of countable rank.

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(ii): In [102, Example A1.65] one finds the construction of a character χ : ZN → T of order 2 (i.e., 2 · χ = 0 in additive notation) which does not factor in the form ϕ

p

ZN −−→ R −−→ T,

p(r) = r + Z.

Then, as an element of G = Hom(ZN , T), the character χ is not in the image Ga of Hom(ZN , p) = expG : Hom(ZN , R) = L(G) → G. Set Z = {0, χ }. Then Z is a closed subgroup of G such that Ga ∩ Z = {0}. Put C = G/Z and let f  : G → C be the quotient morphism whose kernel is Z. The restriction F : Ga → C is injective. Let f : V → C be defined by f (X) = F (expG X). By (i) the corestriction q : V = L(G) → Ga of the exponential function is a quotient morphism, and F = f  q is the canonical epic-monic factorisation of F . Since G is isomorphic to the completion G of Ga and the inclusion Ga → G is the completion morphism γGa : Ga → G, the assertion follows. The significance of 4.11 (ii) is as follows: In the category of all complete topological abelian groups, the completion of a quotient plays the role of a quotient in the category as it has the expected universal properties; nevertheless, it will in general fail to have familiar properties as 4.11 (ii) illustrates. The possible incompleteness of quotients plays a somewhat disturbing role in the general theory of pro-Lie groups. The simplest nontrivial projective limits of finite-dimensional Lie groups are the products RX . The product RN is metrizable and complete, hence every quotient is complete. Corollary 4.11 shows that the “next largest product”, RR already has incomplete quotients, and it is remarkable that there are such quotients whose completion is compact. While we are utilizing the exponential function of compact abelian groups for the purpose of exhibiting incomplete quotients, we record additional information that is relevant in this context. Proposition 4.12. Assume that the corestriction expG : L(G) → Ga of the exponential function of a compact connected abelian group is a quotient morphism. Then (i) Ga has arbitrarily small open arcwise connected identity neighborhoods in the topology induced from that of G. (ii) G is locally connected.  is ℵ1 -free. (iii) G Proof. (i) A quotient morphism is open. But L(G) is a locally convex topological vector space and thus has arbitrarily small arcwise connected neighborhoods of zero which are mapped onto open identity neighborhoods of Ga by expG . (ii) Let W be an identity neighborhood of G. Then there is an identity neighborhood V such that V 2 ⊆ W . By (i), Ga has an open arcwise connected identity neighborhood U satisfying U ⊆ V . Then the closure U of U in G is contained in V ⊆ V V ⊆ W . There is an open set UG in G such that U = UG ∩ Ga , and since Ga is dense in G we have U = UG . Also, since U is arcwise connected, U is connected

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and is an identity neighborhood in G. Thus U is a connected identity neighborhood in G which is contained in W . Thus G is locally connected. (iii) This follows from Proposition 4.6 and (ii) above. Thus the exponential function of a compact connected abelian group can be open onto its image only if the group is locally connected. Proposition 4.13. If the arc component Ga of the zero element of a compact connected abelian group G is locally arcwise connected, then the corestriction expG : L(G) → Ga of the exponential function is open. Proof. On the group G there is a filter basis N (G) converging to 1 and consisting of closed compact subgroups N such that G/N is a finite-dimensional torus. By [102, Theorem 7.66 (iv)], there are arbitrarily small identity neighborhoods of G of the form N ⊕ V where V = expG U for an open n-cell neighborhood U in a finite-dimensional vector subspace F of L(G) such that (n, X)  → n + expG X : N × U → N ⊕ V is a homeomorphism. Now Na ⊕ V = (N ⊕ V )a is the arc component of 0 in Ga ∩ (N ⊕ V ). Since Ga is locally arcwise connected, arc components of open sets of Ga are open in Ga , and thus Na ⊕ V is open in Ga . Therefore Ga has arbitrarily small identity neighborhoods of the form Na ⊕ V . However, these are of the form Na ⊕ V = expN L(N ) ⊕ expG U = expG (L(N ) ⊕ U ) = expG (L(N ) ⊕ U ). Now we know that G = limN ∈N (G) G/N and just as in the proof of 3.1 we conclude that therefore L(G) = limN ∈N (G) L(G)/L(N ) and that L(G) has arbitrarily small neighborhoods of the form L(N )⊕U . This proves that expG is an open morphism. In fact one can show, that strong local connectedness is both necessary and sufficient for the exponential function of a compact abelian group to be surjective onto its image. Indeed the following theorem holds. Theorem 4.14 (Characterisation Theorem for Strong Local Connectivity of Compact Connected Abelian Groups). For a compact connected abelian group G and its zero arc-component Ga , the following conditions are equivalent: (i) (ii) (iii) (iv)

G is strongly locally connected. The exponential function expG : L(G) → G is open onto its image. Ga is locally arcwise connected.  is an S-group, that is, every finite rank pure subgroup of G  is free and is a G direct summand.

Proof. See [106, Theorem 5.1]. We have presented the implications (iv) ⇔ (i) ⇒ (ii) ⇒ (iii) in Propositions 4.7, 4.10, and 4.12. The implication (ii) ⇒ (i) is harder to prove.

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Remarks 4.15. (i) If a topological group G contains a closed central subgroup N such that N ∼ = RR , then G has an incomplete quotient group. (ii) Let {Aj : j ∈ J } be a family of abelian topological groups such that def  card J ≥ 2ℵ0 and each Aj contains a copy of R. Then G = j ∈J Aj has an incomplete quotient group. (iii) Let {Ej : j ∈ J } be a family of Hausdorff topological vector spaces such that def  card J ≥ 2ℵ0 . The E = j ∈J Ej has an incomplete quotient group. (v) If a topological group has a quotient group which has an incomplete quotient group, then it itself has an incomplete quotient group. (vi) Let {Gj : j ∈ J } be a family of topological groups such that card J ≥ 2ℵ0 and each Gj contains closed normal subgroups Mj ⊆ Nj such that Nj /Mj ∼ = R and [G, Nj ] ⊆ Mj (where [A, B] denotes the subgroup generated by the commutators def  aba −1 b−1 , a ∈ A, b ∈ B). Then G = j ∈J Gj has an incomplete quotient group. Exercise E4.1. Verify the assertions in Remarks 4.15 [Hint. (i) From the example in Corollary 4.11 above we find a closed, totally disconnected subgroup D of N such that N/D is incomplete. Since N is central, D is normal in G and then G/D contains N/D, an incomplete group, as a closed subgroup. The remaining assertions are straightforward.] The example in Corollary 4.11 (i) and the examples in Exercise E4.1 show very clearly that Theorem 4.1 (i) cannot be improved to read that quotients of pro-Lie groups are pro-Lie groups; most of our examples are in fact connected. In [34] it is shown that the class of abelian topological groups which are isomorphic to a product C × RX × ZY where C is a compact group and X and Y are countable sets is closed under the passage to closed subgroups and to quotient groups. This points out the fact that quotient groups of RX are complete if X is countable. In this sense the example of RR is minimal in the class of weakly complete topological vector spaces, if one momentarily accepts the Continuum Hypothesis. In [116] it is proved that a closed connected subgroup of RX is a closed vector subspace; consequently it is a direct summand algebraically and topologically (see [102, Theorem 7.30 (iv)]; in particular the quotient is a weakly complete topological vector space. More generally, we shall see shortly sufficient conditions on a pro-Lie group G and a closed normal subgroup N for G/N to be a pro-Lie group. (See Theorem 4.28 below).

The One Parameter Subgroup Lifting Theorem The lifting of one parameter subgroups deals with the following situation: Assume that f : G → H is a quotient morphism and Y ∈ L(H ); under which circumstances is there an X ∈ L(G) such that L(f )(X) = Y ?

The One Parameter Subgroup Lifting Theorem

Lemma 4.16. Assume that

183

ϕ

P ⏐ −−→ R ⏐ ⏐ ⏐ π Y G −−→ H

(1)

f

def

is a pullback of topological groups. Set K = ker ϕ. Then the following conditions are equivalent: (i) (ii) (ii ) (iii) (iv)

K is a semidirect factor and ϕ is surjective. ϕ is a retraction. ϕ|P0 : P0 → R is a retraction, where P0 is the identity component of P . There is an X ∈ L(G) such that L(f )(X) = Y . There is a subgroup R of P such that KR = P and K ∩ R = {1}, and further that ϕ|R : R → R is open.

These conditions imply: (v) There is a closed subgroup R of P such that KR = P and K ∩ R = {1}. Proof. (i) ⇔ (ii): The equivalence of (i) and (ii) is a standard exercise in topological group theory (see e.g. E1.5). (ii) ⇒ (ii ): If a morphism σ : R → P satisfies ϕ  σ = idR , then σ (R) ⊆ P0 as R is connected, and thus its corestriction σ : R → P0 satisfies ϕ  σ = idR . (ii ) ⇒ (ii): Conversely, if σ : R → P0 satisfies ϕ  σ = idR , then its coextension σ : R → P satisfies ϕ  σ = idR . (ii) ⇒ (iii): If X : R → P is a one parameter subgroup satisfying ϕ  X  = idR def

then X = π  X : R → G is a one parameter subgroup of G such that L(f )(X) = f  X = f  π  X = Y  ϕ  X  = Y  idR = Y . (iii) ⇒ (ii): Assume Y = L(f )(X) = f X. Then for all r ∈ R we have f (X(r)) = Y (r). Now the explicit form of the pullback is P = {(g, r) ∈ G × R | f (g) = Y (r)} and ϕ(g, r) = r (see e.g. Theorem 1.5). Hence (X(r), r) ∈ P for all r ∈ R and if we set X (r) = (X(r), r), then X  : R → P is a morphism satisfying ϕ(X  (r)) = r for all r. (i) ⇒ (iv) ⇒ (v) is trivial. (iv) ⇒ (ii): The morphism ϕ|R : R → R is continuous and open. Thus ϕ(R) is an open subgroup of R and therefore equals R. So ϕ|R is surjective, and since K ∩ R = {0} it is also injective. Hence it is an isomorphism of topological groups and thus is invertible; the coextension σ : R → P of (ϕ|R)−1 : R → R satisfies ϕ  σ = idR . Lemma 4.17. If f in the pullback (1) is surjective, then ϕ is surjective. If f is open, then ϕ is open. If f is a quotient morphism so is ϕ. Proof. Surjectivity: if r ∈ R then, since f is surjective, there is a g ∈ G such that f (g) = Y (r).

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Openness: The filter of identity neighborhoods of P has a basis of open sets of the form W = P ∩ (U × I ), where U is an open identity neighborhood of G and I an open interval around 0 in R. Then ϕ(W ) = {r ∈ I | (∃g ∈ U ) f (g) = Y (r)} = I ∩ Y −1 (f (U )). Since f is an open map, f (U ) is an open subset of H and thus by the continuity of Y , the set ϕ(W ) is open. Quotients: This assertion follows from the combination of the preceding two. Lemma 4.18. In the pullback (1), assume that G is a pro-Lie group. Then P is a pro-Lie group. Proof. By Theorem 1.5 (a) we have P = {(g, r) ∈ G × R : f (g) = Y (r)} in the category TopGr of topological groups. Since G and R are pro-Lie groups, the product G× R is a pro-Lie group since by Theorem 3.3 the category proLieGr is closed under the formation of products. Since the groups G, R, and H are all Hausdorff, the subgroup P is closed. Hence by Theorem 3.35 it follows that P is a pro-Lie group. We now are ready for a proof of the lifting of one parameter subgroups. This is not easy because in the absence of countability assumptions, this requires the Axiom of Choice, and the absence of compactness in the present situation forces us to rely on completeness and the convergence of Cauchy filters. The proof will require from the reader a certain facility handling “multivalued morphisms” as a special type of binary relations; but most of what is required will be self-explanatory in the proof. Lemma 4.19 (The One Parameter Subgroup Lifting Lemma). Let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Then every one parameter subgroup Y : R → H lifts to one of G, that is, there is a one parameter subgroup σ of G such that Y = f  σ . Proof. By Lemmas 4.16, 4.17, and 4.18, we may assume that H = R and we have to show that f is a retraction. Let K = ker f . Since {N ∗ = f (N) | N ∈ N (G)} converges to 0 in R, and since there are no subgroups in ]−1, 1[ other than {0} there is an N ∈ N (G) such that f (N) = N ∗ = {0}, and thus N ⊆ K. Then for all N ∈ N (G), N ⊇ N , the morphism f induces a quotient morphism fN : G/N → R, fN (gN ) = f (g), and fN (gN ) = 0 iff f (g) = 0 iff g ∈ K, that is, ker fN = K/N . If we let pN : K → K/N and qN : G → G/N denote the quotient morphisms, then we have a commutative diagram 1 →

N ⏐ ⏐ incl 1 → K ⏐ ⏐ pN  1 → K/N

idN

−−−→

N ⏐ ⏐ incl f incl −−−→ G −−→ R ⏐ ⏐ → 0 ⏐q ⏐id N  R fN incl −−−→ G/N −−→ R → 0

(2)

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185

with exact rows and columns. Due to the fact that the exponential map of a Lie group is a local homeomorphism at 0, an open morphism ψ : L1 → L2 between Lie groups induces an open and therefore surjective morphism L(ψ) between their Lie algebras: L(ψ)

L(L ⏐ 1 ) −−−→ L(L ⏐ 2) ⏐expL expL1 ⏐   2 L1 −−−→ L2 . ψ

Hence there is a morphism σN : R → G/N such that fN  σN = idR . The binary −1  σN : R → G satisfies the following conditions: relation  = qN def

(i) (0) = N and every (r) ⊆ G is a coset modulo N. (ii) The graph of  is a closed subgroup of R × G. (iii) We have a commutative diagram of binary relations of which all but  are functions: idR R −−−→ R ⏐ ⏐ ⏐ ⏐id   R f G −−−→ R (3) ⏐ ⏐ ⏐ ⏐id qN   R G/N −−→ R. fN

A binary relation  : R → G satisfying (i), (ii) and (iii) will be called a multivalued morphism associated with N . The set S of all multivalued morphisms  : R → G associated with some N ∈ N (G) is partially ordered under containment ⊆. By Zorn’s Lemma we find a maximal filter F ⊆ S. It is our goal to show def

that M = {(0) |  ∈ F } is cofinal in N (G). Assuming that this is proved, we note that for each r ∈ R and  ∈ F the subset (r) is a coset N x = xN with N = (0) ∈ N (G), and thus (r)(r)−1 = N x(N x)−1 = N; since M converges to 1, we conclude that {(r) |  ∈ F } is a Cauchy filter basis. Since G is complete, it converges to an element σ (r) ∈ G, giving us a function σ : R → G. As each (r), being a coset modulo N = (0) ∈ M, is closed, we have σ (r) ∈ (r) for all  ∈ F . Consequently, since (3) is commutative for each  ∈ F for N = (0) we have the following commutative diagram for all N ∈ M: R ⏐ ⏐ σ G ⏐ ⏐ qN  G/N

idR

−−−→ f

−−−→ −−→ fN

R ⏐ ⏐id  R R ⏐ ⏐id  R R.

(4)

The upper rectangle shows that f  σ = idR , and the fact that each qN   : R → G is a morphism of topological groups shows that qN  σ : R → G/N is continuous.

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4 Quotients of Pro-Lie Groups

Theorem 2.1 (i) shows that G has arbitrarily small open identity neighborhoods U satisfying U N = U for some N ∈ M. Then if V is a zero neighborhood of R such that −1 (U/N ) = U . This shows that σ is continuous. qN (σ (V )) ⊆ U/N , then σ (V ) ∈ qN Hence σ is the required coretraction for f . Thus the remainder of the proof will show that M is cofinal in N (G). Suppose that this is not the case. Then there exists an N ∈ N (G), N ⊇ N such that M ⊆ N for all M ∈ M ⊆ N (G). Let us temporarily fix M; then M ∩ N ∈ N (G), and thus def

G† = G/(M ∩ N) is a Lie group: G

MNG GG ww GG w GG ww w w GG w w N M GG ww GG w GG ww GG ww G ww M ∩N

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

G† .

This shows that for fixed M everything takes place in the Lie group G† in which def

def

M † = M/(M ∩ N) and N † = N/(M ∩ N) are closed normal Lie subgroups with M † ∩ N † = {1}. Thus μ : M † → G/N, μ(m(M ∩ N )) = mN is a morphism of Lie groups mapping M † bijectively onto MN/N and inducing an isomorphism of Lie algebras L(M † ) → L(MN/N ) ⊆ L(G/N ). Now M † is a closed normal subgroup of the Lie group G† and thus M † /M0† is a discrete normal subgroup of the Lie group G† /M0† . We let M ‡ be the open subgroup of M † containing (M † )0 and being such that M ‡ /M0† = (M † /M0† ) ∩ (G† )0 /M0† . Hence M ‡ /M0† is a discrete normal subgroup of a connected Lie group. Consequently it is finitely generated and thus countable. Therefore M ‡ has countably many components and so μ(M0† ) is an analytic subgroup Man ⊆ G/N agreeing with (MN/N )0 and having Lie algebra L(Man ) = L(MN/N ) = L((MN/N )0 ). (See [102, Theorem 5.52ff.]) Accordingly, {L(MN/N ) | M ∈ M} is a filter basis of finite-dimensional vector subspaces of L(G/N ). Hence there is a smallest element m = L(M# N/N ) in it such that for all M ≤ M# in M we have L(MN/N) = m. Let us abbreviate q(M# ∩N ) : G → G/(M# ∩ N ) by q # : G → G# , further f(M# ∩N) : G# → R by f # , and M# /(M# ∩ N ) by M # . Since L(μ) : L(M # ) → L(M # N/N ) = m is an isomorphism we have ,, q#

M(M # ∩ N) M# ∩ N

- = q # (M0# ) for M ⊆ M# in M. 0

(5)

The One Parameter Subgroup Lifting Theorem

187

There is a  # ∈ F such that M # =  # (0). Then for all  ∈ F contained in  # , the subgroup q # ((0)) of the Lie group G# is contained in q # ( # (0)), satisfies q # ((0)) = M0# , and (f #  q #  )(R) = R. Thus for all r ∈ R we have q # ( # (r)) = q # ((r)) since the right side is contained in the left and both are cosets modulo M # . In the Lie group G# we have the configuration G

M# NG GG v v GG vv v GG vv GG v v M# H N HH ww w HH ww HH ww HH ww M# ∩ N

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

G# .

−1 # Let σ # = σM# ∩N : G# → R be defined by σ = qM #   # . Then  # = qM #  σ and we have a commutative diagram of binary relations def

idR

−−−→

R ⏐ ⏐ #  G ⏐ qM # ⏐ 

f

−−−→

G/M #

−−−→ fM #

R ⏐ ⏐id  R R ⏐ ⏐id  R

(3# )

R.

def

We conclude that S = q # ( # (R)) =  # (R)/(M # ∩ N ) is a closed subgroup of G# whose Lie algebra L(S) cannot be contained in K # = K/(M # ∩ N ) = ker f # . From dim G# /K # = 1 we conclude L(G# ) = L(K # ) + L(S)

and L(S) = L(S) ∩ L(K # ) + R · X

for a suitable element X ∈ L(G# ) satisfying f # (expG# X) = 1. Setting τ : R → S, τ (r) = expG# r · X we obtain a coretraction for f # : G# → R. The binary relation def

 = (q # )−1  τ : R → G is a member of S. Moreover, for all  ∈ F we have q # ()(r) ⊇ τ (r) for all r ∈ R. Hence  ∩  is a member of S. Now the maximality of F shows that  ∈ F . But this implies that M # ∩ N = (0) ∈ M and that is a contradiction to our supposition allowing us a choice of an N such that M ∩ N = M for all M ∈ M. This contradiction finally completes the proof. There are some subtleties here which we should point out. Following Theorem 5.52 in [102] we have seen that the additive group h of a Banach space mapped surjectively onto an abelian Lie group G (which itself is quotient of a Banach space modulo a

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4 Quotients of Pro-Lie Groups

discrete subgroup) such that G has a one parameter subgroup which does not lift to h. This cannot happen if the domain is separable, but it does happen in the category of not necessarily finite-dimensional Lie groups. While being surjective, the morphism in question is not open and the Open Mapping Theorem fails. On the other hand, let G be a free abelian topological group in the sense of Graev (see [71]) generated by the space |R| underlying the topological abelian (additive) group R of real numbers. (For details on free topological groups see for instance [150]; see also [130].) Since the generating space |R| is arcwise connected and contains the identity, G itself is arcwise connected. Further, G is a kω group; hence it is a complete topological group ([150, pp. 379, 380]). Accordingly, G is not metrizable, because if it were it would be a σ -compact Baire space group and thus would be locally compact – which it is not. (See also [151].) Now we use the universal property of G to extend the identity map |R| → R to a morphism of topological groups q : G → R. If we consider |R| as embedded into G (which we may) then the restriction ρ of q to |R| is the identity map. Set K = ker q and define m : |R| × K → G by m(r, k) = rk; then m has an inverse given by m−1 (g) = (ρ −1 q(g), (ρ −1 q(g))−1 g). So m is a homeomorphism such that q m = ρ pr 1 . Thus, topologically, q is equivalent to a projection of a product onto one of the factors and is therefore a quotient map. However, algebraically G is a free group generated by the elements of |R| and thus does not contain any divisible elements, and so L(G) = {0}. Hence there is no morphism X : R → G such that q  X = idR . The quotient morphism q does not lift and L(q) : L(G) → L(R) = R is not surjective. We have seen that the functor L preserves all limits and thus, in particular, all kernels (since ker f for a morphism f of topological groups is nothing but the equalizer of f and the constant morphism). We shall say that a functor F : A → B between categories of topological groups is strictly exact if it preserves kernels and quotients. A morphism e : N → G of topological groups is said to be a strict morphism if its corestriction N → im e is an open morphism, that is, the bijective morphism N/ ker e → im e induced by it is an isomorphism of topological groups. Also recall from Theorem 4.1, that a quotient H of a pro-Lie group G is a proto-Lie group and that it has a completion H which is a pro-Lie group. Now, as a corollary of the One Parameter Subgroup Lifting Lemma we obtain the following theorem. The Strict Exactness Theorem for L Theorem 4.20. Let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Then the following conclusions hold. (i0 ) H is a proto-Lie group and the canonical embedding γH : H → HN (H ) =

lim

N ∈N (H )

H /N

according to 1.29 and 1.30 induces an isomorphism L(γH ) : L(H ) → L(HN (H ) )

The One Parameter Subgroup Lifting Theorem

189

of pro-Lie algebras. In particular L(H ) is a pro-Lie algebra. (i) The pro-Lie algebra morphism L(f ) : L(G) → L(H ) is a quotient morphism. In particular, the functor L : proLieGr → proLieAlg is strictly exact. (ii) If f

e

N −−→ G −−→ H is an exact sequence of morphisms of pro-Lie groups with a strict morphism e, then L(f )

L(e)

L(N ) −−−−→ L(G) −−−−→ L(H ) is an exact sequence of pro-Lie algebras. Proof. (i0 ) From the Quotient Theorem of Pro-Lie Groups 4.1 we recall that HN (H ) = limN∈N (H ) H /N is the completion H of the proto-Lie group H . Let νN : H → H /N denote the limit morphism and qN : H → H /N be the quotient morphism for N ∈ N (H ). From Theorem 1.29 we recall that νN  γH = qN for all N ∈ N (H ). The functor L preserves limits, and so for each N ∈ N (H ) we have a commutative diagram L(γH )

L(H −−−−→ L(H ⏐ ) ⏐ ) ⏐ ⏐ qN  L(ν) L(H /N ) −−−−→ L(H /N ),

(∗)

idL(H /N)

where the L(νN ) are the limit maps. The composition f

qN

qN  f : G −−→ H −−→ H /N is a quotient morphism of pro-Lie groups. The One Parameter Subgroup Lifting Lemma 4.19 tells us that it induces a surjective morphism L(qN  f ) : L(G) → L(H /N). Surjective morphisms of weakly complete topological vector spaces are quotient maps (see Appendix 2, Theorem A2.12 (b)). Since L is a functor, we have L(qN  f ) = L(qN )  L(f ); since L(qN )  L(f ) is surjective, L(qN ) : L(H ) → L(H /N) is surjective. Thus in the diagram (∗), the maps L(νN )  L(γH ) are all quotient maps, and this implies by Theorem 1.20 (i) that L(γH ) has a dense image. Also, L(γH ) has a zero kernel, since L preserves kernels and γH is an embedding. From Theorem 1.30 it follows that L(γH ) is an embedding. Therefore the composition L(f )

L(γH )

L(γH )  L(f ) : L(G) −−−−→ L(H ) −−−−→ L(H ) is a morphism of pro-Lie algebras with a dense image. From Appendix 2, Theorem A2.12 (a) we know that every morphism of weakly complete topological vector spaces with dense image splits and thus is surjective. Since the underlying vector spaces of L(G) and L(H ) are weakly complete by 3.8 and 3.12, the morphism L(γH )L(f ) is

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4 Quotients of Pro-Lie Groups

surjective. Therefore, the embedding L(γH ) is surjective and thus is an isomorphism. This is what we had to show. (i) We observed that L preserves kernels because kernels are limits. In (i0 ) we saw that L(H ) is a pro-Lie algebra, and the One Parameter Subgroup Lifting Lemma 4.19 shows that L(f ) is surjective, hence a quotient since surjective morphisms of weakly complete topological vector spaces are quotients by Theorem A2.12(b). Therefore, L preserves quotients, and so L is strictly exact. This completes the proof of (i). (ii) Let e

f

N −−→ G −−→ H be morphisms of pro-Lie groups such that im e = ker f and e is open onto its image. Set K = ker f , and decompose e as κ  ε where ε : N → K is the corestriction of e to its image and κ : K → G is the inclusion morphism. Then L(e) = L(κ)  L(ε) and L(κ) : L(K) → L(G) is the kernel of L(f ) since L preserves kernels, and L(ε) : L(N) → L(K) is surjective since L preserves quotients. Thus im L(e) = L(K) = ker L(f ).

It is remarkable that we do not require that H is a pro-Lie group; yet we obtain that L(H ) is a pro-Lie algebra. Such a situation can indeed arise as is shown by 4.9 and 4.10 where we have a quotient morphism from RR onto an incomplete group with a totally disconnected kernel inducing on the Lie algebra level the identity map of RR . The identity morphism Rd → R from the discrete group of real numbers to the group of real numbers with its natural topology is a bijective morphism of Lie groups inducing the zero morphism L(Rd ) = {0} → R ∼ = L(R). Thus one cannot drop the assumption in the theorem that e be a strict morphism. Sometimes one has an Open Mapping Theorem available which gives us this assumption for free – for instance if ker f is a Baire space and N is locally compact and σ -compact. In fact, we shall prove an Open Mapping Theorem for almost connected pro-Lie groups later in the book. In that context we shall show elementarily the existence of a connected proto-Lie group 2 ℵ0 H whose Lie algebra is algebraically isomorphic to Rℵ0 , while L(H ) ∼ = R2 . Thus there is no chance of proving (i0 ) of Theorem 4.20 without the assumption that H is in fact a quotient of a pro-Lie group. Corollary 4.21. (i) If N is a closed normal subgroup of a pro-Lie group G, then the quotient morphism q : G → G/N induces a map L(q) : L(G) → L(G/N ) which is a quotient morphism with kernel L(N ). Accordingly there is a natural isomorphism X + L(N) → L(f )(X) : L(G)/L(N ) → L(G/N ). (ii) Let G be a pro-Lie group. Then {L(N ) | N ∈ N (G)} converges to zero and is cofinal in the filter  (L(G)) of all ideals i such that L(G)/i is finite-dimensional. Furthermore, L(G) is the projective limit limN ∈N (G) L(G)/L(N ) of a projective system of bonding morphisms and limit maps all of which are quotient morphisms, and

The One Parameter Subgroup Lifting Theorem

191

there is a commutative diagram L(γG ) G ∼ L(G) −−−−→ L(GN (G) ) = L(limN ∈N ⏐ ⏐ (G) N ) = limN ∈N (G) ⏐ ⏐ expG  L(limN∈N (G) expG/N ) G −−−−→ GN (G) = limN ∈N (G) G/N.

L(G) L(N )

γG

Proof. Assertion (i) is an immediate consequence of the Strict Exactness Theorem 4.20. (ii) We know that L preserves limits. Thus L(γG ) : L(G) → L(GN (G) ) is an isomorphism. By (i) above, L(G/N ) ∼ = L(G)/L(N ) and thus L(G) ∼ =

lim

N ∈N (G)

L(G)/L(N ).

Thus by 1.27 (ii), the filter basis {L(N ) | N ∈ N (G)} of the kernels of the limit maps converges to 0 and the projective system of the L(G)/L(N ) has the natural quotient morphisms as bonding maps; by 1.27 (ii) it follows that the limit maps are quotient morphisms as well. It then follows that this filter basis is cofinal in  (L(G)). (Compare 1.40.) Recall from Definition 2.21 that E(G) denotes the subgroup expG L(G) algebraically generated by expG L(G). Corollary 4.22. (i) For a pro-Lie group G, the subgroup E(G) = expG L(G) is dense in G0 , i.e. E(G) = expG L(G) equals G0 . In particular, a connected nonsingleton pro-Lie group has nontrivial one parameter subgroups. Moreover, if h is a closed proper subalgebra of L(G), then expG h = E(G). (ii) A morphism f : G → H of pro-Lie groups induces a surjective (hence quotient) morphism L(f ) : L(G) → L(H ) in each of the following cases: (a) f is open. (b) f is surjective and G is almost connected. (iii) Assume that a morphism f : G → H of pro-Lie groups induces a surjective morphism L(f ) : L(G) → L(H ); by (ii) above this is the case, in particular, if f is open or if f is surjective and G is almost connected. Then the induced morphism E(f ) : E(G) → E(H ) is surjective, that is E(H ) = f (E(G)). As a consequence, H0 = f (G0 ). If H is a Lie group, then H0 ⊆ f (G). If H is in addition connected, then f is surjective. (iv) A morphism f : G → H of pro-Lie groups into a finite-dimensional Lie group inducing a surjective morphism L(f ) is locally trivial, that is, there is an open identity neighborhood V of H and a continuous map s : V → G such that f (s(h)) = h for all h ∈ V . In particular, for N = ker f , ϕ : f −1 (V ) → N × V ,

ϕ(g) = (gs(f (g))−1 , f (g))

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4 Quotients of Pro-Lie Groups

is a homeomorphism with inverse ψ given by ψ(n, h) = ns(h). (vi) Let G be a pro-Lie group and assume that for all N from a basis of N (G) the quotient G/N is connected. Then G is connected. Proof. (i) First we show that a nonsingleton connected pro-Lie group has nontrivial one parameter subgroups. Let G be a nonsingleton connected pro-Lie group. There is a g ∈ G, g = 1. Since lim N (G) = 1 there is an N ∈ N (G) such that g ∈ / N. Then G/N is a nonsingleton connected Lie group. Thus L(G/N ) = {0}. Then L(G) = {0} by 4.21 (i). Next we let G be an arbitrary pro-Lie group. The closed subgroup E(G) = expG L(G) is fully characteristic, hence normal. By the One Parameter Subgroup Lifting Lemma 4.19, every one parameter subgroup of G/E(G) lifts to one in G which is contained in E(G) by the definition of E(G). Hence L(G/E(G)) = {0}. Thus G/E(G) is totally disconnected by what we just proved, and thus G0 ⊆ E(G) ⊆ G0 . def

For a proof of the last assertion of (i), let us write g = L(G) and let h be closed subalgebra of g such that h = g. Then there is a 0-neighborhood U of g such that h + U = g. By Corollary 4.21 (ii) we find a closed normal subgroup N of G such that def

n = L(N) ⊆ U . Thus h + n = g. Since dim g/n < ∞ and all vector subspaces of a finite-dimensional real vector space are closed we know that h+n is a closed subalgebra of g. If we can prove that expG (h + n) = E(G) then certainly expG h = E(G). Thus by replacing h by h + n and renaming the subalgebra we shall assume without loss of generality that n ⊆ h. By Corollary 4.21 (i) we have a commutative diagram {0} →

n ⏐ ⏐ id {0} → ⏐ n expN ⏐  {1} → N

−−−→

incl

quot

j

q

h −−−→ h/n → {0} ⏐ ⏐ ⏐ ⏐ incl  L(j ) L(q) −−−→ ⏐ g −−−→ L(G/N ⏐ ) → {0} ⏐exp expG ⏐   G/N −−−→ G −−−→ G/N → {1},

and we may naturally identify L(G/N ) with g/n; under this identification, expG/N (h/n) gets identified with q(expG (h)) = (expG (h))N/N. Now expG hN/N = (expG (h))N/N = expG/N (h/n). By the same token, E(G)N/N = expG gN/N = expG/N (g/n) = (G/N )0 since G/N is a Lie group. Since h/n = g/n the analytic subgroup expG/N (h/n) of (G/N )0 is proper. Thus expG hN = E(G)N and since expG h ⊆ E(G), this must mean expG h = E(G). (ii) (a) Let f : G → H be an open morphism of topological groups. Then f (G) is an open, hence closed, subgroup of H (see e.g. [102, Proposition A4.25 (ii)]) and thus a pro-Lie group by Corollary 4.8. The open and surjective corestriction G → f (G) (inducing an isomorphism of topological groups G/ ker f → f (G)) is a quotient morphism between pro-Lie groups and thus induces a quotient morphism

The One Parameter Subgroup Lifting Theorem

193

L(f ) : L(G) → L(f (G)) by the Strict Exactness Theorem 4.20. Since f (G) is open in H , the inclusion j : f (G) → H induces an isomorphism L(j ) : L(f (G)) → L(H ) of topological Lie algebras. Thus L(f ) : L(G) → L(H ) is a quotient morphism. (b) By Theorem A2.12 (b) of Appendix 2, a surjective morphism of weakly complete topological vector spaces is a quotient morphism; so we have to show that L(f ) is surjective, that is, that every one parameter subgroup of H can be lifted to one of G. By Lemmas 4.17 and 4.18, it is no restriction to assume that H = R = L(H ). Assume that X ∈ L(G) is such that L(f )(X) = 0. Then L(f )(R · X) = R = L(H ). It therefore suffices to show the existence of an X ∈ L(G) which is not annihilated by L(f ). However, suppose that L(f )(L(G)) = {0}. Then f (E(G)) = {1H }. By the continuity of f it follows that f (E(G)) = {1H }. By (i) above, E(G) = G0 . Since G is almost connected, G/G0 is compact. Since f (G0 ) = {1H }, there is a morphism F : G/G0 → H = R, F (gG0 ) = f (g). Then f (G) = F (G/G0 ) is a compact subgroup of H = R and therefore is {1H }, and this contradicts the surjectivity of f . (iii) By assumption, L(H ) = L(f )(L(G)), and thus expH L(H ) = expH L(f )(L(G)) = f (expG L(G)). Consequently E(H ) = expH L(H ) = f (expG L(G)) = f expG L(G) = f (E(G)).

Thus H0 = E(H ) = f (E(G)) ⊆ f (G0 ) ⊆ H0 = H0 , and this shows f (G0 ) = H0 . Now assume that H is a connected Lie group. Then expH L(H ) is a neighborhood of 1 and thus E(H ) is an open and hence also closed subgroup of H and thus H0 ⊆ E(H ). Since E(H ) is connected, E(H ) = H0 . Now H = E(H ) = f (E(H )) ⊆ f (G), and thus H0 ⊆ f (G). If H = H0 , then f is surjective. (iv) The morphism L(f ) : L(G) → L(H ) is a surjective morphism of weakly complete topological vector spaces and therefore splits by Theorem A2.12 (a), that is, there is a morphism of weakly complete topological vector spaces σ : L(H ) → L(G) such that L(f )  σ = idL(H ) . Since H is a finite-dimensional Lie group, there is an open zero neighborhood U of L(H ) and an open identity neighborhood V of H such that expH |U : U → V is a homeomorphism. Define s : V → G by s = expG  σ  (expH |U )−1 . Let h ∈ V . Then f (s(v)) = (f  expG σ )((expH |U )−1 (h)) = (expH L(f )  σ )((expH |U )−1 (h)) = expH ((expH |U )−1 (h)) = h, as asserted. The remainder is an easy exercise by verifying that ψ  ϕ and ϕ  ψ are the respective identity functions. (v) Let qN : G → G/N denote the quotient morphism. By (ii) we have G/N = E(G/N) = qN (E(G)). Thus G = E(G)N for all N ∈ N (G) and so G0 = E(G) = E(G) = G. There is a connected commutative infinite-dimensional Banach Lie group G whose exponential function maps a proper closed hyperplane h in its Lie algebra g surjectively onto the group (see [102, Chapter 5, paragraph following Theorem 5.52]); this illustrates that 4.22 (i) is not exactly a trivial matter.

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4 Quotients of Pro-Lie Groups

Let H = Tp be the p-adic solenoid (see Example 1.20 (ii)), let G = L(H ) = R, and let f : G → H be the exponential function expH : L(H ) → H . Then L(f ) : L(G) → L(H ),

L(G) = R

is an isomorphism, but f is far from surjective. In fact, im f = Ha is the identity  arc component of H and as abelian groups, π0 (H ) = H /Ha and Ext p1∞ · Z, Z are isomorphic groups of continuum cardinality. (See Chapter 8 in [102].) The relation H0 = f (G0 ) for a quotient morphism f such as in 4.22 (iii) cannot be improved as the example of the following quotient morphism of locally compact abelian groups shows: Let G = R × Zp for the group of p-adic integers Zp , let H = G/{(n, −n) | n ∈ Z} ∼ = Tp and let f be the corresponding quotient morphism. Note that H is compact and connected. (See [102, Exercise E1.11 following Definition 1.30]. We consider Z as a subgroup of Tp as well.) Then G0 = R × {0}, and f (G0 ) = Ha = H = H0 . However, we want to point out, that the proof of 4.22 (iii) does show more than what was asserted. Observe that for any topological group G we can write E(G) = expG L(G) and E(G) = E(G). If G is a proto-Lie group, then its completion is a pro-Lie group G∗ = GN (G) and we have g ⊆ g∗ (writing G ⊆ G∗ ). If equality g = g∗ holds then expG g = expG∗ g∗  is dense in G∗ and is contained in G, and so is dense in G. Therefore: Remark 4.22 (iii)∗ . Assume that a morphism f : G → H of topological groups induces a surjective function L(f ) : L(G) → L(H ). Then the induced morphism E(f ) : E(G) → E(H ) is surjective, that is E(H ) = f (E(G)). If G0 = E(G) and H0 = E(H ), then H0 = f (G0 ). In particular, if G is a pro-Lie group and H is a proto-Lie group whose pro-Lie algebra coincides with the pro-Lie algebra of its completion, then H0 = f (G0 ). The vanishing of the Lie algebra means the absence of nontrivial one parameter subgroups. This property should be discussed here. Just for the record we repeat at this point what we showed in Proposition 3.30. In the light of the Pro-Lie Group Theorem 3.34 that result can be slightly reformulated. Corollary 4.23 = 3.30. For a pro-Lie group G the following statements are equivalent: (a) (b) (c) (d)

L(G) = {0}. G is totally disconnected. G is zero-dimensional. G is prodiscrete.



This generalizes the classical result that a totally disconnected finite-dimensional Lie group is discrete.

Sufficient Conditions for Quotients to be Complete

195

Sufficient Conditions for Quotients to be Complete In the structure theory of locally compact groups, the concept of “almost connected groups” is quite helpful, but it applies perfectly well to arbitrary topological groups.We introduced it in our preface and in Definition 1(ii) in the overview chapter, and we reiterated it many times since. For easy reference we recall: Definition 4.24 = 1(ii). A topological group G is called almost connected if G/G0 is compact. Clearly, all compact groups and all connected topological groups are almost connected. If G is almost connected and N is a closed normal subgroup of G, then G0 N /N ⊆ (G/N)0 , whence (G/N )/(G/N )0 is a quotient group of (G/N )/(G0 N/N) ∼ = G/G0 N ∼ = (G/G0 )/(G0 N /G0 ), which is in turn a quotient group of the compact group G/G0 . Thus quotients of almost connected groups are almost connected. For the next steps we observe, that the filter basis N (G) that we use to define pro-Lie groups may be “thinned out” to a cofinal filter basis M(G) if G is almost connected. We say that M and N in N (G) are close if M ∩ N is open in M and N . This relation is reflexive and symmetric. We claim that it is transitive: Indeed, if N1 ∩ N2 is open in Nj , j = 1, 2 and N2 ∩ N3 is open in Nj , j = 2, 3 then N1 ∩ N2 ∩ N3 is open in N2 and then in N1 ∩ N2 and N2 ∩ N3 ; therefore the triple intersection is also open in N1 and N3 whence N1 ∩ N3 is open in N1 and N3 . Therefore closeness is an equivalence relation, and we write M ∼ N if M and N are close. Now we define def

M(G) = {N ∈ N (G) | (M ⊆ N and M ∼ N ) ⇒ |N/M| < ∞}.

(6)

Lemma 4.25. (i) If G is an almost connected pro-Lie group, then M(G) as defined in (6) is cofinal in N (G). In particular, M(G) is a filter basis. (ii) If M(G) is cofinal, then γ : G → GM(G) = limM∈M(G) G/M is an isomorphism. Proof. (i) Let N ∈ N (G) and M ∼ N in N (G) such that N ⊇ M. The quotient def

G/M → G/N has the discrete kernel D = N/M and so induces a covering map of the identity components (G/M)0 → (G/N )0 whose kernel is K = (N/M) ∩ (G/M)0 = (N ∩G0 M)/M. Since G is almost connected, the Lie group G/M is almost connected, and thus (G/M)/(G/M)0 ∼ = G/G0 M is finite; consequently D/K is finite. The group K is central (see e.g. [102, A4.27]), in particular abelian, and it is finitely generated. (See [102, Proposition 5.75], applied to a maximal connected abelian subgroup H : all of these contain the center (see [86, p. 189], Theorem 1.2, or [85, p. 280, Satz III.7.11]).)

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Let tor K denote the finite torsion subgroup of K. D = N/M    K    tor K   {1}

 D/K finite 

K/ tor K ∼ = Zn(M)

 finite

The torsion-free rank n(M) of K/ tor K is a natural number. We claim that n(N ) is the maximal rank of free abelian subgroups of D: Indeed let F be a free abelian subgroup of D. Then F K/K ∼ = F /(F ∩K) is finite, hence F ∩K is a free abelian group such that rank(F ∩ K) = rank F (see e.g. [102, Theorem A1.10]). But rank(F ∩ K) ≤ n(M), and this establishes the claim. Let GN denote the simply connected covering group of (G/N )0 . Then for each M ∼ N, M ⊆ N there is a covering morphism GN → (G/M)0 . The set {n(M) | M ⊆ N, M ∼ N} is bounded by the torsion-free rank of π1 (GN ). Let us pick an M ⊆ N in the ∼-equivalence class of N with maximal n(M). Now let M  ⊆ M be a member in the ∼-class of M. Then M  ∼ N and M  ⊆ M, and thus n(M  ) = n(M) by the maximality of n(M). We claim that |M/M  | is finite. Suppose that this is not the case. Then in the sequence of coverings (G/M  )0 → (G/M)0 → (G/N )0 the kernel of the first one has a positive torsion-free rank n , and the kernel of the second has torsion-free rank n(M). Hence the torsion-free rank n(M  ) of the composition is n + n(M) > n(M), contrary to the maximality of n(M). This contradiction proves our claim that |M/M  | is finite. Hence M ∈ M(G) and M ⊆ N and this shows that M(G) is cofinal in N (G). (ii) By the Cofinality Lemma 1.21 we have lim

M∈M(G)

G/M =

lim

N ∈N (G)

G/N = GN (G) .

Since G is pro-Lie we have G ∼ = GN (G) . For almost connected pro-Lie groups, we shall see an alternative approach to M(G) in 9.45. We shall now introduce a topology which we call the Z-topology and which is reminiscent of the Zariski topology on algebraic varieties and, in particular, on algebraic groups. This topology will be defined on almost connected pro-Lie groups. While it is a T1 -topology (singletons are closed sets) it gives rise to one of those rare occasions where we choose to use topologies which are not Hausdorff. When the topology is restricted to a Lie group which is an algebraic group, it will in general be coarser than the Zariski topology. Recall that a subbasis for the set of closed sets C of a topology O(X) on a set X is a set B ⊆ C such that every closed set is the intersection of sets each of which is a finite

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union of members of B. We say that the topology O(X) is generated by B. Every set of subsets of a set X can serve as a subbasis for a topology. Recall that a collection of subsets of a set is said to satisfy the finite intersection property if every finite subcollection has a nonempty intersection, that is, iff it generates a filter. Let X be a topological space. By the Alexander Subbasis Theorem (see e.g. [131, p. 139, Theorem 6]), the topology O(X) generated by a subbasis B for the set of closed sets of X is compact iff every set of sets from B satisfying the finite intersection property has a nonempty intersection. Note in passing that a proof of the Alexander Subbasis Theorem requires the Axiom of Choice. Definition 4.26 (The Z-topology on a pro-Lie group). Let G be a pro-Lie group and def

set B(G) = {gN | g ∈ G, N ∈ M(G)}. The topology Z(G) on G generated by B(G) as a subbasis for the set of closed sets is called the Z-topology. We shall often abbreviate Z(G) by Z. Note that we are not saying that G with the Z-topology is a topological group. Proposition 4.27. The Z-topology Z on a pro-Lie group G has the following properties: (i) All left and right translations are homeomorphisms in the Z-topology. (ii) Inversion x  → x −1 : G → G is a homeomorphism in the Z-topology.  (iii) If M(G) = {1} then Z is a T1 -topology. This is the case if G is almost connected. (iv) If G is almost connected, then Z is a compact T1 -topology. Proof. (i), (ii) It suffices to observe that the subbasis B(G) is bijectively mapped onto itself by left and right translations (since x(gN ) = (xg)N respectively, (gN )x = (gx)N) and by inversion (since (gN )−1 = N −1 g −1 = g −1 N ). (iii) By (i) it suffices to show that {1} is closed, i.e., that for each g = 1 there is a closed set containing 1 but not g. Thus let g = 1. Since M(G) = {1} there is an N ∈ M(G) such that g ∈ / N . Then N is the required closed set. (iv) We assume that G is an almost connected pro-Lie group and show that the Z-topology is compact. We proceed in steps. Step (a). Assume that G is a Lie group, in which case {1} ∈ M(G). Now let F = {gj Nj | j ∈ J } be a filter basis of subbasic sets with suitable Nj ∈ M(G). We like to show that it has a nonempty intersection and then apply the Alexander Subbasis Theorem to prove that G is compact for the Z-topology. Define F0 = {gj (Nj )0 | j ∈ J }. Since the collection of finite dimensional vector spaces {L(Nj ) | j ∈ J } satisfies the descending chain condition, there is a j0 ∈ J such that for all j ≥ j0 we have L(Nj ) = L(Nj0 ) and hence (Nj )0 = (Nj0 )0 . Thus there is no harm in assuming that the identity components of the members of F all agree with some M ∈ M(G). But then we pass to the quotient G/M. To simplify notation we then assume that M is a singleton and that all members of F are discrete normal subgroups of the Lie group G. Now we use our hypothesis that G is almost connected in its group topology. This property is preserved by the reductions we made. Hence G/G0 is finite since in a Lie

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group the identity component is open. Then there is a g ∈ G such that gG0 ∩ F = Ø for all F ∈ F . We may consider the filter basis g −1F instead  and assume in the end that {G0 ∩ F : F ∈ F } is a filter basis F  . Since F  ⊆ F . Therefore there is no harm in assuming F  = F . Thus we are dealing with a connected Lie group G and a filter basis F of cosets modulo discrete normal subgroups D ∈ M(G). Since {1} ∈ M(G) and {1} is open in each D ∈ M(G), D ∼ {1} for all D. By the definition of M(G), this means that D is finite. Thus the filter basis F consists of finite sets and so has a nonempty intersection. An application of the Alexander Subbasis Theorem completes the proof of the compactness of G in this case. Step (b). Assume that G is a strict projective limit of its Lie group quotients G/N , def  N ∈ M(G) and assume that H = N ∈M(G) G/M is compact in its Z-topology. We claim that then G is compact in its Z-topology. For a proof consider M, N ∈ M(G), M ⊇ N, let SMN denote the subgroup of G/M×G/N of all pairs (gM, hN ) with hN ⊆ gM. Then hM = hN. We claim that (G/M ×G/N )/SMN = (G/M ×{N })SMN /SMN is isomorphic to G/M. Indeed G/M × G/N is an almost connected Lie group, and thus is σ -compact and local compact, and G/M is a locally compact. Then by the Open Mapping Theorem for Locally Compact Groups (see for instance [79, p. 42, Theorem 5.29]) applies and  establishes the claim. Hence SMN ∈ M(G/M × G/N). Now let PMN ⊆ H = P ∈M(G) G/P be  the subgroup of all (gP P )P ∈N (G) such that gN ∈ gM M. Then PMN ∼ = SMN × P ∈M(G)\{M,N } G/P and thus H /PMN ∼ = G/M and therefore PMN ∈ M(H ). In particular PMN is Z-closed in H . Hence limP ∈M(G) G/P = M,N ∈N (G),N ⊆M PMN is a Z-closed subgroup of H . Since H is Z-compact by assumption, it follows that G ∼ = limN ∈M(G) G/N is Z-compact. After Steps (a) and (b) the proof of Claim (iv) will be finished if we prove the next and final step:  Step (c). G is a product j ∈J Gj of almost connected Lie groups Gj . We claim that G is compact in the Z-topology. Observation (i). If M ∈ M(G), then there is a unique largest cofinite subset EM of J such that j ∈EM Gj ⊆ M, where we have identified the partial product in an unmistakable fashion with a subgroup of G. Proof of Observation (i). Since G/M is a Lie group and thus has no small subgroups, we find an open neighborhood U of M such that U M = U and every subgroup of G contained in U is contained in M. Let V be a basic identity neighborhood for the product topology of G such that V ⊆ U . Then there is a finite  subset EM ⊆ J and identity neighborhoods Wj in Gj for j ∈ EM such that V = j ∈J Vj for " Wj if j ∈ EM , Vj = Gj if j ∈ J \ EM .    are two Then j ∈J \EM Gj ⊆ V ⊆ U and therefore j ∈EM Gj ⊆ M. If EM and EM such sets, then       Gj = Gj Gj ⊆ M.  j ∈J \EM ∩EM

j ∈J \EM

 j ∈J \EM

Sufficient Conditions for Quotients to be Complete

199 def

Thus  we may assume that EM was selected to be the smallest set such that P (M) = j ∈J \EM Gj , {1} × P (M) ⊆ M.  Now let N(M) be the projection of M onto j ∈EM Gj along P (M) ⊆ M. Then M = N(M) × P (M) where N(M) is a closed normal and almost connected normal subgroup of G (incidentally not containing any of the factorsGj , j ∈ EM ) such that M = N(M) × P (M). For a subset I ⊆ J let us write GI = j ∈I Gj , and let Fin(J ) denote the set of all finite subsets of J . Then we have Observation (ii). The set M(G) is the set of all M = N × GJ \E ,

E ∈ Fin(J ),

N ∈ M(GE ).

In particular, each M ∈ M(G) contains a member of M(G) of the form {1}×GJ \E , E ∈ Fin(J ).  Observation (iii). In the Z-topology, G = j ∈J Gj is compact for any infinite family of almost connected Lie groups Gj . In order to show that G is compact in the Ztopology, by the Alexander Subbasis Theorem it is sufficient to show that a filter basis B of subbasic closed sets, that is, cosets B = g(B)M(B) ∈ B,

g(B) ∈ G, M(B) ∈ M(G),

has a nonempty intersection. Notice that M(B) = B −1 B is uniquely determined by B and B ⊇ C implies M(B) ⊇ M(C), while the representative g(B) is unique only modulo M(B). If g ∈ B∩C for B, C ∈ B, then B = gM(B) and C = gM(C), whence B ∩ C = g(M(B) ∩ M(C)); in general M(G) is not closed under finite intersections. As we have seen in Observation (ii), for each B ∈ B, the normal subgroup M(B) of G is of the form M(B)EB × GJ \EB for a suitable minimal finite set EB ∈ Fin(J ) and a suitable subgroup M(B)EB ∈ M(GEB ). We write g(B) as g(B)EB × gJ \EB ∈ GEB × GJ \EB . Then we have B = g(B)EB M(B)EB × GJ \EB .

(∗)

The set of all filter bases B  containing B and having cosets g(B  )M(B  )B  as members is inductive with respect to containment ⊆. Hence it contains maximal ones by Zorn’s  Lemma. The intersection of any of these maximal filter bases is contained in B. It is therefore no restriction of generality if, in order to simplify notation, we assume that B itself is maximal. Clearly B is infinite. Let E ∈ Fin(J ); we consider the projection fE : G → GE and inspect the filterbasis fE (B) of cosets modulo normal subgroups in the almost connected Lie group GE . Equation (∗) permits us the comparison B = g(B)EB M(B)EB × GE\EB × GJ \(E∪EB ) , ker fE = {1E∩EB } × GEB \E × {1E\EB } × GJ \(E∪EB ) ,

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with obvious identifications of the partial products of the Gj . Accordingly we see that fE (B) = fE(E∪EB ) (g(B)EB M(B)EB × GE\EB ). Note that g(B)EB M(B)EB × GE\EB = (g(B)EB × {1E\EB })·(M(B)EB × GE\EB ) is a coset modulo the closed normal subgroup M(B)EB × GE\EB . However, M(B)EB ∈ M(GEB ), and thus the component (M(B)EB )0 is open in M(B)EB and therefore M(B)EB ∼ (M(B)EB )0 ; hence M(B)EB has finitely many components and therefore M(B)EB × GE\EB has finitely many components. The image fE(E∪EB ) ((M(B)EB × GE\EB )0 ) is an analytic subgroup of the Lie group GE and thus fE (M(B)) = fE(E∪EB ) (M(B)EB × GE\EB ) is a finite extension of an analytic subgroup of GE . In view of the fact that the set of finite extensions of analytic subgroups in an almost connected Lie group satisfies the finite descending chain condition, we have seen that (†) for each E ∈ Fin(J ) the filterbasis fE (B) contains a smallest element AE , that is, there is a C ∈ B such that for all D ∈ B AE = fE (C) ⊆ fE (D). We now claim that (‡) for each B ∈ B there is a unique element hB ∈ GEB such that def

B = {hB } × GJ \EB ∈ B and B ⊆ B. For a proof of this claim let B ∈ B. Then by (∗) we have B = g(B)EB M(B)EB × GJ \EB . By (†) the filter basis fEB (B) has a minimal element AEB ⊆ g(B)EB M(B)EB . Let a ∈ AE . Then g(B)EB M(B)EB = aM(B)EB and a ∈ fEB (C) for all C ∈ B. Hence, if for C ∈ B we set Ca = fE−1 (a) ∩ C, then B def

(∀C ∈ B) Ca = Ø,

and Ca−1 Ca = GJ \EB ∩ M(C) ⊇ GJ \(EB ∪EC ) ∈ M(G).

In particular, since fEB |GEB is (up to natural identification) the identity of GEB , (a) ∩ B = {a} × GJ \EB . If C ∈ B, then Ba ∩ C = fE−1 (a) ∩ B ∩ C. we have Ba = fE−1 B B Since B is a filterbasis, there is a D ∈ B such that D ⊆ B ∩ C. Then Ø = Da = fE−1 (a) ∩ D ⊆ fE−1 (a) ∩ B ∩ C = Ba ∩ C. Thus {Ba } ∪ B is contained in a filter basis B B  B of cosets modulo closed normal subgroups of G which are members of M(G). By the maximality of B we conclude B  = B. This means B ⊇ Ba ∈ B, and if we set

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201

hB = a and B = Ba , then this is the assertion (‡). For a proof of the uniqueness of hB , assume that B contains also {hB } × GJ \EB ∈ B. Since B is a filterbasis, {hB } × GJ \EB ∩ {hB } × GJ \EB = Ø. This entails hB = hB and thus secures uniqueness. From (∗) we obtain for each pair C ⊆ B in B the comparison B = g(B)EB M(B)EB × GEC \EB × GJ \EC , × GJ \EC , C = g(C)EC M(C)EC

(∗∗) (∗∗∗)

Accordingly, we have the comparison B = {hB } × GEC \EB × GJ \EC , × GJ \EC , C = {hC }

(++) (+++)

From (++) and (+++) we have (∀B, C ∈ B) B ⊇ C ⇒ fEB EC (hC ) = hB .

(#)

Next we show that the updirected family {EB : B ∈ B} is cofinal in Fin(J ), that is, for each E ∈ Fin(J ) there is a B ∈ B such that E ⊆ EB . Since E is finite and {EB : B ∈ B} is up-directed, this is equivalent to saying that each j ∈ J is contained  in some EB , that is B∈B EB = J . By way of contradiction, suppose that this is not the case. Then there is a j ∈ J such that j ∈ / EB for all B. Then (∗) takes the following form: B = gEB MEB × Gj × GJ \(EB ∪{j }) . Then the filter basis of all sets B  = gEB MEB × {1j } × GJ \(EB ∪{j }) can be enlarged to contain B, and that contradicts the maximality of B. Thus {EB : B ∈ B} is cofinal in Fin(J ) as asserted.  Then condition (#) implies that there is a family  h = (hj )j ∈J ∈ G = j ∈J Gj such that hEB = (h j )j ∈EB ∈ G EB and thus h ∈ B∈B B. Since B ⊇ B for all B ∈ B we conclude h ∈ B. Thus B = Ø which is what we had to show to complete Observation (iii) and thereby Step c. In Step (c) we have shown that arbitrary products of families of almost connected pro-Lie groups are Z-compact. In Step (b) we have shown that if this is the case, then for any almost connected pro-Lie group G, the arbitrary strict projective limits G∼ = GM(G) = limN ∈M(G) G/M is Z-compact. This completes the proof of Claim (iv) of the proposition. The proof of assertion (iv) is delicate in so far as we have to keep clearly in focus that we want to prove compactness for the Z-topology. While it is true that an arbitrary

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product of almost  connected pro-Lie groups Gj is compact in the Tychonoff product topology of  j (Gj , Z(Gj )) this contributes nothing to the fact, proven in (iv), that  ( j Gj , Z( j Gj )) is compact, as this topology of this space in general is finer than the product topology. Exercise E4.2. Verify the details of the following examples. (i) On the groups G = Z and G = R the Z-topology is the cofinite topology, that is the topology whose closed sets are all finite sets and the set G. But Z fails to be compact in the topology having as a subbasis for its closed sets the collection {gN : g ∈ G, N ∈ N (Z)}. $ # [Hint. First observe that M(G) = {0}, Z . Regarding the second assertion, the filter basis {1 + p + p2 + · · · + p n−1 + pn Z | n ∈ N} consists of cosets of groups in N (Z) and has empty intersection in Z.] (ii) Let Rc be R with the cofinite topology. The product topology on R2c has a basis of open sets of the form R2 \ L where L is a finite union of straight lines each of which is either horizontal or vertical. Thus the diagonal of R2 is dense in R2c . It is, however, closed in R2 for the Z-topology of R2 . The diagonal is the graph of the identity morphism R → R. Thus graphs of morphisms continuous in the ordinary topology are not closed in the product space for the Z-topologies on the range and the domain in general. With this preparation we can prove the first part of the following theorem giving one set of sufficient conditions for a quotient group of a pro-Lie group to be a pro-Lie group. We complement this result by alternative sufficient conditions which are easily obtained by citing relevant literature, and by one condition whose sufficiency we shall establish much later in the book in Chapter 9. The Quotient Theorem for Pro-Lie Groups Revisited Theorem 4.28. Assume that G is a pro-Lie group and K is a closed normal subgroup. Then G/K is a pro-Lie group if at least one of the following conditions is satisfied: (i) (ii) (iii) (iv)

K is almost connected and G/G0 is complete. K satisfies the First Axiom of Countability. K is locally compact. G is almost connected and K is the kernel of a morphism whose image is a pro-Lie group.

Proof. (i) We accomplish the proof in two parts. In the first part we assume that G/G0 def

is compact, that is, that G is also almost connected. By Theorem 4.1, H = G/K is a proto-Lie group. Let f : G → H be the quotient morphism and let C be a Cauchy filter of H . We have to show that C converges. Since G is almost connected, M(G) is cofinal in N (G) by 4.6. For each N ∈ M(G) let N ∗ = f (N) and let pN ∗ : H → H /N ∗ be the quotient morphism. Then the image pN ∗ (C) in the Lie group H /N ∗ is a Cauchy filter and thus has a limit hN . In

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203

fact, (hN )N∈M(G) ∈ H{N ∗ |N ∈M(G)} = limN ∈M(G) H /N ∗ ∼ = HN (H ) ; indeed C has to def

converge to a point in the completion of H . Now let FN = (pN ∗ f )−1 (hN ). Then {FN | N ∈ N (G)} is a filter basis consisting of cosets of closed normal subgroups KN of G where K = ker f . We claim that KN ∈ M(G). Now KN ∈ N (G). Let M ⊆ KN in N (G) such that M ∼ KN, that is, M is open in KN. Then M ∩ K is open in K. Since K is almost connected, KM/M ∼ = K/(M ∩ K) is finite. We must show that KN/M is finite; for this purpose it is now no loss of generality, after replacing M by KM if necessary, to assume that K ⊆ M. Then M is open in MN, whence MN is closed and thus KN = MN . Now MN/M ∼ = N/(M ∩ N ) is discrete, that is M ∩ N is open in N whence M ∩ N ∼ N . Now N ∈ M(G) implies that KN /M = MN/M ∼ Since = N/(M ∩ N) is finite. Thus KN ∈ M(G) as claimed.  G is compact in the Z-topology by Proposition 4.8 (iv), there is a g ∈ N ∈N (G) FN . Then pN ∗ (f (g)) = hN for all N ∈ N (G) from which we deduce that f (g) = lim C. Thus every Cauchy filter in H converges and so H is complete. By (i), H is a proto-Lie group, and thus by the definition of pro-Lie groups (3.25) we have shown that H is a pro-Lie group. In the second part of the proof we assume that G/G0 is complete. As a quotient of K/K0 , the factor group K/(K ∩ G0 ) ∼ = (K/K0 )/((K ∩ G0 )/K0 ) is compact. Moreover, in the factor group G/(K ∩ G0 ), the subgroup G0 /(K ∩ G0 ) is closed. Hence the product (G0 K/(K ∩ G0 )) · (K/(K ∩ G0 )) = G0 K/(K ∩ G0 ) is closed in G/(K ∩ G0 ) and therefore G0 K is closed in G and so is a pro-Lie group by the Closed Subgroup Theorem 3.35. The factor group G0 K/G0 is a continuous homomorphic image of the compact group K/(K ∩ G0 ) under the morphism k(K ∩ G0 )  → kG0 and consequently is a compact group. Therefore G0 K is an almost connected proLie group. Then by Part 1 of the proof, G0 K/K is a pro-Lie group. Further G/G0 is complete by assumption. Since K/(K ∩ G0 ) is compact, its homomorphic image G0 K/K is compact. Hence G/G0 K ∼ = (G/G0 )/(G0 K/G0 ) is complete by (iii) below. Now the factor group G/K has the complete normal subgroup G0 K/K such that (G/K)/(G0 K/K) ∼ = G/G0 K is complete, and it is therefore complete by [176, p. 225, 12.3]. Thus the proof of (i) is complete. (ii) and (iii): In view of [176, p. 242, Lemma 13.13], both of these are special cases of the sufficient condition given in [176, p. 206, Theorem 11.18]. (iv) will be established later in Chapter 9 in Lemma 9.57. Of course, if G itself satisfies the First Axiom of Countability, then (ii) follows. But there are simpler alternative arguments in this case. Recall that a topological group is metrizable if and only if it satisfies the First Axiom of Countability. (See for instance [102, Theorem A4.16].) Quotients of complete metrizable topological groups are complete because Cauchy sequences can be lifted. (See [26, Chap. 9, §3, no 1, Proposition 4].) The assertion that G/N is a pro-Lie group then follows from 4.1 (i). Naturally, pro-Lie groups often fail to be metrizable. Dierolf and Roelcke explain in [176] that there is a common cover for the two conditions of metrizability and local compactness; metrizable and locally compact spaces are almost metrizable, and it is true that for a complete topological group G and

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an almost metrizable closed normal subgroup N the factor group G/N is complete. For the concept of almost metrizability we refer to [176, p. 241, Definition 13.11]. We mention that it follows from the work of Wigner [201] that there is yet another sufficient condition for the conclusion of Theorem 4.28 to hold: (ii∗ ) K is isomorphic to an arbitrary product of groups each of which satisfies the First Axiom of Countability. This condition trivially implies (ii) and is implied by (ii∗∗ ) K is isomorphic to an arbitrary product of Lie groups. We shall see in Chapter 8 that a simply connected reductive Lie group is a product of simply connected simple Lie groups and copies of R (see Theorem 8.14 and Corollary 8.15). (ii∗∗ ) is implied by (ii∗∗∗ ) K is a simply connected reductive pro-Lie group. As we already noted above in comments subsequent to the example in Corollary 4.11, in general, the quotient of a complete topological group is not necessarily complete; in the case of additive groups of topological vector spaces this was observed at an early stage ([132], [75]). Grosso modo, the completeness of quotient groups of a complete topological group G is a delicate matter, and this is why, in order to find sufficient conditions appropriate for the category proLieGr, somewhat protracted arguments were needed in the proof that Condition (i) of Theorem 4.28 is sufficient for completeness. The Axiom of Choice is buried in the proof of Z-compactness, e.g. in the Alexander Subbasis Theorem. The arguments we cited from the book of Dierolf and Roelcke are all but trivial. Only the case that G is assumed to be metrizable in 4.28 (ii) is simpler, but this is also the most restrictive condition. The following question is in a sense a converse to that which is answered by the Closed Subgroup Theorem 1.34 and the Quotient Theorem 4.1: Let G be a topological group and N a closed normal subgroup of G such that both N and G/N are pro-Lie groups. Under what circumstances is G a pro-Lie group? Among the following three examples, the first two illustrate that this fails to be the case under the rather simple circumstances of semidirect products. The third example however, showing that every pro-Lie group has a tangent bundle, illustrates that often semidirect products do work well also for pro-Lie groups. Examples 4.29. (i) Let L be a nontrivial compact Lie group and let Z act automordef

phically on P = LZ by the shift. Set G = P  Z. Then N = P × {0} is a compact normal pro-Lie subgroup, and G/N is discrete, and thus a Lie group. But G is not a pro-Lie group.

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(ii) Let D be a discrete group, and A a compact nontrivial group of automorphisms of D. Then the compact group AN acts automorphically on the discrete group V = D (N) via (an )n∈N · (dn )n∈N = (an dn )n∈N . Set G = V  AN . Then N = V × {1} is a discrete normal subgroup, thus in particular a normal Lie subgroup. The factor group G/N ∼ = AN is compact and therefore a pro-Lie group. But G is not a pro-Lie group. def (iii) For each pro-Lie group G, the semidirect product T (G) = L(G) Ad G is a well-defined pro-Lie group. Proof. Exercise E4.3. Exercise E4.3. Prove the assertions made in the description of the Examples 4.29 (i) and (ii). [Hint. (i) G has arbitrarily small identity neighborhoods of the form, U = LN\F × {1} for some finite set F ⊆ Z (where we identify LA in an obvious fashion with a subgroup of ZN for any subset A of Z). Assume that F = Ø and x = (xn )n∈Z ∈ LN\F are such that xn = 1. If xn−m ∈ F then a shift of (x, 1) is outside U . Hence U does not contain any nondegenerate normal subgroups of G. (ii) G has arbitrarily small identity neighborhoods U = {1} × AN\F with a finite subset F of N. If a = (an )n∈N ∈ AN with an = 1, then there is a d ∈ D such that / {1} × A, and thus if an (d) = d. Then (d, 1)(1, an )(d −1 , 1) = (d · an (d)−1 , an ) ∈ v = (dj )j ∈N with dj = 1 for j = n, then (v, 1)(1, a)(v −1 , 1) ∈ / {1} × AN . So U does not contain any nondegenerate normal subgroup. (iii) Any identity neighborhood of T (G) contains one of the form U ×V where U is a zero neighborhood of L(G) and V is an identity neighborhood of G. In 4.21 (ii) we see that there is a closed ideal j of L(G) such that j ⊆ U and dim L(G)/j < ∞. Likewise there is a normal subgroup of G such that N ⊆ V and G/N is a Lie group. Then j×N is a normal subgroup of T (G) such that j×N ⊆ U ×V and T (G)/(j×N ) ∼ = L(G)/jG/N is a Lie group. Thus T (G) is a pro-Lie group, as asserted.] In the proof of (iii), the reader will notice very quickly that j and N may be chosen in such a way that j = L(N ) so that in fact, setting M = L(N ) × N, by what we shall see in Corollary 4.22, T (G)/M ∼ = L(G/N ) AdG/N G/N . Now T (G/N ) = L(G/N) AdG/N G/N is the tangent bundle of the Lie group G/N . Thus T (G) is approximated by the tangent bundles of the Lie group quotients G/N of G. We record some basic facts on the natural relationship between pro-Lie groups and proto-Lie groups. Exercise E4.4. The category of proto-Lie groups and morphisms of topological groups between them is denoted by protoLie. For a topological group G we write L(G) = GN (G) = limN ∈N (G) G/N. Recall that we have a morphism γG : G → L(G) (see 1.29, 1.39) Prove the following facts: (i) For any topological group G, the topological group L(G) is a pro-Lie group. (ii) γG is an embedding if and only if G is a proto-Lie group and an isomorphism if and only if G is a pro-Lie group.

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(iii) For each morphism f : G → H of proto-Lie groups into a pro-Lie group H there is a unique morphism f  : L(G) → H such that f = f   γG . protoLie γG

proLieGr

G ⏐ ⏐ ∀f 

−−→

L(G) ⏐ ⏐  f

L(G) ⏐ ⏐  ∃!f

H

−−→

H

H.

idH

(iv) The assignment G  → L(G) on proto-Lie groups extends to a functor L : protoLie → proLieGr which is left adjoint to the inclusion functor proLieGr → protoLie. (See e.g. [102, Appendix 2, Theorem A3.28ff.].) [Hint. (i) We have L(G) = limN ∈N (G) G/N where G/N is a Lie group for each N by the definition of N (G). In particular, L(G) is a complete topological group. The limit maps νN : L(G) → G/N are quotient morphisms inducing isomorphisms  | N ∈ N (G)}  → G/N (with N  = ker νN ) by 1.29 (iii). Then by 1.27 (ii) {N L(G)/N  is is a filter basis of closed normal subgroups of L(G) converging to 1 and L(G)/N a Lie group for all N ∈ N (G). Now Proposition 3.27 shows that L(G) is a pro-Lie group. (ii) This follows from 3.26. (iii) Assume now that f : G → H is a morphism of proto-Lie groups. Since H is a pro-Lie group, it is complete. Since G is a proto-Lie group, it has a completion by 3.26 and γG : G → L(G) is the completion morphism. Hence there is a unique extension of f to a morphism f  : L(G) → H . (iv) This is now a consequence of (iii) and Theorem A3.28 of [102]. On a larger scale, the inclusion functor from the category pro-Lie groups proLieGr into the category TopGr of topological groups has a left adjoint L : TopGr → proLieGr given by the construction of Exercise E4.5.; in Chapter 2 this functor was derived from more category theoretical principles in 2.25 (iii). There is a full subcategory of the category protoLie which arises naturally from the fact that a quotient group of a pro-Lie group is a proto-Lie group. The following exercise comments on this category. Exercise E4.5. Let Q denote the full subcategory of TopGr of all topological groups which are quotient groups of pro-Lie groups. Prove the following facts about this category: (i) Q is a (full) subcategory of protoLie. (ii) Q is closed in TopGr under the formation of products and retracts. (iii) Q is closed under the formation of closed subgroups. (iv) Q is closed under the formation of quotient groups. (v) The restriction of the functor P : protoLie → proLieGr to Q is left adjoint to the inclusion functor.

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[Hint. (i) is a consequence of the Quotient Theorem for Pro-Lie Groups 4.1 (ii): Show that a product of quotient morphisms is a quotient morphism and that the composition of quotient morphisms is a quotient morphism; use these facts to show that Q is closed in TopGr under the formation of products and retracts. (iii) If H /N is a closed subgroup of G/N, then H is a closed subgroup of G and if G is a pro-Lie group, then H is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35. Assertions (iv) and (v) are straightforward.] Exercise E4.6. Recall that we say that a topological group has no small normal subgroups if there is an identity neighborhood not containing any subgroups except the singleton one. Let G be a pro-Lie group, N a closed normal subgroup, and qN : G → G/N the associated quotient morphism. Then we have, on the level of sets, an injective function Hom(qN , H ) : Hom(G/N, H ) → Hom(G, H ), Hom(qN , H )(f ) = f  qN . If G is a pro-Lie group and N ⊆ M in N (G), giving an associated quotient morphism qMN : G/N → G/M, then we have an associated injective function Hom(qMN , H ) : Hom(G/M, H ) → Hom(G/N, H ),

M ⊆ N, M, N ∈ N (G).

This is an injective system of sets and functions, and we obtain an injective limit colimN∈N (G) Hom(G/N, H ) for which the maps Hom(qN , H ) : Hom(G/N, H ) → Hom(G, H ) provide an injective function iN (G) : colim Hom(G/N, H ) → Hom(G, H ). N ∈N (G)

We shall identify colimN ∈N (G) Hom(G/N, H ) with a subset of Hom(G, H ), which means identifying a morphism f : G/N → H with its canonical composition f qN ∈ Hom(G, H ). Prove the following statement: Proposition A. A proto-Lie group is a Lie group if and only if it has no small subgroups. [Hint. Let U be an identity neighborhood of G which contains no nondegenerate subgroups. Since G is a proto-Lie group, there is an N ∈ N (G) that is contained in U and thus satisfies N = {1}.] Proposition B. If G is a pro-Lie group, and H is a group without small subgroups, then colim Hom(G/N, H ) = Hom(G, H ). N ∈N (G)

[Hint. Let f : G → H be a member of Hom(G, H ) and let U be an identity neighborhood of G which does not contain a nondegenerate subgroup of H . Then f −1 (U ) is an identity neighborhood of G which, according to lim N (G) = 1 contains some N ∈ N (G). Then f (N) is a subgroup of H contained in U and thus is {1}, and thus f factors in the form f = f   qN . Therefore f ∈ Hom(G/N ) after our identification of f and f  .]

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Quotients and Quotient Maps between Pro-Lie Groups In Theorem 4.1 (iv) we proved the next best thing to an Open Mapping Theorem without having made a special definition. Let us do this here. Definition 4.30. (i) A morphism f : G → H of topological groups is called relatively open if the set {f (U ) | U is an identity neighborhood of G} is a basis for the filter of identity neighborhoods of H . (ii) A morphism f : G → H from a topological group to a pro-Lie group will be called pro-open, if for each M ∈ N (H ) the function g  → f (g)M : G → H /M is open. An open morphism is a relatively open morphism, and if the range is a pro-Lie group, it is pro-open. Let G be the additive group of real numbers endowed with the following topology: consider on Q the topology induced by the natural topology of R and let the open identity neighborhoods of Q form a basis for the identity neighborhoods of R. This is a metrizable group topology on R. If H = R with its natural topology, and if f : G → H is the identity map then f is a surjective morphism of topological groups which is relatively open but fails to be open. If H = Tp is the p-adic solenoid p

p

p

lim{T ←− T ←− T ←− · · · } then H is a pro-Lie group. Let Zp be the subgroup . / 1 p 1 p 1 p lim Z/Z ←− 2 Z/Z ←− 3 Z/Z ←− · · · . p p p This group and the group Zp of p-adic integers are isomorphic. Let D be the group Zp with its discrete topology and set G = D × R. Then G is an abelian finite-dimensional Lie group (hence a pro-Lie group) with uncountably many components. Define f : G → H by f ((zn + Z)n∈N , r) = (zn + p −n r + Z)n∈N . Then f is a surjective morphism of topological groups whose kernel is isomorphic to Z (see [102, Example E1.11 following Definition 1.30]). Let πn : Tp → T be the n-th projection. Then 3 4 2 ε ε (πn f )({0}×] − ε, ε[) = − n , n + Z Z ⊆ T, p p and thus f is pro-open. But it fails to be relatively open, let alone open. In a remark following the proof of Theorem 4.1 we mentioned a nondiscrete proLie group topology τ on the countable group Z(N) making the identity map from the discrete group Z(N) to (Z(N) , τ ) a bijective morphism between two countable pro-Lie groups that fails to be open.

Quotients and Quotient Maps between Pro-Lie Groups

209

Lemma 4.31. (The Pro-Open Mapping Theorem for Pro-Lie Groups) (i) Let G be a pro-Lie group with arbitrarily small elements N ∈ N (G) such that the Lie group G/N has countably many components. This is the case if G has only countably many components. If f : G → H is a surjective morphism of topological groups onto a pro-Lie group H , then f is pro-open. (ii) If f : G → H is a pro-open morphism of pro-Lie groups such that f (N ) ∈ N (H ) for all N ∈ N (), then f is relatively open. (iii) If, in addition to the hypotheses of (i) above, f is locally closed (in the sense that for any identity neighborhood W of G there is an identity neighborhood U contained in W such that f (U ) is closed), then f is open. Proof. Exercise E4.7. Exercise E4.7. Verify Lemma 4.31. [Hint. (i) This was in effect proved in Theorem 4.1 (iv). (ii) We have to show that for arbitrarily small identity neighborhoods U of G the closure of the image f (U ) is an identity neighborhood of H . Since G is a pro-Lie group, by Theorem 1.27 (i), we may assume that we are given an open identity neighborhood U of G for which there is an N ∈ N (G) such that U N = N U = U . Then by (i), f (U )N ∗ /N ∗ is an identity neighborhood of H /N ∗ and thus f (U )N ∗ = f (U )f (N ) ⊆ f (U )f (N) = f (U N) = f (U ) is an identity neighborhood of H . (iii) is an immediate consequence of (i) and (ii).] This is all we shall say here in the line of the Open Mapping Theorem. Notice that a morphism is locally closed if it is a closed map; this is the case for instance if it is a proper map. Definition 4.32. A topological group P is called procountable, if the filter basis of all open subgroups N such that P /N is countable converges to the identity.  An arbitrary product P = j ∈J Dj of any family of countable discrete groups Dj is procountable. Indeed, a complete topological group is procountable iff it is isomorphic to a closed subgroup of a product of a family of countable discrete groups. Lemma 4.33. Let P be a procountable group and f : P → H a morphism of topological groups into a connected pro-Lie group G. Then f (P ) is totally disconnected. def

Proof. Let A = f (P )0 . We have to show that A is singleton. From the Closed Subgroup Theorem 3.35 we know that A is a connected abelian pro-Lie group and that f −1 (A) is a procountable subgroup of P . For the purposes of the proof we may assume that P = f −1 (A), that is, that A = f (P ). Suppose now that A is nonsingleton. Then we have a quotient morphism q : A → L where L is a nonsingleton connected Lie def

group. Let j : A → A be the inclusion morphism. Then F = q  j  f : P → L is morphism into a Lie group with a dense connected image F (P ). Now let U be a zero neighborhood in L such that {0} is the only subgroup contained in U . Since P is procountable, there is an open normal subgroup N contained in F −1 (U ) such that

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4 Quotients of Pro-Lie Groups

P /N is countable. Then F (P ) = q(A) is a homomorphic image of P /N and is, therefore, a dense countable connected subgroup of L. But any countable subspace of a connected metrizable space is totally disconnected (Exercise E4.9 below). Hence q(A) is singleton. So L is singleton, contrary to what we have supposed. Exercise E4.8. A countable subset of a metrizable space is totally disconnected. [Hint. Let (X, d) be a metric space and C a countable subset. We may assume that X has at least two different points. Let c ∈ C and define f : X → R+ = [0, ∞[ by f (x) = d(c, x). Then f is continuous, f (X) ⊆ R+ is connected and is therefore a nondegenerate interval Ic containing 0 = f (c). The set f (C) is countable; hence Ic \f (C) is uncountable and clusters toward 0. There is a sequence (rn )n∈N in Ic \f (C) such that rn < n1 . Then the open balls of radius rn in C form a basis of open-closed neighborhoods of c in C.] We note that, more generally, a connected space on which the continuous real valued functions separate the points, has to have a cardinality that is at least that of the continuum. There are countable connected Hausdorff spaces. (See for instance [52, pp. 352, 353].)

Postscript There are two substantial results in this chapter. The first is that the functor L preserves quotient morphisms. More precisely, let f : G → H be a quotient morphism of topological groups and assume that G is a pro-Lie group. Let Y ∈ L(H ) be a one parameter group of H . Then there is a one parameter group X ∈ L(G) such that f  X = L(f )(X) = Y , that is, the morphism L(f ) : L(G) → L(H ) induced on the Lie algebra level is surjective. In particular, if f is a quotient morphism between pro-Lie groups then L(f ) is a quotient morphism of pro-Lie algebras. Accordingly, the functor L : proLieGr → proLieAlg not only preserves all limits, but it also preserves quotient morphisms. This is a nontrivial fact which, not surprisingly, has substantial consequences. Its proof requires the Axiom of Choice. It is important to ask whether a quotient of a pro-Lie group is a pro-Lie group. Unfortunately, the answer is in the negative, because quotient groups of complete topological groups are not necessarily complete and may not even have a completion. Indeed Sipacheva [184] showed that every topological group is a quotient group of a complete topological group. However, the quotient of a pro-Lie group is always a proto-Lie group, has a completion, and this completion is a pro-Lie group. Moreover, if f : G → H is a morphism of pro-Lie groups, and if q : G → Q is the natural dense morphism from G to the pro-Lie group completion of G/ ker f , then there is a unique morphism of pro-Lie groups f  : Q → H such that f = f  q. This indeed looks like a good universal property for a quotient object Q even though q is not surjective as we expect a quotient morphism to be. Unfortunately, f  may not be injective either. Perhaps the failure of such a category as the category of pro-Lie groups to behave neatly

Postscript

211

with respect to passing to quotient objects makes it plausible that there may be no easy category theoretical approach to quotients on the level of objects and arrows. Surely the simplest connected pro-Lie groups are the products of reals RX . The quotients of these are complete if X is countable, and we show in this chapter that RX has incomplete quotients as soon as card X ≥ 2ℵ0 . Surprisingly, this all happens in a context which we think we are familiar with, to wit, the context of compact connected abelian groups and their exponential function. So, in contrast with quotients of locally compact groups which are always locally compact, and in contrast with quotients of classical Lie groups which are always Lie groups, quotients of pro-Lie groups often fail to be pro-Lie groups – and this happens already for the additive groups of weakly complete topological vector spaces. In such circumstances it is important to have sufficient conditions under which quotients of pro-Lie groups are complete. Classically, first countable complete topological groups have complete quotients. We demonstrate a result which pertains to connected and indeed more generally to almost connected pro-Lie groups. Here we have used the terminology that a topological group G is almost connected if the factor group G/G0 modulo the identity component is compact. Indeed we show that the quotient group G/N of a pro-Lie group G modulo an almost connected closed normal subgroup N is a pro-Lie group again provided that the protodiscrete factor group G/G0 is complete. (see Lemma 3.31). The proof of this fact is not superficial. It uses the Axiom of Choice as well. We do not know an example of a pro-Lie group G such that G/G0 fails to be complete. The literature on topological groups such as [176] provides us with the information that a factor group G/N of a pro-Lie group G is a pro-Lie group whenever N is locally compact. In Chapter 9 in Theorem 9.60 we shall prove a fairly good Open Mapping Theorem for pro-Lie groups which applies whenever the domain is almost connected. We mentioned this in Theorem 4.28. In the meantime we include a weaker result showing that a surjective morphism from a pro-Lie group G with countably many components onto a pro-Lie group is “pro-open” where the pro-openness of a morphism between pro-Lie groups is a substitute for openness.

Chapter 5

Abelian Pro-Lie Groups

We saw in [102] that compact abelian groups are not simply an informative, important, and interesting special case of more general compact groups, but rather an integral part of the structure of these groups. Even at the superficial level we know that the closure of any cyclic group or of any one parameter subgroup in a locally compact group is either isomorphic to Z, respectively, to R, or else is a compact abelian subgroup. A similar situation persists in the case of pro-Lie groups. We know that in any pro-Lie group the closure of any cyclic subgroup or any one parameter subgroup is an abelian pro-Lie group. Further every locally compact abelian group is an abelian pro-Lie group. A topological group is called a weakly complete vector group if it is isomorphic to the additive group of a weakly complete topological vector space, and therefore is isomorphic to RJ for some set J . (See Corollary A2.9 in Appendix 2.) We shall see that the structure of connected abelian pro-Lie groups reduces completely to that of weakly complete vector groups and compact connected abelian groups for which we have a rather exhaustive structure and Lie theory (as shown, for instance, in [102, Chapters 7 and 8]). As the structure theory of pro-Lie groups unfolds we shall have to recall results which we prove in this chapter. There are certain aspects of the duality theory of locally compact abelian groups which extend to abelian pro-Lie groups although this larger class does not have a perfect duality theory outside the domain of connected groups. We shall discuss duality theory as well. Prerequisites. We need the general theory of pro-Lie groups up to and including Chapter 4 and elements of the duality theory of topological abelian groups such as it is, for instance, presented in [102, Chapter 7]. In particular, the theory of weakly complete topological vector spaces and their duality theory is assumed to be known (see Appendix 2).

Examples of Abelian Pro-Lie Groups We begin by offering some orientation of the class of abelian pro-Lie groups by presenting a list of examples. Let us firstly recall (see for instance [199]) the following basic types of locally compact nondiscrete fields: (a) The field R of real numbers. (b) The field Qp of p-adic rationals for some prime p. (c) The field GF(p)[[X]] of Laurent series in one variable with the exponent valuation over the field with p elements.

Examples of Abelian Pro-Lie Groups

213

All other nondiscrete locally compact fields are finite extensions of these; in cases (a) and (b) the characteristic is 0 and in case (c) it is finite. Of course, every field F with the discrete topology is a locally compact field.  ∞ 1  Let Z(p∞ ) = n=1 p n · Z /Z denote the Prüfer group of all elements of p-power order in T = R/Z. We consider on the group Z of integers, the group Q of rationals, and the Prüfer group Z(p∞ ) as endowed with their discrete topologies. Examples 5.1. Let J be an arbitrary infinite set. The following examples are abelian pro-Lie groups. (i) All locally compact abelian groups. (ii) All products of locally compact abelian groups, specifically: (a) the groups RJ ; (b) the groups (Qp )J ; (c) the groups QJ for the additive group of rational numbers Q with its discrete topology; (d) the groups ZJ for the group Z of integers with its discrete topology; (e) the groups Z(p∞ )J . An infinite product of noncompact locally compact groups is not locally compact, so, for infinite J , none of the groups in (ii)(a)–(ii)(e) is locally compact, but if J is countable, they are Polish (that is, completely metrizable and second countable). A countable product of discrete infinite countable sets in the product topology is homeomorphic to the space R \ Q of the irrational numbers in the topology induced by R (see [26, Chap. IX, §6, Exercise 7]). So QN , ZN , and Z(p∞ )N are abelian prodiscrete groups on the Polish space of irrational numbers. These elementary examples show, that the category of abelian pro-Lie groups is considerably larger than that of locally compact groups. The groups in (ii)(a)–(ii)(c) are divisible and torsion-free, and the groups in (ii)(e) are divisible and have a dense torsion group. Recall from the Closed Subgroup Theorem for Pro-Lie Groups 3.35 that every closed subgroup of a pro-Lie group is a pro-Lie group, and from the Completeness Theorem for Pro-Lie Groups 3.36 that the product of any family of pro-Lie groups is a pro-Lie group. It is then clear that from even simple examples a wealth of examples can be easily constructed. There is a less obvious but very instructive example which we present separately. This example is based on the example which we discussed in Chapter 4 right after the Quotient Theorem for Pro-Lie Groups 4.1: see Proposition 4.2 and the discussion leading up to this proposition and all the information presented subsequently through the Characterisation Theorem for Strong Local Connectivity of Compact Connected Abelian Groups 4.14. This discourse is due to Poguntke and the authors as described in [106]. However, for the present example we need only what we shall specify explicitly here.

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5 Abelian Pro-Lie Groups

If J is any set and j ∈ J , then δj : J → R is defined by " 1 if x = j , δj (x) = 0 otherwise. For any subset S ⊆ RJ let S denote the subgroup algebraically generated by S as is usual. Proposition 5.2. The free abelian group Z(N) has a nondiscrete topology making it into a prodiscrete abelian group F in such a fashion that the following conditions are satisfied: (i) There is an injection j : F → W mapping F isomorphically (algebraically and topologically) onto a closed subgroup of the weakly complete vector group W of topological dimension 2ℵ0 (that is, W ∼ = RR ) such that W/F is an incomplete group whose completion is a compact connected and locally connected group, and that the R linear span span F is dense in W . def

(ii) The subset B = {δn : n ∈ N} satisfies F = B and j (B) is unbounded in W . (iii) If K ⊆ F is any compact subset, then there is a finite subset M ⊆ N such that k ∈ K ⊆ Z(N) implies that the support supp(k) = {m ∈ N : k(m) = 0} is contained in M. In particular, every compact subset of F is contained in a finite rank subgroup of F . Proof. (i) Let G be the character group of the discrete group ZN . Then G is a compact, connected and locally connected but not arcwise connected group. Let Ga denote the arc component of the identity element in G. It was proved in [106] that the corestriction of the exponential function expG : L(G) → Ga is a quotient map. Let W = L(G) ∼ = ℵ0 Hom(ZN , R) ∼ = R2 and F = ker expG ⊆ L(G) = W . The exponential function of (locally) compact abelian groups is extensively discussed in [102, Proposition 7.36ff.]; in [102, Theorem 7.66], the kernel of the exponential function is denoted by K(G). Now F as a closed subgroup of a pro-Lie group is a pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups. It is totally disconnected (see [102, Theorem 7.66 (ii)]), and so by Lemma 3.31 F is a prodiscrete group. By [102, Proposition 7.35 (v)(d)], the linear span span F = span K(G) is dense in L(G) = W . Since Ga does not contain any copy of a cofinite-dimensional closed vector subspace of L(G), the function expG : L(G) → Ga cannot induce a local isomorphism and thus its kernel F = K(G) is not discrete. Furthermore, from [102, Theorem 7.66 (ii)] we observe that F ∼ = Hom(ZN , Z), algebraically, and from [49, p. 53 Corollary 15 and p.560, Corollary 24], we get that  : Z(N) → Hom(ZN , Z), ((pm )m∈N )((zm )m∈N ) = m∈N pm zm is an isomorphism of abelian groups and that, accordingly, F is a free group algebraically generated by (B).

Weil’s Lemma

215

(ii) We now prove that (B) ⊆ Hom(ZN , Z) ⊆ Hom(ZN , R) = W is unbounded. Note that (δn ) : ZN → Z is simply the evaluation evn : ZN → Z, evn (f ) = f (n). Since ZN is considered with the discrete topology, the topology on W ∼ = Hom(ZN , Z) N is that of pointwise convergence, that is, the topology induced from ZZ . Let s : N → Z be an arbitrary element of ZN . Then pr s : Hom(ZN , Z) → Z is given by pr s (ϕ) = ϕ(s) and thus pr s ((B)) = {evn (s) : n ∈ N} = {s(n) : n ∈ N} = im s. Therefore the projection of B into the s-component of Hom(ZN , Z) ⊆ Hom(ZN , R) ∼ = W is bounded if and only if s is bounded. Since there are unbounded elements s ∈ ZN , the set j (B) is unbounded in W . (iii) Let K ⊆ Z(N) be a compact subset and suppose that it fails to satisfy the claim. Then there is a sequence of elements kn ∈ K such that Mn = max supp(kn ) is a strictly increasing sequence. Now we define a function s : N → Z recursively as follows: Set s(1) = 0 and s(m) = 0 for m ∈ / {Mn : n ∈ N }. Assume that s(m) has been defined 5 m for 1 ≤ m ≤ Mn in such a way that σm = M j =1 km (j )s(j ) ≥ m. Now solve the inequality def

n + 1 ≤ σn+1 =

Mn 6

kn+1 (j )s(j ) + kn+1 (Mn+1 )s(Mn+1 )

j =1

#5 $ ). The s-projection pr ((K)) = k(n)s(n) : k ∈ K ⊇ {σn = for s(M n+1 s n∈ N 5 k (m)s(m) : m ∈ J } contains arbitrarily large elements σ ≥ n and thus cannot n m∈N n be bounded, in contradiction to the compactness of K. We realize that (ii) is implied by (iii); we preferred to give separate proofs for better elucidation of this remarkable example. The group F cannot be metrizable, because as a complete abelian group its underlying space would be a Baire space and thus as a countable topological group would have to be discrete. It is therefore noteworthy that there are pro-Lie groups whose underlying space is not a Baire space. As a countable group, it is trivially σ -compact, that is the countable union of compact sets. We observe in conclusion of our brief discussion of examples of abelian pro-Lie groups that we have seen prodiscrete abelian groups on the space R \ Q of irrationals and on a countable nondiscrete space.

Weil’s Lemma In the domain of locally compact groups, Weil’s Lemma says: Let g be an element of a locally compact group and g the subgroup generated by it. Then one (and only one) of the two following cases occurs (i) n  → g n : Z → g is an isomorphism of topological groups. (ii) g is compact.

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Exercise E5.1. Prove Weil’s Lemma. [Hint. We prove minutely more: Let E = R or E = Z and consider a morphism X : E → G. There is no loss in assuming that G = X(E), that is, that G is abelian and X has a dense image and so X is an epimorphism of abelian topological groups. Let K = ker X. The closed subgroup K of E is either cyclic or R. Thus E/K is singleton, a circle, or finite, or isomorphic to E. Since X factors through the quotient map E → E/K with a morphism X : E/K → im X, E ⏐ ⏐ quot E/K

X

−−→ −−→

A  ⏐ ⏐incl im X,

X

E/K and thus im X is compact in the first two cases. In these cases G is compact and we are in Case (ii) since im X is dense in G. We may and shall assume henceforth that X is injective and thus is a monic of topological abelian groups. There are elementary proofs of Weil’s Lemma in the literature, see for instance [102, Proposition 7.43]. A proof using the duality of locally compact groups and a : G →E  is a monic and epic of locally bit of Lie theory is as follows. The dual X compact abelian groups and thus is injective and has dense image. Since R has no  has no nonsingleton compact subgroups and thus, nonsingleton compact subgroups, G in particular, has no small subgroups. Hence it is of the form Rp × Tq × D for a  is an injective torsion-free discrete group D. Then p + q ≤ 1 (considering that X p+q  linear map from R → R). Case p + q = 0: Then A is discrete and so A is  maps A 0 ∼ compact. Case p = 1: Here the injective map X = R isomorphically onto ∼  E = R, and D = 0 follows. Case q = 1: Similarly, A = E = T and thus A ∼ = Z.] Weil’s Lemma for Pro-Lie Groups Theorem 5.3. Let E be either R or Z and X : E → G a morphism of topological groups into a pro-Lie group. Then one (and only one) of the two following cases occurs: (i) r  → X(r) : E → X(E) is an isomorphism of topological groups. (ii) X(E) is compact. def

Proof. The closed subgroup A = X(E) is an abelian pro-Lie group by the Closed Subgroup Theorem for Pro-Lie Groups 3.35. Thus we may assume that G is abelian and the image of X is dense. Now let N ∈ N (G) and pN : G → G/N the quotient map. Then G/N is an abelian Lie group for a morphism pN X : E → G/N with dense image. By Weil’s Lemma for locally compact groups, either pN  X is an isomorphism of topological groups or else G/N is compact. If M ⊇ N in N (G) and pN  X is an isomorphism, then pM  X and the bonding map G/M → G/N are isomorphisms as well. Thus there are two mutually exclusive cases:

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217

(A) (∀N ∈ N (G)) pN  X is an isomorphism of topological groups and all bonding maps G/M → G/N are isomorphisms. (B) There is a cofinal subset Nc (G) ⊆ N (G) such that G/N is compact. In Case (A), the limit G ∼ = limN ∈N (G) G/N is isomorphic to E. In Case (B) let N ∈ N (G) and let ↑N = {P ∈ N (G) : P ⊆ N }. Then for all P ∈ N (G) the quotient G/P is compact. By the Cofinality Lemma 1.21 (ii) we know G∼ = limP ∈↑N G/P . Since all G/P are compact for P ∈ ↑N , the group G is compact in this case, and the theorem is proved. Definition 5.4. (i) Let G be a topological group. Then comp(L)(G) denotes the set of all X ∈ L(G) such that expG R · X is compact. A one parameter subgroup X ∈ comp(L)(G) is called a relatively compact one parameter subgroup. Furthermore, comp(G) denotes the set {x ∈ G : x is compact}. An element x ∈ comp(G) is called a relatively compact element of G. (ii) A topological abelian group G is said to be elementwise compact if G = comp(G). It is said to be compact-free if comp(G) = {0}. In any topological abelian group G, the set comp(G) of relatively compact elements is a subgroup, since g, h ∈ comp(G) implies that gh is contained in the compact subgroup gh. A discrete abelian group is elementwise compact if and only if it is a torsion group; it is compact-free if and only if it is torsion-free. If f : G → H is a morphism of abelian pro-Lie groups, then f (comp(G)) ⊆ comp(f (G)) ⊆ comp(H ); accordingly f (comp(G)) ⊆ comp(H ). The quotient morphism f : Z → Z/2Z shows that in general f (comp(G)) = comp(H ) even for quotient morphisms. Theorem 5.5. Let G be an abelian pro-Lie group. Then (i) comp(G) is a closed subgroup of G and therefore an elementwise compact proLie group in its own right. (ii) comp(G) ∼ = limN ∈N (G) comp(G/N ). (iii) For N ∈ N (G), let CN ⊆ G be the closed subgroup of G containing N for  which CN /N = comp(G/N ). Then comp(G) = N ∈N (G) CN and comp(G) = limN∈N (G) CN /N (iv) The factor group G/ comp(G) is compact-free.  Proof. (i) Since G ∼ = limN ∈N (G) G/N  ⊆ N ∈N (G) G/N we may assume that G is a closed subgroup of a product P = j ∈J Lj of abelian Lie groups Lj . Any abelian Lie group is of the form L = Rp × Tq × D for a discrete group D (see for instance [102, Corollary 7.58 (iii)]). Then comp(L) = {0}× Tq ×tor D for the torsion subgroup tor D of D. Hence comp(L) is closed. Thus pr j (comp(G)) ⊆ pr j (comp(G)) ⊆ comp(Lj ) as comp(Lj ) is closed. Now let g = (gj )j ∈J ∈ comp(G) ⊆ P . Then gj ∈ comp(Lj ) def  and thus gj  is compact and g ∈ K = j ∈J gj . As a product of compact groups, K is compact and thus g ⊆ K is compact and thus g ∈ comp(G). Hence comp(G) is closed and thus a pro-Lie group by the Closed Subgroup Theorem 3.35.

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(ii) The assignment G → comp(G) defines a functor from the category of abelian pro-Lie groups to itself. If G is a pointwise compact abelian pro-Lie group and H is any abelian pro-Lie group, then any morphism of topological groups f : G → H factors through comp(H ), that is, is of the form f = inclcomp(H ),H f  with a unique morphism f  : H → comp(H ). That is, comp is a right adjoint to the inclusion functor of the full subcategory of pointwise compact abelian pro-Lie groups in the category AbproLieGr of abelian pro-Lie groups. See for instance [102, Proposition A3.36]. Therefore it preserves limits (see for instance [102, Theorem A3.52]). Since comp preserves limits, it preserves, in particular, projective limits. (iii) follows from the Closed Subgroup Theorem for Projective Limits 1.34 (v). (iv) Let g comp(G) be nonzero in G/ comp(G) that is, g ∈ / comp(G). Then by / comp(G/N ) and then (ii) there is an N ∈ N (G) such that g ∈ / CN . Then gN ∈ (gN) comp(G/N ) is not relatively compact in (G/N )/ comp(G/N ). Since the quotient morphism G → G/N maps comp(G) into comp(G/N ), there is an induced quotient morphism F : G/ comp(G) → (G/N )/ comp(G/N ) given by F (g comp(G)) = (g/N) comp(G/N ). As this element is not relatively compact, the element g comp(G) cannot be relatively compact in G/ comp(G). Thus G/ comp(G) is compact-free. The examples 5.1 (ii)(b) and (e) are elementwise compact abelian pro-Lie groups which are not locally compact since J is infinite. The examples 5.1 (ii)(a), (c), and (d), and the example in Proposition 5.2 are compact-free abelian pro-Lie groups. In the first part of the following definition we recall when a topological group G with identity component G0 is called almost connected (see Definitions 1 and 4.24). Definition 5.6. Let G be a topological group. (i) G is said to be almost connected if G/G0 is compact. (ii) G is said to be compactly generated if there is a compact subset K of G such that G = K. (iii) G is said to be compactly topologically generated if there is a compact subset K of G such that G = K. Lemma 5.7. Let f : G → H be a morphism of topological groups and assume, firstly, that G is almost connected and, secondly, that f (G) is dense in H . Then H is almost connected. In particular, each quotient group of an almost connected group is almost connected. Proof. Since f (G0 ) is connected and contains the identity, f (G0 ) ⊆ H0 and therefore the morphism ϕ : G/G0 → H /H0 , ϕ(gG0 ) = f (g)H0 is well-defined. By assumption G/G0 is compact. So the continuous image ϕ(G/G0 ) is compact and thus, since H /H0 is Hausdorff, is closed in H /H0 . Since f (G) is dense in H , it follows that ϕ(G/G0 ) is dense in H /H0 . Therefore, H /H0 = ϕ(G/G0 ), and so H /H0 is compact. Proposition 5.8. Assume that G is an abelian pro-Lie group satisfying at least one of the two conditions (i) G is almost connected;

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(ii) G is topologically compactly generated. Then comp(G) is compact and therefore is the unique largest compact subgroup of G. Proof. We may assume that G = comp(G) and must show that G is compact. Then by Theorem 5.5 (ii) we have G = limN ∈N (G) comp(G/N ); it therefore suffices to verify that comp(G/N ) is compact for all N. Let N ∈ N (G). The Lie group G/N is isomorphic to L = Rp × Tq × D with a discrete group D (compare the proof of Theorem 5.5 (i)), and thus comp(L) = {0} × Tq × tor(D) for the torsion group tor(D) of D. Thus we have to verify that tor(D) is finite. In case (i), G/N ∼ = L is an almost connected Lie group by Lemma 5.7 and thus D is itself finite. In case (ii), G/N ∼ = L has a dense subgroup generated by a compact set, and this is then true for the discrete factor D. Then D is finitely generated and thus is isomorphic to the direct sum of a finite group and a finitely generated free group (see for instance [102, Theorem A1.11]). Thus tor(D) is finite in this case as well. Lemma 5.9. Assume that G is an abelian pro-Lie group. Then L(comp(G)) = comp(L)(G)), and there is a closed vector subspace W such that (X, Y )  → X + Y : W × comp(L)(G)) → W ⊕ comp(L)(G) = L(G) is an isomorphism of weakly complete topological vector spaces. Proof. By Definition 5.4, a one parameter subgroup X : R → G is in comp(L)(G) iff X(R) is compact iff X(R) ∈ comp(G). Thus comp(L)(G) = L(comp(G)). Since comp(G) is a closed subgroup of G by Theorem 5.5 (i), L(comp(G)) is a closed vector subspace of L(G). Then it has an algebraic and topological vector space complement, by Theorem A2.11 (i) of Appendix 2.

Vector Group Splitting Theorems Lemma 5.10. Let G be an abelian pro-Lie group, and assume that comp(G) is compact. Then the following conclusions hold. (i) There is a weakly complete vector group W and a compact abelian group C which is a product of circle groups such that G may be considered as a closed subgroup of W × C such that G ∩ ({0} × C) = comp(G). (ii) G/ comp(G) is embedded as a closed subgroup into the weakly complete vector group W . Proof. (i) For each N ∈ N (G) we have an embedding iN : G/N → W (N )×C(N ) for a finite-dimensional vector group W (N) and a finite-dimensional torus C(N ). Hence

220 G∼ =

5 Abelian Pro-Lie Groups

lim

N∈N (G)

G/N ⊆





i=

N∈N (G) iN

G/N −−−−−−−→

N ∈N (G)



W (N ) × C(N ) = W × C

N ∈N (G)

 for a  weakly complete vector group W = N ∈N (G) W (N ) and a compact group C∼ = N∈N (G) C(N). Since i is an embedding we may write G ⊆ W × C and assume that G is closed. Then comp(G) ⊆ comp(W × C) = {0} × C. But conversely, as every element in {0} × C is relatively compact, we have G ∩ ({0} × C) ⊆ comp G. (ii) By (i) above we may assume G ⊆ W × C for a weakly complete vector group W and torus C and {0} × C is the maximal compact subgroup of W × C. Hence comp(G) ⊆ {0} × C, and since C is compact, comp(G) ⊆ {0} × C. Let p : P → W be the projection of W × C onto W with kernel {0} × C. Then p is a proper and hence closed morphism of topological groups; therefore p|G : G → p(G) is a quotient morphism onto a closed subgroup of W . Since ker(p|G) = comp(G) we have G/ comp(G) ∼ = p(G) and the lemma follows. We remark in passing that an abelian Lie group, being isomorphic to a group × Tq × D for some discrete group (see for instance [102, 1 Exercise E5.18], or Corollary 7.58), is isomorphic to a subgroup of a connected abelian Lie group if and only if it is algebraically generated by a compact subset. If a topological abelian group is isomorphic to the additive topological group of a weakly complete topological vector space, it is called a weakly complete vector group. Rp

Lemma 5.11. Let G be an abelian pro-Lie group such that comp(G) = {1}. Then expG : L(G) → G0 is an isomorphism of topological groups. Proof. It is no loss of generality to assume that G = G0 , and we shall do that from now on. If X ∈ ker expG , that is X(1) = 0, then expG R.X = X(R) is a homomorphic image of R/Z and is therefore compact. Hence X(R) ⊆ comp(G) = {1} and thus X = 0. So expG is injective. Let N ∈ N (G). Since G is connected, G/N is a connected abelian Lie group, and thus the hypotheses of Lemma 5.10 are satisfied. Thus by Lemma 5.9 we may and will now assume that G is a closed subgroup of a weakly complete vector group V and we let i : G → V denote the inclusion map. We may identify V with L(V ) and expV with idV . There is a commutative diagram L(i)

L(G) −−→ ⏐ expG ⏐  G −−→ i

V ⏐ ⏐id  V V.

The morphism L(i) implements an isomorphism of L(G) onto a closed vector subspace of V . This implies that the corestriction expG : L(G) → expG L(G) is an isomorphism of topological groups, and thus expG L(G) is a closed vector subspace of V via i. By Corollary 4.22, exp L(G) is dense in G = G0 . Since expG L(G) is closed, we have G = expG L(G), and expG : L(G) → G is an isomorphism.

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Lemma 5.12 (Vector Group Splitting Lemma for Connected Abelian Pro-Lie Groups). Let G be an almost connected abelian pro-Lie group. Then there is a closed subgroup V of G such that expG |L(V ) : L(V ) → V is an isomorphism of topological groups and that (X, g)  → (expG X) + g : L(V ) × comp(G) → V ⊕ comp(G) = G is an isomorphism of topological groups. In particular, every connected abelian pro-Lie group is isomorphic to RJ × C for some set J and some compact connected abelian group C. Proof. By Theorem 5.5 (i), comp(G) is a closed subgroup. def

Let q : G → H = G/ comp(G) be the quotient morphism. By the Strict Exactness Theorem for L 4.20 we have a commutative diagram of strict exact sequences incl

L(q)

incl

q

0 → comp(L)(G) −−−→ L(G) −−−→ L(H ⏐ ⏐ ⏐ ) →0 ⏐ ⏐ ⏐exp expG | comp(L)(G) expG  H 0→ comp(G) −−−→ G −−−→ H → 0. By Lemma 5.10, the morphism L(q) splits, that is, there is a morphism of weakly complete topological vector spaces σ : L(H ) → L(G) such that L(q)  σ = idL(H ) . By 5.5 (iv) we have comp(H ) = {1}. Now we use the fact that comp(G) is compact by Proposition 5.8 to conclude that H is complete and thus is a pro-Lie group by Theorem 4.28 (iii). Since G is connected, H is connected. Then Lemma 5.11 implies that expH : L(H ) → H is an isomorphism. We define s : H → G by s = expG  σ  −1 −1 −1 exp−1 H . Then qs = qexpG  σ expH = expH  L(q) σ  expH = expH  expH = idH . Thus q splits. Now set V = s(H ) and let μ : V × comp(G) → G be defined by μ(v, g) = v + g and ν : G → V × G by ν(g) = (g − s(q(g)), s(q(g))). Then μ and ν are inverses of each other, and this completes the proof of the lemma. By applying the Vector Group Splitting Lemma 5.12 to the identity component G0 of an abelian pro-Lie group we see at once that: In any abelian pro-Lie group G, the identity component has a unique largest compact subgroup comp(G0 ) and G0 is isomorphic to a direct product of G0 / comp(G0 ) and comp(G0 ), where G0 / comp(G0 ) is isomorphic to the additive group of the weakly complete topological vector space L(G)/ comp(L)(G) and comp(G). Thus connected abelian pro-Lie groups are already reduced to weakly complete vector groups (Example 5.1 (ii)(a)) and compact connected groups. Definition 5.13. Let G be an abelian pro-Lie group. Let V be any closed subgroup of G such that (i) V is isomorphic to the additive topological group of a weakly complete topological vector space,

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(ii) (v, c)  → v +c : V ×comp(G0 ) → G0 is an isomorphism of topological groups. Then V is called a vector group complement. A vector group complement in a connected abelian pro-Lie group is not unique, but all vector group complements are isomorphic to G/ comp(G). We shall in fact want to know whether we can find a complement that is invariant under the action of a compact group of automorphisms of G. We therefore inspect the issue a bit more closely. Since all of this happens in the identity component of an abelian pro-Lie group, we shall not restrict the generality if we assume that G is in fact connected. Assume that  is a topological group and G is a pro-Lie group. Assume that (ω, g)  → ω · g :  × G → G is an automorphic action, that is, each g  → ω · g is an automorphism of topological groups and 1 · g = g, ω1 ω2 · g = ω1 · (ω2 · g) for all g ∈ G, and ω1 , ω2 ∈ . Since L is a functor, we have an action (ω, X)  → ω · X of  def on g = L(G) such that exp(ω · X) = ω · (exp X). By this definition if X : R → G is a one-parameter subgroup of G, then ω.X : R → G is defined by (ω · X)(r) = ω · X(r). If (ω, g)  → ω · g :  × G → G is a continuous function then we claim that (ω, X)  → ω · X :  × g → g is also continuous: Indeed, recall that on g = Hom(R, G) we consider the topology of uniform convergence on compact subsets and let C ≥ 0, ω ∈ , X ∈ g, and U an identity neighborhood of G. Let V be an identity neighborhood of G such that ω · V ⊆ U and consider a Y ∈ g such that (∀r ∈ [−C, C]) Then

X(r)−1 Y (r) ∈ V .

(∗)

(ω · X)(r)−1 (ω · Y )(r) = ω(X(r)−1 Y (r)) ⊆ ω · V ⊆ U

for all r ∈ [−C, C]. By the continuity of ϕ :  × R → G, ϕ(ρ, r) = (ω · X)(r)−1 (ρω · Y )(r), for each r ∈ [−C, C], there is an identity neighborhood r of  and an open neighborhood Wr of r ∈ R such that ϕ(r × Wr ) ⊆ U . Since [C, C] is compact, we find  def  r1 , . . . , rn ∈ [−C.C] such that [−C, C] ⊆ nm=1 Wrm . Let  = nm=1 rm . Then ϕ( × [−C, C]) ⊆ U . That is, for all ω ∈ ω and all Y satisfying (∗) we have (ω · X)(r)−1 (ω · Y )(r) ∈ V . And this proves our claim. We shall say that (ω, g)  → ω · g :  × G → G is a continuous action iff it is an action and a continuous function.

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If G is a topological abelian group and C a closed subgroup, then there is a complementary subgroup V of G such that G = V ⊕ C algebraically and topologically iff the morphism p : G → G, p(v + c) = c satisfies p 2 = p and C = im p; conversely, if such a function is given then ker p is a complementary subgroup for C. Proposition 5.14. (i) Let V be any vector space complement of a connected abelian pro-Lie group G and let us write G = V × C with C = comp(G). For any morphism of topological groups f : V → C, the subgroup graph(f ) = {(v, f (v)) : v ∈ V } is also a vector group complement. All vector group complements are obtained in this way. If f : V → C is given, then αf : G → G, αf (v, c) = (v, c + f (v)) is an automorphism of G mapping V ×{0} to graph(f ), while Pf : G → G, Pf (v, c) = (0, c−f (v)) is an idempotent endomorphism of G whose kernel is a vector space complement and whose image and fixed point set is C. (ii) Let  be a compact group and assume that there is a continuous automorphic action  × G → G on the connected abelian pro-Lie group G. Then there is a vector space complement in G that is invariant under . Proof. (i) Exercise E5.2 below. (ii) Since C is a characteristic subgroup, it is invariant under the action of , and it follows that L(C) is invariant in g = L(G). The vector subspace L(C) has a vector space complement in the weakly complete vector space g. (See Chapter 7, Theorem 7.7 (iv).) Thus we have an idempotent endomorphism of topological vector spaces q : g → g, im q = L(C). Since  is compact and acts continuously and linearly on the locally convex complete vector space g, the integral 7 p= ωqω−1 dω : g → g ω∈

exists and is an idempotent equivariant endomorphism of topological vector spaces, that is, it satisfies p(ω · X) = ω · p(X) and im p = L(C). (See [102, Example 3.37 (ii).]) def

Then W = ker p is an  invariant vector subspace of g such that g = W ⊕ L(C) algebraically and topologically. So, using the terminology of (i) above, writing G = V × C and identifying L(G) with L(V ) × L(C), we can now invoke (i) for L(G) and write W = {(X, Y + λ(X)) : X ∈ L(V ), Y ∈ L(C)} for a suitable linear morphism ϕ : L(V ) → L(C). The exponential function expV : L(V ) → V is an isomorphism. We define f : V → C by f (expV X) = expC ϕ(X) and get a vector space complement expG W = {(v, f (v)) : v ∈ V } ⊆ V × C, and by (i) again, this an algebraic and topological complement for C, that is, G = expG W ⊕ C. Moreover, if ω ∈ , then ω · (X, ϕ(X)) = (X  , ϕ(X )) for some X , whence ω · (expV X, f (expV X)) = ω · expG (X, ϕ(X)) = expG ω(X, ϕ(X)) = expG (X , ϕ(X )) = (expV X , f (expV X )) ∈ Vϕ . Thus expG W is -invariant.

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5 Abelian Pro-Lie Groups

Exercise E5.2. Prove Remark 5.14 (i). [Hint. The function α : G0 → G0 , α(v, c) = (v, c + f (v)) is an automorphism of topological groups mapping V × {0} onto graph(f ). If W ⊆ G0 = V × C is a vector group complement, then since W is a vector group complement there is a projection pr W : G0 → W with kernel {0} × C. Let pr C : G0 → C denote the projection with kernel V × {0}. Define f : V → C by pr C (pr W (v, 0) − (v, 0)). Then f is a morphism of topological groups; if w = (v, c) ∈ W then w = pr W (v, 0) and w − (v, 0) = (0, c) whence c = f (v). The assertions on αf and Pf are straightforward.] We now work towards removing the hypothesis of almost connectivity from the Vector Group Splitting Lemma 5.12. Lemma 5.15. Let W be a weakly complete topological vector space and G a proto-Lie group. Assume that f : W → G is a bijective morphism of topological abelian groups. Then f is an isomorphism of topological groups.  denote the completion of G. Then G  is a connected abelian pro-Lie group; Proof. Let G by theVector Group Splitting Lemma 5.12, it is therefore of the form V ⊕C algebraically and topologically for a weakly complete topological vector subgroup V and a compact  → V denote the projection onto V along C. The function subgroup C. Let pr V : G def

ϕ = pr V f : W → V is a dense morphism of weakly complete vector groups and is therefore a quotient morphism of weakly complete topological vector spaces by Theorem A2.12 (a) of Appendix 2. Thus there are closed vector subspaces V1 and V2 of W such that W = V1 ⊕ V2 algebraically and topologically such that V2 = ker ϕ and ϕ  i : V1 → V , where i : V1 → W is the inclusion, is an isomorphism of weakly  σ = f i(ϕi)−1 complete topological vector spaces. Now the morphism σ : V → G, −1 −1 satisfies pr V σ (v) = pr V f  i  (ϕ  i) = ϕ  i  (ϕ  i) = idV . This means that  = σ (V ) ⊕ C, σ (V ) = f (V1 ). In order to simplify notation, after replacing V by G def

σ (V ), if necessary, we may actually assume that V = f (V1 ). Then D = f (V2 ) ⊆ C, and we have G = V × D with a dense subgroup D of C. We recall that G is a protoLie group; then D ∼ = G/V is a connected proto-Lie group and a dense subgroup of a compact group. Let N ∈ N (D); then D/N is a Lie group and a dense subgroup of C/N . A Lie group is complete, and thus D/N is closed in C/N, that is, D/N = C/N for all N ∈ N (D).  Hence G is complete and f : W → V × C Therefore D = C and thus G = G. is bijective. We can write W = W1 × W2 such that f = f1 × f2 where f1 is an isomorphism from W1 onto V and f2 : W2 → C is a bijective morphism of topological abelian groups from a weakly complete vector group W2 onto a compact group C. In particular, every point in C is on a one parameter subgroup, that is exp : L(C) → C is surjective. But C, as a bijective image of a real vector space, is torsion-free and divisible.  is divisible and torsion-free and so is a rational vector space, that is, a direct Thus C . If C = Q J for some sum of copies of Q, and therefore C is a power of copies of Q

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225

)J as L preserves limits, and expC may set J , then L(C) may be identified with L(Q J ∼ ) → Q  is not surjective, because the be identified with (expQ  ) . But expQ  : R = L(Q  nontrivial compact homomorphic images of R are isomorphic to R/Z ∼ = T and T ∼ =Q  . Thus J = Ø and C = {0}. This means that G = V and since  T ∼ =Q∼ = Z ∼ =Q W → V is an isomorphism of topological groups. f ∼ = Note that Lemma 5.15 is a very special example of an Open Mapping Theorem (see 9.60 and its proof). Lemma 5.16. Assume that G is an abelian topological group with closed subgroups G1 and H such that G1 is either (a) a weakly complete topological vector subgroup, or (b) a compact subgroup. Assume further that G1 + H = G, that G1 ∩ H = {0}, and that G/H is a proto-Lie group. Then μ : G1 × H → G, μ(v, h) = v + h is an isomorphism of topological groups. Proof. Clearly μ is a bijective morphism of topological groups. We must show that its inverse is continuous. The morphism β : G1 → G/H , β(v) = v + H is a continuous bijection. In Case (a), by Lemma 5.15 β is open. In Case (b) it is a homeomorphism since G1 is compact. That is, β −1 : G/G0 → G0 is continuous in both cases. Let def

q : G → G/G0 be the quotient map. Then α = β  q : G → G0 is a morphism of topological groups, and μ−1 (g) = (α(g), g − α(g)) is likewise a morphism of topological groups. Notice that by the Quotient Theorem for Pro-Lie Groups 4.1, the quotient G/H is a proto-Lie group if G is a pro-Lie group. Proposition 5.17. Assume that G is an abelian proto-Lie group and that G1 is a closed connected subgroup which is a finite-dimensional Lie group. Then there is a closed subgroup H such that the morphism (v, h)  → v + h : G1 × H → G is an isomorphism of topological groups. Proof. We claim that it is no loss of generality to assume that G is a pro-Lie group.  be the completion of G. Since the subgroup G1 is a finite-dimensionIndeed, let G al Lie group, it is locally compact. Locally compact subgroups are closed (see for instance [102, Corollary A4.24.]). Thus G1 is also a closed subgroup of the pro If we can show that there is a closed subgroup H  of G  such that Lie group G.  → G  is an isomorphism of topological groups, then (v,  h) → v +  h : G1 × H def  we get a subgroup of G such that (v, h)  → g + h : G1 × H → G setting H = G ∩ H is an isomorphism of topological groups. This proves our claim; from here on we shall assume that G is a pro-Lie group.

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The subgroup G1 , being a finite-dimensional Lie group, has no small subgroups. Hence there is a zero neighborhood U such that {0} is the only subgroup contained in G1 ∩ U . Now let N ∈ N (G) be contained in U . Then N ∩ G1 ⊆ U ∩ G1 = {0}. Recall that G/N is a Lie group and (G1 + N )/N is isomorphic to G1 /(G1 ∩ N ) ∼ = G1 by the Closed Subgroup Theorem for Projective Limits 1.34 (iv). So (G1 + N )/N is isomorphic to Rp ⊕ Tq and is a closed subgroup of G/N which is isomorphic to Rm ⊕ Tn ⊕ D for a discrete subgroup D. It is therefore a direct summand algebraically and topologically, that is, there is a closed subgroup H of G containing N such that H /N is a complementary summand for (G1 +N )/N . Thus G1 +H = G and G1 ∩H ⊆ def

G1 ∩ N = {0}. Then if C = comp(G1 ) = {0}, it follows from Lemma 5.16 (a), that (v, h) → v + h : G1 × H → G is an isomorphism of topological groups. Therefore, in the general case, G/C ∼ = G1 /C × (H + C)/C and thus there is a vector subgroup V of G such that G ∼ = V × (C + H ); it remains to be observed that C + H ∼ = C × H. But that is Lemma 5.16 (b). Next we need a lemma on weakly complete topological vector spaces. Recall that an affine subspace A of a vector space W is a subset of the form A = g + V for some vector subspace V . The affine subspace is linear iff g ∈ V . Lemma 5.18. Let W be a weakly complete topological vector space and F a filter basis of closed affine subspaces. (a) Assume that F is a closed vector subspace of W and all A ∈ F are linear. Then F+



F =



F + H.

(∗)

If, in addition, dim E/F < ∞, then   F+ F = (F + H ).

(∗∗)

H ∈F

H ∈F

(b) Assume that all A ∈ F are linear. Then the following conditions are equivalent: (i)  lim F = 0. (ii) F = {0}.  (c) F = Ø. (d) Assume that F ⊆ N (W ). Then the following conditions are equivalent: (i) lim F = 0. (ii) F is a basis of N (W ). Proof. See Theorems A2.13 and A2.14 of Appendix 2. In terms of a terminology that has been used for situations like this, (c) can be expressed in the following form: Weakly complete topological vector spaces are linearly compact.

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We shall prove next that every weakly complete vector subgroup in an abelian pro-Lie group splits. Another way of expressing this in a category theoretical fashion is this: Weakly complete vector groups and torus groups are relative injectives for the class of embeddings in the category of abelian pro-Lie groups. For a full understanding recall that for a class E of monomorphisms, an object I in a category is called a relative injective for E , if for any monic ι : A → B from E and any morphism f : A → I there is a morphism F : B → I such that f  ι = f . idI

I −−→  ⏐ f⏐ A −−→ ι

I  ⏐ ⏐F B.

(See for instance [102, paragraph following 9.77].) By definition, an object is an injective, if it is a relative injective for the class of all monomorphisms. Exercise E5.3. A pushout in a category is a pullback in the opposite category (see Definition 1.1 (iii)). Show Proposition A. Let I be an object in a category and consider the following two statements: (i) I is a relative injective for E . (ii) Any monic ι : I → B from E is a coretraction, that is there is an ε : B → I such that ε  ι = idI . Then (i) ⇒ (ii) and if the category has pushouts pushing forward E -monics, then the two conditions are equivalent. This is the case in the category AbTopGr of abelian topological groups and in the category AbproLieGr of abelian pro-Lie groups for the class E of closed embeddings. Proposition B. Any product of relative injectives is a relative injective. [Hint. Proof of A: For a proof of (i) ⇒ (ii) simply take A = I and f = idI in the definition. For a proof of the reverse implication let ι : A → B be a monomorphism from E and f : A → I any morphism into an object I satisfying (ii). Form the pushout ϕ

I −−→  ⏐ f⏐ A −−→ ι

P  ⏐β ⏐ B.

(∗)

Since pushouts push forward E -monics and ι is an E -monic, the morphism ϕ is an E -monic as well. Hence by (ii) there is an ε : P → I such that ε  ϕ = idI . Set F = εβ. Then F  ι = εβι = εϕf = idI f = f . ϕ

ε

ι

idB

I −−→ P   −−→ ⏐ ⏐β f⏐ ⏐ A −−→ B −−→

I  ⏐ ⏐F B.

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Finally, we recall from Theorem 1.5 (a), that the pushout (∗) is formed by creating first the coproduct P  of I and B and then the coequalizer e : P  → P of the two morphisms copr I f and copr B ι (assuming that coproducts and coequalizers exist). In the category AbTopGr of abelian topological groups P  = I × B and copr I (x) = def

(x, 0), copr B (b) = (0, b). The coequalizer e is the cokernel of δ = copr I f − copr B ι, that is the quotient morphism e : I × B → (I × B)/Q with Q = δ(A) = {(f (a), −a) : a ∈ A}, where we have assumed A = A ⊆ B and ι to be the inclusion map. Notice that Q is closed in I × B: If (x, b) = limj (f (aj ), −aj ) for a net (aj )j ∈J in A, then b ∈ A = A and x = limj f (aj ) = f (−b) since f is continuous. (Indeed, Q is in essence the graph of f .) Now ϕ : I → (I × B)/Q is given by ϕ(x) = (x, 0) + Q. Then x ∈ ker ϕ iff (x, 0) ∈ Q, that is, (x, 0) = (f (a), −a) for some a ∈ A. Then a = 0 and as a consequence x = f (a) = f (0) = 0. Thus ϕ maps I bijectively onto ((I × {0}) + Q)/Q = (I × A)/Q. Since Q is closed in B, the group I × A is closed in I × A. We claim that the morphism ψ : I → (I × A)/Q, ψ(x) = (x, 0) + Q is open. So let U be a symmetric identity neighborhood of I . Find a symmetric identity neighborhood V of I such that V + V ⊆ U . Now we let W = f −1 (V ); this is a symmetric identity neighborhood of A since f is continuous. Now we show that ψ(U ) contains (V × W ) + Q/Q and this will show that ψ(U ) is an identity neighborhood of (I × A)/Q and thus prove the asserted openness of ψ. Let (v, w) ∈ V × W . We claim that there are elements u ∈ U and a ∈ A such that (u, 0) + (f (a), −a) = (v, w). Indeed, we let a = −w. Then u − f (w) = v, implies u = v + f (w) ∈ V + V ∈ U and this proves the claim. In the category AbproLieGr everything remains the same except that in order to form the pushout in the category AbproLieGr we have to form the completion P of (I × B)/Q, because (I × B)/Q is a proto-Lie group but not necessarily a pro-Lie group (see Chapter 4, the Quotient Theorem for Pro-Lie Groups 4.1). Thus ϕ : I → P is the composition of the closed embedding morphism γ : I → (I × B)/Q and the dense embedding morphism η : (I × B)/Q) → P . But I is a pro-Lie group and is therefore complete, and since ϕ(I ) = η(γ (I )) is an isomorphic copy of I it is a complete and therefore closed subgroup of B. Hence ϕ is a closed embedding in this case as well. Proof of B. Let {Ij : j ∈ J } be a family of relative injectives and ι : A → B a mono from E . Assume that f : A → j ∈J Ij is any morphism. Fix j since Ij is an E injective, there is an Fj : B → Ij such that Fj  ι = pr j f . Then the universal  property of the product (see Definition 1.4 (i)) give us a morphism F : B → j ∈J Ij such that pr j F = Fj . Then pr j F ι = Fj  ι = pr j f and this implies F  ι = f by the uniqueness in the universal property of the product.] Theorem 5.19. Assume that G is an abelian pro-Lie group with a closed subgroup G1 and assume that there are sets I and J such that G1 ∼ = RI × TJ . Then G1 is a

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homomorphic retract of G, that is, a direct summand algebraically and topologically. So, G ∼ = G1 × G/G1 . Proof. By Exercise E5.3, Proposition A, applied to the category AbproLieGr of abelian pro-Lie groups, we are claiming that G1 is a relative injective for the class of closed homomorphic embeddings in AbTopGr. By Exercise E5.3, Proposition B above, it suffices to prove this for each of the cases (i) G1 = R, and (ii) G1 = T. For each of these cases Proposition 5.17 applies and proves the claim. It may be a useful exercise to present an independent and category theory free proof of the preceding theorem in the case that G1 ∼ = RI is a weakly complete topological vector space. It proceeds as follows: Let S be the set of all closed subgroups S of G satisfying the following conditions: (i) S ∩ G1 is a vector group. (ii) (∀g ∈ G) S ∩ (g + G1 ) = Ø. We claim that (S, ⊇) is an inductive poset. For a proof of the claim let T be a chain def  in S and set T = T . We  have to show that T satisfies (i) and (ii). (i) We note T ∩ G1 = S∈T S ∩ G1 , and all S ∩ G1 are closed vector groups; hence their intersection is a closed vector group. (ii) Let g ∈ G; we must show that T ∩ (g + G1 ) = Ø. Now for each S ∈ T we find an sS ∈ S ∩ (g + G1 ) by (ii). Then g + G1 = sS + G1 . We claim that (∗) S ∩ (g + G1 ) = sS + (S ∩ G1 ). Indeed if s ∈ S ∩(g+G1 ), then s ∈ g+G1 = sS +G1 and thus s −sS ∈ S ∩G1 and thus s ∈ sS +(S∩G1 ). Conversely, if s ∈ S∩G1 , then sS +s ∈ S∩(sS +G1 ) = S∩(g+G1 ). By (i) we know that S ∩ G1 is a (closed) vector subgroup VS of G1 . Thus from (∗) we obtain (∗∗) Ø = (S − g) ∩ G1 = sS − g + VS , where sG − g ∈ G1 . Now the family {sS − g + VS : S ∈ T } is a filter basis of closed affine subspaces of the vector group G1 . By Lemma 5.18 (c), there is a  t∈ (sS − g + VS ) ∈ G1 , S∈T



 and thus t + g ∈ S∈T (sS + VS ) = S∈T S ∩ (g + G1 ) = T ∩ (g + G1 ). This completes the proof that (S, ⊇) is inductive. Using Zorn’s Lemma, let H be a minimal member of S. We claim that H ∩G1 = {0}. Suppose that the claim were false. def

Then H1 = H ∩ G1 is a nonzero weakly complete vector group. Let N be a vector subgroup of H1 , such that H1 /N is a finite-dimensional vector group (for instance, one isomorphic to R). Now G/N as a quotient of a pro-Lie group, is a proto-Lie group by Theorem 4.1. Then by Proposition 5.17, there is a closed subgroup S of H containing N such that (H1 /N ) + (S/N ) = H /N, and (H1 /N ) ∩ (S/N ) = {N }. Thus

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H1 ∩ S = N is a vector subgroup and G1 + S = G1 + H = G and so the subgroup S of H /N satisfies (i) and (ii) above. The minimality of H then entails S = H and thus N = H1 ∩ S = H1 and that is a contradiction to the choice of N. This proves our claim that there is a closed subgroup H of G such that G = G1 + H and G1 ∩ H = {0}. Thus the function (v, h)  → v + h : G1 × H → G is a bijective morphism of topological groups by Lemma 5.16. Recall from the Vector Group Splitting Lemma 5.12 and Definition 5.13 that every abelian pro-Lie group has a vector group complement. Vector Group Splitting Theorem for Abelian Pro-Lie Groups Theorem 5.20. Let G be an abelian pro-Lie group and V a vector group complement. Then there is a closed subgroup H such that: (i) (v, h)  → v + h : V × H → G is an isomorphism of topological groups. (ii) H0 is compact and equals comp G0 and comp(H ) = comp(G); in particular, comp(G) ⊆ H . (iii) H /H0 ∼ = G/G0 , and this group is prodiscrete. (iv) G/ comp(G) ∼ = V × S for some prodiscrete abelian group S without nontrivial compact subgroups. (v) G has a characteristic closed subgroup G1 = G0 + comp(G) which is isomorphic to V × comp(H ) such that G/G1 is prodiscrete without nontrivial compact subgroups. (vi) The exponential function expG of G = V ⊕ H decomposes as expG = expV ⊕ expH where expV : L(V ) → V is an isomorphism of weakly complete vector groups and expH = expcomp(G0 ) : L((comp(G0 )) → comp(G0 ) is the exponential function of the unique largest compact connected subgroup; here L(comp(G0 )) = comp(L)(G) is the set of relatively compact one parameter subgroups of G. (vii) The arc component Ga of G is V ⊕Ha = V ⊕comp(G0 )a = im L(G). Moreover, if h is a closed vector subspace of L(G) such that exp h = Ga , then h = L(G). Proof. (i) By Theorem 5.19, H exists such that (i) is satisfied. (ii) Let us write G = V × H . Then G0 = V × H0 . Since V × {0} is a vector group complement, G = (V × {0}) ⊕ comp(G0 ) algebraically and topologically. Then the projection of G0 onto H0 along V maps the compact subgroup comp(G0 ) onto H0 . Thus H0 is compact. So {0} × H0 ⊆ comp(G0 ), and since G0 = V × H0 , the factor group comp(G0 )/({0} × H0 ) is isomorphic to a subgroup of V . Since V as a vector group has no nontrivial compact subgroup, comp(G) = {0} × H0 follows. (iii) Retaining the convention G = V × H after (i), we have G0 = V × H0 . Then G/G0 =

V × H ∼ (V × H )/(V × {0}) ∼ = = H /H0 . V × H0 (V × H0 )/(V × {0})

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By the Closed Subgroup Theorem 3.35, H is a pro-Lie group. Since H0 is compact, H /H0 is complete by Theorem 4.28 (iii). Thus by Lemma 3.31 H /H0 is a prodiscrete group (iv) Again we write G = V × H and have comp(G) = {0} × comp(H ). Thus G/ comp(G) =

V ×H ∼ = V × (H / comp(H )). {0} × comp(H )

By Lemma 5.5 (iii), H / comp(H ) is compact-free. By (iii) above, H /H0 is prodiscrete. As a quotient of the prodiscrete group H /H0 , the quotient H / comp(H ) ∼ = (H /H0 )/(comp(H )/H0 ) is a protodiscrete group by Proposition 3.30 (b). (v) comp(G) is a closed characteristic subgroup and by (iv) the factor group G/ comp(G) decomposes into a direct product V ×S in which V ×{0} is the connected component and thus is characteristic. The kernel G1 of the composition of the quotient morphism G → G/ comp(G) and the projection G/ comp(G) → S is a closed characteristic subgroup equal to G0 + comp(G) and G/G1 ∼ = S. Applying (i) to G1 we get G1 ∼ = V × comp(G). (vi) follows immediately from (i) and Definition 5.4 (i) in view of the fact that for any topological vector space V the exponential function expV : L(V ) → V , expG X = X(1), is an isomorphism of topological vector spaces as all one parameter subgroups are of the form X = r  → r · v for a unique vector v = vX . See also Lemma 5.15. (vii) As G = V ⊕ H is a direct product decomposition we have Ga = Va ⊕ Ha . But V , as the additive topological group of a topological vector space is arcwise connected, and Ha = (H0 )a = comp(G0 )a . By [102, Theorem 8.30 (ii)], we have comp(G0 )a = L(comp(G0 )) = L(H ). Thus from (vi) we get Ga = expG L(G). Now let h be a closed subalgebra of L(G). If h = L(G), then expG h = E(G) = Ga by Corollary 4.21 (i). In the context of conclusions (iv) and (v) we recall from 5.10 (ii) that every prodiscrete group without compact subgroup embeds into a weakly complete vector group. But Proposition 5.2 shows that such embeddings may have bizarre properties. For locally compact abelian groups, 5.20 (i) yields a core result of their structure theory; it is presented practically in every source book on locally compact abelian groups (see for instance [102, Theorem 7.57]). The examples in 5.1 (ii)(b)–(e) illustrate certain limitations of this main result. The examples in 5.1 (ii)(b) and (ii)(e) show what prodiscrete pointwise compact groups may look like; neither has a compact open subgroup and therefore both fail to be locally compact. The examples in (ii)(b) are torsion-free and divisible, the examples in (ii)(e) have a dense proper torsion subgroup and are divisible. The examples in 5.1 (ii)(c) and (d) are compact-free; those in (ii)(c) are divisible, those in (ii)(d) have no nondegenerate divisible subgroups. Thus unlike in the locally compact case, we cannot expect that inside the factor H , the subgroup comp(H ) is open, or, equivalently, that the factor group G/G0 has an open compact subgroup.

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It is easy to mix these examples. There are compact abelian groups in which the component does not split (see [102, Example 8.11]). Theorem 5.20 completely elucidates the structure of the identity component G0 , it largely clarifies the structure of G1 (although comp(G) is best understood in the locally compact case), and it reduces the more subtle problems on G to the compactfree prodiscrete factor group G/G1 . One should recall the example in Proposition 5.2 which typically might occur as a prodiscrete factor group. Corollary 5.21. Let G be an abelian pro-Lie group and V a vector group complement. Then the following statements are equivalent: (i) (ii) (iii) (iv)

G/G0 is locally compact. G/V is locally compact. comp(G/V ) is locally compact and open in G/V . There is a locally compact subgroup H of G containing comp(G) as an open subgroup such that such that (v, h)  → v + h : V × H → G is an isomorphism of topological groups.

Proof. (i) ⇔ (ii): We have G/G0 ∼ = (G/V )/(G0 /V ) and G0 /V is compact by Lemma 5.12. The quotient of a locally compact group is locally compact, and the extension of a locally compact group by a locally compact group is locally compact. (iv) ⇒ (iii): Since comp(G) = comp(H ) this is clear. (iii) ⇒ (ii): Trivial. (ii) ⇒ (iv): The locally compact abelian group G/V has a compact identity component (G/V )0 = comp(G/V )0 and thus has a compact open subgroup C. Let K be the full inverse image of C in G. Then K is an almost connected open subgroup of G to which the Vector Group Splitting Theorem applies. Thus K is the direct product of V and the unique maximal compact subgroup comp(K) of K. Then (G/ comp(K))0 = K/ comp(K) ∼ = V , this is an open divisible subgroup of G/ comp(K). But then G/ comp(K) is the direct sum of (K/ comp(K))0 and a discrete group H / comp(K) with a closed subgroup H of G containing comp(K) as an open subgroup. In particular, comp(K) ⊆ comp(H ) and comp(H ) is open in H . From (G/ comp(K)) ∼ = (K/ comp(K)) × (H / comp(K)) and K∼ = V × comp(K) we derive G∼ = V × H. As we have comp(G) = comp(H ), the implication (ii) ⇒ (iv) is proved.

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Compactly Generated Abelian Pro-Lie Groups The Vector Group Splitting Theorem tells us that each abelian pro-Lie group G is built up in a lucid fashion from a weakly complete vector group and a more special abelian pro-Lie group H which is turn is an extension of the characteristic closed subgroup comp(G) = comp(H ) by a prodiscrete compact-free factor group H / comp(H ) ∼ = G/(G0 + comp(G)). The examples 5.1 (ii)(b) and (ii)(e) (and the groups easily manufactured from these by passing to products, subgroups and quotients) indicate that we are not to expect very explicit information on comp(G) without further hypotheses, and a similar statement holds for prodiscrete compact-free groups (see 5.1 (ii)(c), (ii)(d) and 5.2). A topological space is called a Polish space if it is completely metrizable and second countable. It is said to be σ -compact, if it is a countable union of compact subspaces. It is said to be separable if it has a dense countable subset.  Countable products of Polish spaces are Polish. For instance, any product n∈N Ln of a countable sequence of second countable Lie groups is a Polish pro-Lie group; this applies in particular to RN or ZN . A mixture of topological and algebraic properties of topological groups is exemplified by the concepts introduced in 5.6, to which we return presently. Remark 5.22. (i) Every almost connected locally compact group is compactly generated. (ii) Every compactly generated topological group is σ -compact. (iii) A topological group whose underlying space is a Baire space and which is σ -compact is a locally compact topological group. (iv) A σ -compact Polish group is locally compact. (v) A compactly generated Baire group is locally compact. Proof. (i) Let K be a compact neighborhood of the identity. Then K is an open subgroup which has finite index in G. Let F be any finite set which meets each coset modulo K. Then K ∪ F is a compact generating set of G. def

(ii) If K is a compact generating set of G, C = KK −1 is a compact generating then ∞ −1 set satisfying C = C; then G = C = n=1 C n . (iii) A Baire space cannot be the union of a countable set of nowhere dense closed subsets. A topological group containing a compact set with nonempty interior is locally compact. (iv) By the Baire Category Theorem (see [26, Chapter 9, §5, no 3, Théorème 1]), every Polish space is a Baire space. (v) is clear from the preceding. The following remarks are straightforward from the definitions, from Proposition 5.8, and the Vector Group Splitting Theorem 5.20. Remark 5.23. Let G ∼ = V × H be an abelian pro-Lie group with a vector group complement V and H as in Theorem 5.20. Then the following statements are equivalent:

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(i) G is Polish iff both V and H are Polish. (ii) G is σ -compact iff V, comp(G), and H / comp(H ) are σ -compact. (iii) G is compactly generated iff V and H / comp(H ) are compactly generated and comp(G) is compact. (iv) G is separable iff V and H are separable. These simple remarks lend some urgency to a more detailed understanding of the situation of weakly complete topological vector spaces; we shall turn to this topic in the next section. Remark 5.24. For a discrete abelian group, the following statements are equivalent: (i) (ii) (iii) (iv)

G is finitely generated free. G is isomorphic to a closed additive subgroup of Rn for some natural number n. G is isomorphic to a closed additive subgroup of RJ for some set J . G is isomorphic to a closed additive subgroup of a weakly complete topological vector space.

Proof. For the equivalence of (i) and (ii) see for instance [102, Theorem A1.12 (i)]. Trivially (ii) ⇒ (iii) ⇒ (iv). Assume (iv), that is, that G is a closed discrete subgroup of a weakly complete vector group W . Since G is discrete, there is an identity neighborhood U1 of W such that W ∩ U1 = {0}. Let U be an open identity neighborhood of W such that U + U + U + U ⊆ U1 . Since lim N (W ) = 0 there is a V ∈ N (W ) such that V ⊆ U and thus U + V ⊆ U + U . By replacing U by U + V where necessary we assume that U + V = U and U + U ⊆ U1 . If u ∈ (G − U ) ∩ U then u = g − u for some 0 = g ∈ G and u ∈ U ; thus g = u + u ∈ G ∩ U + U ⊆ G ∩ U1 = {0}. Thus V is the complement of (G \ {0}) − U in G + V . So G ∼ = (G + V )/V is a discrete hence closed subgroup of the finite-dimensional vector space W/V . This result provides an alternative proof of the fact that the prodiscrete free abelian group F of infinite rank in 5.2 cannot be discrete. Proposition 5.25. Let G be an abelian compact-free pro-Lie group. (i) If N (G) has a basis M of subgroups such that G/M is compactly generated, then G is isomorphic to a closed subgroup of a product RI × ZJ and hence also of a weakly complete topological vector space RK . (ii) If G is compactly generated, then (i) applies. (iii) The group G is compactly generated and Polish iff it is isomorphic to Rm × Zn iff it is locally compact. Proof. (i) We assume that for N ∈ M we have G/N = VN ⊕ FN ⊕ tor G/N where VN is a finite-dimensional vector group, FN is finitely generated free and tor G/N is the finite torsion  group of G/N. It follows that G∼may  be identified with a closed subgroup of P = N∈M G/N = V ×F ×C where V = N ∈M VN is a weakly complete vector   ∼ ∼ group, F = N ∈M FN and C = N ∈M tor G/N . Then comp(P ) = {0} × {0} × C

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235

and G ∩ comp(P ) = {0} since G is compact-free. The projection P → V × F is a proper, hence closed morphism, with kernel comp(P ), mapping G onto a closed subgroup of F which is isomorphic to G/(G ∩ comp(P )) ∼ = G. Since V is a product of copies of R and F is a product of copies of Z, assertion (i) is proved. (ii) If G is compactly generated and N ∈ N (G) then N is a closed subgroup such that G/N is compactly generated, hence is of the form specified in the proof of (i). (iii) If G is Polish and compactly generated, then it is locally compact by 5.22 (i),(iv) and thus, being compact-free, is isomorphic to Rm × Zn . Corollary 5.26. Any compactly generated, compact-free prodiscrete group is isomorphic to a closed subgroup of a group ZJ . If it is not of finite rank, then it is not isomorphic to a subgroup of ZN . This situation is illustrated by the example in Proposition 5.2. In the proof of Proposition 5.8 it was only needed that the Lie group quotients of the abelian Lie group G in question were compactly generated. This together with Proposition 5.25 yields at once: Corollary 5.27. If G is an abelian pro-Lie group whose Lie group quotients are compactly generated, then comp(G) is compact and G/ comp(G) is embeddable in a weakly complete vector group. This applies, in particular, to all compactly generated abelian pro-Lie groups.

Weakly Complete Topological Vector Spaces Revisited We begin with an observation showing that the idea of topologically compactly generated pro-Lie groups may not be very restrictive. Remark 5.28. A weakly complete vector group is topologically compactly generated. A group of the form ZJ for any set J is topologically compactly generated. Proof. See Remark A2.15 of Appendix 2. Lemma 5.29. For a weakly complete topological vector space W , the following statements are equivalent: (A) (B) (C) (D)

W W W W

is σ -compact. is locally compact. is finite-dimensional. is compactly generated.

Proof. See Proposition A2.17 of Appendix 2. ∼ RJ by CorolLet W be a weakly complete topological vector space. Then W = lary A2.9 of Appendix 2. The cardinal card J is called the topological dimension of W . (See [103].)

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Lemma 5.30. For a weakly complete topological vector space W , the following statements are equivalent: (i) W ∼ = RJ with card J ≤ ℵ0 . (ii) W is locally compact or is isomorphic to RN . (iii) W is finite-dimensional or is isomorphic to RN . (iv) W is second countable. (v) W is first countable. (vi) W is Polish. Proof. See Appendix 2, Proposition A2.18. Lemma 5.31. For a weakly complete topological vector space W , the following statements are equivalent: (a) W is separable. (b) W contains a dense vector subspace of countable linear dimension over R. (c) W is isomorphic as a topological vector space to RJ with card J ≤ 2ℵ0 . They are implied by the equivalent statements of Lemma 5.30. Proof. See Proposition A2.19 of Appendix 2. For the following theorem we recall the definition of the characteristic closed subgroup G1 = G0 + comp(G) ∼ = V × comp(G) of an abelian pro-Lie group in Vector Group Splitting Theorem 5.20 (v) for a vector group complement V (see Definition 5.13 (ii)). Theorem 5.32 (The Compact Generation Theorem for Abelian Pro-Lie Groups). (i) For a compactly generated abelian pro-Lie group G the characteristic closed subgroup comp(G) is compact and the characteristic closed subgroup G1 is locally compact. (ii) In particular, every vector group complement V is isomorphic to a euclidean group Rm for some m ∈ N0 = {0, 1, 2, . . . }. (iii) The factor group G/G1 is a compactly generated prodiscrete group without compact subgroups. If G/G1 is Polish, then G is locally compact and G∼ = Rm × comp(G) × Zn . (iv) Assume that G is a pro-Lie group containing a finitely generated abelian dense subgroup. Then comp(G) is compact and G ∼ = Rm comp(G) × Zn , m, n ∈ N. In particular, G is locally compact. (v) A finitely generated abelian pro-Lie group is discrete. Proof. By Theorem 5.20, G ∼ = V × H such that H0 is compact. The factors V and H are compactly generated as homomorphic images of G. By Lemma 5.29, V ∼ = Rn for some nonnegative integer n. By Proposition 5.8, comp(H ) is compact and by 5.20, comp(G) = comp(H ) and H0 ⊆ comp(H ). Thus G1 ∼ = V × comp(G) is locally compact. Also, H / comp(H ) is totally disconnected, and by Theorem 4.28 (iii) this

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quotient is a pro-Lie group and hence is prodiscrete by Proposition 4.23. If the factor group G/G1 is Polish, then it is finitely generated free by 5.25 (iii), and the remainder of (iii) follows. For a proof of (iv) notice first that G is obviously abelian, containing a dense finitely generated abelian subgroup. All quotient groups of G satisfy the hypothesis of (iv), notably each vector group complement of G has a dense finitely generated subgroup and therefore has a dense finite-dimensional vector group; finite-dimensional vector groups are locally compact and therefore complete. Thus each vector group complement is isomorphic to Rm for some m ∈ N. def Also G/G1 contains a dense finitely generated subgroup. Set F = G/G1 ; then comp(F ) = {0}, and by 5.10 (ii), F may be considered as a closed subgroup of a weakly complete vector group V . Then R ⊗ F is algebraically isomorphic with the R-linear span E of F . So dimR E is finite. Every finite-dimensional topological vector space over R is locally compact and is isomorphic to Rp . Any closed free subgroup of Rp , however, is discrete (see for instance [102], Appendix 1, the Closed Subgroups and Quotients Theorem A1.12). This shows that G/G1 is discrete and thus is finitely generated free. Since free discrete factor groups are always direct summands, it now follows from the Vector Group Splitting Theorem 5.20 that G ∼ = Rm × comp(G) × Zn . So comp(G) contains a finitely generated dense subgroup g1  ⊕ · · · ⊕ gk . (For the structure of finitely generated abelian groups see for instance [102], Appendix 1, the Fundamental Theorem of Finitely Generated Abelian Groups A1.11.) By Weil’s Lemma for Pro-Lie Groups 5.3, for each j = 1, . . . , k, either gj ∈ comp(G) or def gj  ∼ = Z. Hence gj  is compact for all j and then K = g1  + · · · + gk  is compact, hence contained in comp(G). On the other hand, K being closed and containing g1 , . . . , gk , contains comp(G). Thus K = comp(G) and this shows that comp(G) is compact. This completes the proof of (iv). Conclusion (v) follows now from (iv): If the group G is finitely generated, then it is countable. A countable locally compact group is discrete by the Baire Category Theorem (or by the existence of Haar measure). Hence a finitely generated locally compact group is discrete. The observation 5.32 (iv) is (in an obvious sense) a generalization of Weil’s Lemma for Pro-Lie Groups 5.3. It is not known whether a compactly generated abelian prodiscrete compact-free group is finitely generated free. By Proposition 5.2, the free abelian group of countably many generators has a nondiscrete nonmetric pro-Lie group topology and fails to be compactly generated.

The Duality Theory of Abelian Pro-Lie Groups The structure theory results we discussed permit us to derive results on the duality of abelian pro-Lie groups. Recall that this class contains the class of all locally compact abelian groups properly.

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 = Hom(G, T) denote its dual with For any topological abelian group G we let G the compact open topology. (See e.g. [102, Chapter 7].) There is a natural morphism   given by ηG (g)(χ ) = χ(g) which may or may not of abelian groups ηG : G → G be continuous; information regarding this issue is to be found for instance in [102], notably in Theorem 7.7. We shall call a topological abelian group semireflexive if   is bijective and reflexive if ηG is an isomorphism of topological groups; ηG : G → G in the latter case G is also said to have duality (see [102, Definition 7.8]). There is an example of a nondiscrete but prodiscrete abelian torsion group due to Banaszczyk [5, p. 159, Example 17.11], which is semireflexive but not reflexive (see also Chapter 14, Example Ac.2. Therefore we know that the category of abelian pro-Lie groups is not self-dual under Pontryagin duality. Recall the following observations. Proposition 5.33. The dual of a weakly complete vector group V is naturally isomorphic to the additive group of its vector space dual V  endowed with the finest locally convex vector space topology. The dual of the additive group of a real vector space E with the finest locally convex vector space topology is naturally isomorphic to the additive group of its vector space dual E  with the weak ∗-topology, that is the topology of pointwise convergence. Both a weakly complete vector group and the additive group of a real vector space given its finest locally convex topology are reflexive. Proof. See Appendix 2, Theorem A2.8.  is naturally Proposition 5.34. Let A and B be topological abelian groups. Then A×B   isomorphic to A × B, and if A and B are reflexive, then A × B is reflexive. Proof. Exercise . Exercise E5.4. Prove Proposition 5.34. [Hint. The proof is straightforward.] Proposition 5.35. Let G be an abelian pro-Lie group and let V be a vector group complement. Then G is reflexive, respectively, semireflexive iff G/V is reflexive, re is isomorphic to a product E × A spectively, semireflexive. The character group G where E is the additive group of a real vector space with its finest locally convex topology and A is the character group of an abelian pro-Lie group whose identity component is compact. Proof. By the Vector Group Splitting Theorem 5.20, G is isomorphic to V × H for a weakly complete vector group V and a pro-Lie group H with compact identity component H0 . The vector group V is reflexive by Proposition 5.33 and its dual is the additive group of a vector space with its finest locally convex vector space topology. The assertion now follows from Proposition 5.34.

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This result reduces the issue of duality of abelian pro-Lie groups to the class of abelian pro-Lie groups whose identity component is compact. Keep in mind for the following corollary that every connected topological group is almost connected. Thus it applies to all connected abelian pro-Lie groups. Theorem 5.36. Every almost connected abelian pro-Lie group is reflexive, and its character group is a direct sum of the additive topological group of a real vector space endowed with the finest locally convex topology and a discrete abelian group. Pontryagin duality establishes a contravariant functorial bijection between the categories of almost connected abelian pro-Lie groups and the full subcategory of the category of topological abelian groups containing all direct sums of vector groups with the finest locally convex topology and discrete abelian groups. Proof. By the Vector Group Splitting Theorem 5.20 according to which an abelian pro-Lie group is almost connected if and only if it is a direct product of a weakly complete vector group V and a compact group C. By Proposition 5.33, V is reflexive and its Pontryagin dual is a vector group obtained as the additive topological group of a real vector space which is given its finest locally convex topology. By the Pontryagin duality between the categories of compact abelian groups and discrete abelian groups (see for instance [102], notably Chapters 7 and 8), C is reflexive and its dual is a discrete abelian group. Then the preceding Proposition 5.35 proves the first assertion. The remaining duality statement is then automatic. Exercise E5.5. Prove the following result. Proposition. If G is an abelian pro-Lie group then the characters of G separate the   ηG (g)(χ ) = χ(g) is injective. points, and the canonical morphism ηG : G → G, [Hint. If g = 0, find N ∈ N (G) not containing g. Let q : G → G/N be the quotient morphism. On the abelian Lie group G/N find a character χ not annihilating q(g) = 0. Then χ  q is a character of G not annihilating g. The kernel of ηG consists of all g which are annihilated by all characters and this is singleton.] The following lemmas deal with not necessarily abelian pro-Lie groups. Lemma 5.37. Assume that G is a pro-Lie group. Then G0 is the intersection of open subgroups. Proof. By Proposition 3.31, G/G0 is protodiscrete. Hence the filter basis N (G/G0 ) consists of open normal subgroups and converges to the identity. If g ∈ G \ G0 , then gG0 = G0 and so there is a normal subgroup N of G such that N/G0 ∈ N (G/G0 ) not containing gG0 . But then N is an open normal subgroup not containing g. A pair (G, H ) consisting of a topological group G and a subgroup is said to have enough compact sets if for each compact subset K of G/H there is a compact subset C of G such that (CH )/H ⊇ K. (See [102, Definition 7.12 (ii)].)

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Lemma 5.38. Let G be a topological group and N a closed normal subgroup. (i) Assume that the quotient morphism G → G/N has local cross sections. Then for every N ∈ N (G), the pair (G, N ) has enough compact sets. (ii) If N ∈ N (G), then G → G/N has local cross sections provided G is a pro-Lie group or G is protodiscrete. Proof. (i) Assume that the quotient morphism qN : G → G/N has local cross sections. Then there is an open identity neighborhood U satisfying U N = U and a continuous function σ : U/N → G such that (qN |U )  σ : U/N → U/N is the identity. Let V be a compact identity neighborhood in the Lie group G/N which is contained in U/N and let V ◦ denote its interior.  Now let K be a compact subset of G/N . Then K ⊆ k∈K kV ◦ . Thus by the compactness of K we find elements k1 , . . . , kn ∈ K such that K ⊆ k1 V ∪ · · · ∪ kn V . −1 We pick elements gj ∈ qN (kj ), j = 1, . . . , n and set C = g1 σ (V ) ∪ · · · ∪ gn σ (V ). Then C is compact as a finite union of compact sets, and qN (C) =

n  j =1

qN (gj )qN σ (V ) =

n 

kj V ⊇ K.

j =1

(ii) If N ∈ N (G) and G is a pro-Lie group then qN has local cross sections by Theorem 4.22 (iv). If G is protodiscrete, and N ∈ N (G), then G/N is discrete; so local (and even global cross) sections exist trivially. Lemma 5.39. Let G be an abelian proto-Lie group and let N ∈ N (G). Assume that the pair (G, N) has enough compact sets. Then:  → N ⊥ , λG,N (χ )(g) = χ(g+N ), is an isomorphism (i) The morphism λG,N : G/N of topological groups. (ii) N ⊥ is a compactly generated locally compact abelian group. (iii) The group N ⊥ is compact if and only if N is open. Proof. (i) If the pair (G, N ) has enough compact subsets, the claim that λG,N is an isomorphism of topological groups follows at once from [102, Lemma 7.17 (i)]. (ii) Since G/N is an abelian Lie group, it is isomorphic to Rp × Tq × D for a  is discrete group D (see for instance [102, Corollary 7.58 (iii)]). Thus N ⊥ ∼ = G/N p q   isomorphic to R × Z × D, where D is a compact abelian group and every compact  for a discrete abelian group. Now Rp × D  is an almost abelian group occurs as D connected locally compact abelian group, and every almost connected locally compact abelian group is of this form by the Vector Group Splitting Lemma. (iii) follows from the fact that G/N is discrete iff N is open and the facts that the character group of a discrete group is compact and vice versa.  its charProposition 5.40. Assume that G is an abelian proto-Lie group and A = G acter group. Then:

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(i) A is the directed union of the family of locally compact subgroups N ⊥ , N ∈ N (G), each of which is algebraically generated by a compact subset. (ii) If G is protodiscrete then A = comp(A). Proof. (i) Let U be a zero-neighborhood of T in which every subgroup is singleton. If χ : G → T is a character of an abelian pro-Lie group G, then all subgroups contained in χ −1 (U ) are annihilated. Since lim N (G) = 0, there is an N that  ∈ N (G) such ⊥ = χ (N ) ⊆ U and thus χ (N) = {0}, that is, χ ∈ N ⊥ . Thus G N ∈N (G) N . Since N (G) = {N : N ∈ N (G)} is a filter basis, {N ⊥ : N ∈ N (G)} is an upwards directed family of closed subgroups of A. By Lemma 5.39, every N ⊥ is a locally compact and compactly generated subgroup. (ii)A pro-Lie group G is totally disconnected iff it is prodiscrete by Proposition 4.23. If this condition is satisfied, then N (G) is a basis for the filter of identity neighborhoods. Thus by 5.39 (iii), every N ⊥ is compact. Hence by (i), we get (ii). def

The character group A = (ZN ) of a power of copies of Z (see Example 5.1 (ii)(b)) is a sum of copies of the circle group T with the compact open topology; it is an arcwise connected topological abelian group. From Theorem 5.20 we know connected abelian pro-Lie groups are isomorphic to V × C for a weakly complete vector group V and a compact connected group C, and thus we can say without even detailing the topology on A that this group is not a pro-Lie group. It does, however, satisfy A = comp(A).  its dual. Then Proposition 5.41. Let G be an abelian pro-Lie group and A = G G⊥ = comp(A). 0 0 is Proof. (i) By 3.31, G/G0 is protodiscrete, and thus by Proposition 5.40 (ii), G/G ⊥  the union of its compact subgroups. Now λ : G/G0 → G0 , λ((χ )(g)) = χ(g + G0 ) is a bijective morphism of topological abelian groups. Since a bijective morphism of topological groups maps each compact subgroup isomorphically onto its image, G⊥ 0 is ⊆ comp(A). a union of compact subgroups. Thus G⊥ 0 ⊥ ⊥  (ii) κ : A/G⊥ 0 → G0 , κ(χ +G0 ) = χ (h) maps A/G0 continuously and bijectively   onto A|G0 ⊆ G0 . But by Theorem 5.36 we know that G0 ∼ = E ×D where E is a vector space with its finest locally convex vector space topology, and D, as the character group of a compact connected abelian group is a discrete and torsion-free abelian group (see 0 ) is singleton. Therefore A|G0 and thus A/G⊥ [102, Corollary 8.5]). Hence comp(G 0 do not contain any nondegenerate compact subgroups. It follows that comp(A) ⊆ G⊥ 0. . Therefore comp(A) = G⊥ 0

The Toral Homomorphic Images of an Abelian Pro-Lie Group From the duality of compact abelian groups we cite some results which we shall immediately apply to abelian pro-Lie groups. Let C denote a compact abelian group.

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 contains a free subgroup of maximal rank such Then the (discrete) character group C  that C/F is a torsion group. The subgroup F is not unique. Indeed, one way of get → Q ⊗ C,  f (x) = 1 ⊗ x, ting F is as follows. We consider the morphism f : C where the tensor product is taken over the ring of integers. Then ker f is exactly  of C,  and the tensor product Q ⊗ C  is a rational vector the torsion subgroup tor C   contains a space which is spanned over Q by f (C). Thus by Zorn’s Lemma, f (C)   Q-basis B, whose Z-span in Q ⊗ C is a free subgroup B whose rank is dimQ (Q ⊗ C), def  Let A = f −1 (B) be the full inverse image of this the torsion-free rank of C. ∼ free abelian group. Then A/ tor C = B is a free abelian group. Since free groups  ⊕ F . Then are projective, there is a free subgroup F of A such that A = tor C f |F : F → B is an isomorphism. The factor group A/F is torsion as is the group ∼   is torsion.   which is a torsion group. Thus C/F C/A ⊆ (Q ⊗ C)/B = f (C)/B def  The character group T = F is a torus, that is, a product of circles, and the dual of the   gives a surjective morphism f: C  → T by duality, inclusion morphism j : F → C  ∼ and C = C. Hence we have a surjective morphism of compact groups q : C → T such  R) → Hom(T, R) is an isomorphism, since C → Q⊗R that Hom( q , T) : Hom(C,  and T → Q ⊗ T induce isomorphisms  R) → Hom(C,  R), Hom(Q ⊗ C, Hom(Q ⊗ T , R) → Hom(T , R). Since L(C) = Hom(R, C), respectively, L(T ) = Hom(R, T ), is isomorphic to  R), respectively, Hom(T , R), we conclude that L(q) : L(C) → L(T ) is an Hom(C, def

isomorphism. This is also seen from the fact that D = ker q is the character group of  , a torsion group and is, therefore, a totally disconnected compact abelian group. C/F Thus the exact sequence q 0 → D → C −−→ T → 0 gives rise to an exact sequence L(q)

0 → L(D) = {0} → L(C) −−−→ L(T ) → 0. For further details see [102, Chapter 8, Definition 8.12–8.20]. In particular, it is shown in [102, Corollary 8.18], that for a connected compact abelian group G, the subgroups D may be chosen arbitrarily small. These considerations motivate Definition 5.42. Let G be an abelian topological group. A closed subgroup D of G will be called a cotorus subgroup if G/D is a torus and the quotient morphism q : G → G/D induces an isomorphism L(q) : L(G) → L(C/D) of Lie algebras. Any group isomorphic to G/D for a cotorus subgroup D of G will be called a toral homomorphic image of G We noted in the discussion preceding this definitions that, in a compact abelian group, cotorus subgroups always exist, and that they are zero-dimensional. We will now show that an abelian pro-Lie group G always has cotorus subgroups, and indeed arbitrarily small ones if G is connected.

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Proposition 5.43. An abelian pro-Lie group G always has at least one cotorus subgroup D, and D is prodiscrete. If G is connected, then D may be chosen arbitrarily small. Proof. By Theorem 5.20, G has a unique largest compact connected subgroup comp G0 = (comp G)0 . Let D1 be a cotorus subgroup of comp G0 . Set G1 = G/D1 . Then (G1 )0 ∼ = R I × comp(G1 )0 ∼ = RI × TJ for suitable sets I and J , I ∪J ∼ and L(G) = R . Now we apply Theorem 5.19 and deduce that G1 is the direct product of (G1 )0 and G1 /(G1 )0 ∼ = G/G0 . Thus we obtain a quotient morphism q : G → RI × TJ × G/G0 . Now we set D = q −1 (ZI × {0} × G/G0 ). Then G ∼ R I × TJ × G/G0 ∼ I = T × TJ . = I D Z × {0} × G/G0 Moreover D/D1 ∼ = ZI × (G/G0 ). That is, we have an exact sequence 0 → D1 → G → ZI × (G/G0 ) → 0, yielding an exact sequence 0 → L(D1 ) → L(G) → L(Z)I × L(G/G0 ) → 0 by Theorem 4.20 and the limit hence product preservation of L. Now L(D1 ) = {0} and L(G/G0 ) = {0} by Proposition 3.30. It follows that L(D) = {0} and that therefore, in view of 4.20, L(G) → L(G/D) is an isomorphism. So D is a cotorus subgroup of G. If G is connected, we may take D = D1 and since D1 may be taken arbitrarily small by [102, Corollary 8.18], we are finished. Corollary 5.44. If G is an abelian pro-Lie group then there is a quotient morphism q : G → TK for some set K such that L(q) is an isomorphism and there is a commutative diagram L(q) L(G) −−−→ R⏐K ⏐ ⏐exp K expG ⏐   T G −−−→ TK . q

Proof. This is just a reformulation of Proposition 5.43. Later, in Chapter 6 and Chapter 8 we shall see that the additive group of L(G)  the simply connected universal group of G. In [102, Chapter 9] it is is written as G, discussed that the fundamental group π1 (C) of a compact group is naturally isomorphic  Z). Thus there are many reflexive topological abelian groups G (definition to Hom(C,  R) agrees with in this chapter in the discussion preceding 5.33) for which Hom(G,  = Hom(C,  R) ∼ C = Rdim C where dim C is the cardinal representing the torsion for which  Among these there are those like G = C  or (Q ⊗ C) free rank of C. (dim C) ∼  π1 (G) = {0} while the torus homomorphic image T = F = (Z ) ∼ = Tdim C has

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the largest possible fundamental group π1 (T ) = Hom(T, Z) ∼ = Hom(Z(dim C) , Z) = dim C dim C ∼ Hom(Z, Z) . =Z To some extent this can be generalized to an arbitrary (not necessarily commutative) pro-Lie group G by considering cotorus subgroups D of the center Z(G). Proposition 5.45. Let G be a pro-Lie group and D a cotorus subgroup of its center. def Then there is a quotient map q : G → H = G/D such that L(q) : L(G) → L(H ) is an isomorphism and Z(H )0 is a torus. Moreover, if G is connected, Z(H ) itself is connected and is a torus, and D may be chosen arbitrarily small. Proof. By Theorem 4.20 we have an exact sequence L(q)

0 → L(D) → L(G) −−−→ L(H ) → 0. Since D is prodiscrete by Proposition 5.43, we conclude L(D) = {0} (see 4.23). Since bijective morphisms between weakly complete topological vector spaces are isomorphisms, we infer that L(q) is an isomorphism. Now let G be connected; consider the center Z(H ) of H and its full inverse image def

A = q −1 (Z(H )). Since Z(H ) is the center of H , we have comm(G × A) ⊆ D. Now comm(G × {x}) is connected for each x and contains 1. But D is totally disconnected, and thus comm(G × A) = {1}, that is, A ⊆ Z(G). Therefore Z(H ) = Z(G)/D = Z(H )0 since D is a cotorus subgroup of Z(G). Then, by Proposition 5.43, D may be chosen arbitrarily small. The main purpose of the cotorus subgroups, however, is that they permit us to prove for pro-Lie groups the existence of good substitutes for universal covering morphisms of Lie groups. The Resolution Theorem of Abelian Pro-Lie Groups Theorem 5.46. Let G be an abelian pro-Lie group and write G = V × H with a vector group complement V according to the Vector Group Splitting Theorem for Abelian ProLie Groups 5.20. Let D be a cotorus subgroup of H . Then D is a prodiscrete group and the morphism δ : L(G) × D → G,

δ(X, x) = (expG X)x,

is a quotient morphism inducing an isomorphism of Lie algebras and having the prodiscrete kernel {(X, exp −x) : exp X ∈ D} ∼ = exp−1 D ∼ = exp−1 (D ∩ comp(G0 )). Proof. Since L(G) = L(V ) × L(H ) and expV : L(V ) → V is an isomorphism, the assertion on G it true if it is true on H . Thus without loss of generality we now assume that V = {0}. Then G0 = (comp G)0 is compact and G = G0 D with a compact def

intersection  = D ∩ G0 , so that the morphism ϕ : G0 × D → G,

ϕ(g0 , d) = g0 d

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is a quotient morphism with kernel {(−d, d) : d ∈ } ∼ = . The compact subgroup  is a cotorus subgroup of G0 . So the morphism μ : L(G) ×  → G0 , μ(X, x) = (exp X)x, is a quotient morphism by the Resolution Theorem for Compact Groups ([102, Theorem 8.20]). So we conclude that ν : L(G) ×  × D → G,

ν(X, d1 , d) = (expG X)d1 d

which essentially is the composition ϕ  ν of two quotient morphisms is itself a quotient morphism. Since  is compact, the map κ :  × D → D,

κ(d1 , d) = d1 d

is a quotient morphism. So idL(G) ×κ : L(G) ×  × D → L(G) × D is a quotient map. Now ν = δ  (idL(G) ×κ) and under these circumstances, δ is a quotient morphism. So by Theorem 4.20, L(δ) is a quotient morphism. The assertion about ker δ is straightforward; so ker δ is totally disconnected, and thus L(ker δ) = {0} by 4.23. Hence L(δ) is an isomorphism. A quotient morphism with a prodiscrete kernel is the next best thing we can have to a covering morphism. We have two quotient morphisms with a totally disconnected kernel at work, permitting us to formulate the following “Sandwich Theorem”. The Sandwich Theorem for Abelian Pro-Lie Groups Corollary 5.47. Let G and D be as in Theorem 5.46 and write  = D ∩ comp(G)0 . Then there are quotient maps with totally disconnected kernels δ

ρ

L(G) × D −−→G−−→

G0 D ×  

such that G0 / ∼ = RI × TJ for suitable sets I and J where L(G) ∼ = RI ∪J and that the composition D G0 × ρ  δ : L(G) × D →   is the obvious map, namely, the product of the exponential function L(G) → G0 / and the quotient morphism D → D/. Proof. The information on δ was established in Theorem 5.46, and the information on ρ in Proposition 5.43 and its proof.

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Postscript By contrast with the category of locally compact abelian groups, the larger category AbproLieGr of abelian pro-Lie groups is closed in the category of all abelian topological groups and continuous group morphisms under the formation of arbitrary limits and the passing to closed subgroups. Each locally compact abelian group is a proLie group and thus arbitrary products of locally compact abelian groups and arbitrary closed subgroups of such products are abelian pro-Lie groups. Therefore, the category AbproLieGr of abelian pro-Lie groups is rather large. In the first section we recorded some individual examples so that the reader can form a first impression of the type of topological abelian groups we face. This addresses in particular those readers whose intuition is trained by studying locally compact abelian groups which are the daily bread of classical harmonic analysis and whose structure was exposed in [102], Chapters 7 and 8. The most remarkable of the prodiscrete examples presented here is the example of the free abelian group on countably infinitely many generators which supports a nonmetric totally disconnected pro-Lie group structure with bizarre properties (Proposition 5.1). One of the most prominent examples of connected abelian pro-Lie groups is the additive topological group of the dual vector space of an arbitrary real vector space, where the dual E  = HomR (E, R) of a real vector space is given the topology of pointwise convergence; this topology is also called the weak ∗-topology. It makes the dual into a complete topological vector space which is called a weakly complete topological vector space. Therefore its additive topological group is called a weakly complete vector group, and whenever it occurs as a subgroup of a topological group, it is called a weakly complete vector subgroup. Weakly complete vector groups have a perfect duality theory as is discussed in Appendix 2. The character group and the vector spaces dual of a topological vector space are naturally isomorphic. The compact open topology on the dual of a weakly complete vector space is the finest locally convex topology on the vector space dual. Since the character group of a weakly complete vector group is a vector space and thus is a direct sum of copies of R, every weakly complete vector group is isomorphic to a vector group of the form RJ for some set J . Thus a weakly complete vector group is locally compact if and only if it is finite-dimensional. The transfinite topological dimension of RJ as discussed in [103] is card J . Therefore the topological dimension of a weakly complete vector group agrees with the linear dimension of its dual. Pontryagin duality establishes a dual equivalence between the category of compact abelian groups and the category of (discrete) abelian groups. Therefore an abelian pro-Lie group is a direct product of a weakly complete vector group and a compact abelian group if and only if its dual is a direct sum of a real vector space given its finest locally convex topology, and some discrete abelian group. The main result of this chapter is that every abelian pro-Lie group G is isomorphic to the direct product W ×H of a weakly complete vector subgroup W of G and an abelian pro-Lie group H whose identity component H0 is compact and is the intersection of

Postscript

247

open subgroups. In particular, a connected abelian pro-Lie group is the direct product of a vector group and a compact group. Thus for such abelian pro-Lie groups the structure theory is completely reduced to the theory of weakly complete topological vector spaces on the one hand and to the familiar compact group situation on the other hand. From this result we derive that a necessary and sufficient condition for a connected abelian pro-Lie group to be locally compact is that it is algebraically generated by a compact set. In a somewhat loose way one could describe the situation by saying that abelian pro-Lie groups are very well understood if they are connected (or even not far from being connected) while the area of totally disconnected, that is, prodiscrete abelian groups, is a wide open field. We have observed that almost connected abelian pro-Lie groups are reflexive and have a lucid duality theory which is completely known due to the known vector space duality of weakly complete topological vector spaces and the duality of compact abelian groups. The Lie theory of abelian pro-Lie groups is likewise lucid: Since L(G) = L(G0 ) for any topological abelian group, it deals with the connected part of G only, and if G0 is an abelian pro-Lie group then the Vector Group Splitting Theorem says that G0 is algebraically and topologically the direct sum G0 = V ⊕ comp(G) of a weakly complete vector subgroup complement V and the unique largest compact subgroup comp(G); hence L(G) = L(V ) ⊕ L(comp(G), where expV : L(V ) → V is an isomorphism of a weakly complete vector group and where the exponential function expcomp(G) : L(comp(G)) → comp(G) is very well understood from [102], Chapters 7 and 8. From the theory of the exponential function of compact groups we know that, except for the case of a torus group G, the exponential function expG is rarely if ever surjective. In [102, Chapter 8, Theorem 8.20] it was shown in the so called “Resolution Theorem for Compact Abelian Groups” that this defect could be compensated for by finding a suitable totally disconnected closed subgroup D and a quotient morphism δ : L(G) × D → G such that L(δ) is an isomorphism of Lie algebras. It is noteworthy, that the Resolution Theorem remains intact for arbitrary abelian pro-Lie groups. In the end we have a sandwich theorem of the following nature: There is a totally disconnected subgroup D intersecting G0 in a compact totally disconnected subgroup , and there are quotient morphisms G D ×   inducing isomorphisms on the Lie algebra level such that the composition is the product of the exponential function δ

ρ

L(G) × D −−→ G −−→

expG

−−−→ L(G) ⏐ ⏐ ∼ = RI ∪J

G0  ⏐ ⏐∼ =

−−−→ RI × TJ quot

248

5 Abelian Pro-Lie Groups

and the quotient map D → D/. These results highlight once more the role of weakly complete topological vector spaces and compact groups play in the structure theory of abelian pro-Lie groups. We shall see more resolution theorems in the nonabelian environment in Chapter 12.

Chapter 6

Lie’s Third Fundamental Theorem

The concept of a prosimply connected pro-Lie group is introduced. It coincides with the purely topological concept of a simply connected topological group when the proLie group is in fact a Lie group. It is proved that for every pro-Lie algebra there is a prosimply connected pro-Lie group whose Lie algebra it is, and this prosimply connected pro-Lie group is unique up to isomorphism. In fact, all of this is done functorially and the universal properties arising in this way are studied. Prerequisites. We need the general theory of pro-Lie groups such as presented in Chapters 3 and 4. We also require from the reader a certain familiarity with basic category theory; the concept of an adjoint functor which was used extensively in Chapter 2 will be used essentially here as well.

Lie’s Third Fundamental Theorem for Pro-Lie Groups Consider a pro-Lie group G and the filter basis N (G) of closed normal subgroups N such that G/N is a Lie group. Then G ∈ N (G) and the singleton G/G is a simply connected Lie group. Let us define def

N S(G) = {N ∈ N (G) : G/N is simply connected}. Recall from Chapter 1, Definition 1.12ff. that a subset C of a directed set is cofinal if for each d ∈ D there is a c ∈ C such that d ≤ c. Definition 6.1. A pro-Lie group G is called prosimply connected if N S(G) is cofinal in N (G). We notice that for a prosimply connected pro-Lie group G, the set N S(G) is a filter basis and that G = limN ∈N S(G) G/N by the Cofinality Lemma 1.13 (or 1.21). From Corollary 4.22 (iii) it follows that a prosimply connected pro-Lie group is connected. A pro-Lie group G is a (finite-dimensional) Lie group if and only if {1} ∈ N (G). Accordingly, a Lie group is prosimply connected if and only if it is simply connected. For the simply connected Lie group R, the filter basis N (R) consists of R itself and all cyclic subgroups including {0}. For {0}, R = N ∈ N (R) the quotient R/N is a circle group and is therefore not simply connected. Therefore it would not have been a good idea, in Definition 6.1 to postulate that G/N is simply connected for all N ∈ N (G). Simple connectivity may be defined in several reasonable but nonequivalent ways, including firstly the way it was done in [102, Definition A2.6]. According to this

250

6 Lie’s Third Fundamental Theorem

definition, a space X is called simply connected if it has the following universal property: For any covering map p : E → B between topological spaces, any point e0 ∈ E and any continuous function f : X → B with p(e0 ) = f (x0 ) for some x0 ∈ X there is a continuous map f: X → E such that p  f = f and f(x0 ) = e0 . Let us say that a space is loopwise simply connected if it is connected and π1 (X) = 0. (See e.g. [102, Definition 8.60.]) Examples 6.2. (i) Any product of simply connected finite-dimensional Lie groups is simply connected in any sense and is also a prosimply connected pro-Lie group. (ii) Let G be a compact connected abelian group; then G is a pro-Lie group (see [102, Corollary 2.43]). Every Lie group quotient G/N , N ∈ N (G) is a torus. Thus G is never prosimply connected.  Z). Thus if G is the 1-dimensional By [102, Theorem 8.62], π1 (G) ∼ = Hom(G, ∼ , then G solenoid Q = Q and π1 (G) = {0}, that is, G is loopwise simply connected. However, no compact connected abelian group is simply connected (see [102, Theorem 9.29]). While the definition of prosimple connectedness of a pro-Lie group G does not agree with any of the traditional concepts of simple connectivity, we shall show in Corollary 8.15 in a later chapter that it does agree with the concept of simple connectivity expressed in terms of the lifting property described in the paragraph preceding 6.2 above (see e.g. [102, Definition A2.6]), and the one expressed by the vanishing of the fundamental group π1 (G). Proposition 6.3. Let G be a pro-Lie group. (a) Assume N ∈ N (G). Consider the following two statements: (i) N ∈ N S(G). (ii) N is connected. Then (i) ⇒ (ii), and if G is prosimply connected, both statements are equivalent. (b) Assume that N (G) contains a cofinal subset C of connected subgroups N of G. Then G is locally connected. (c) Any prosimply connected pro-Lie group is locally connected. Proof. We begin by proving (a). (i) ⇒ (ii): Let N ∈ N S(G); then G/N is a simply connected Lie group. Since the identity component N0 of N is characteristic and closed, it is normal in G, and G/N0 is a pro-Lie group by the Quotient Theorem for Pro-Lie Groups Revisited 4.28 (i). Now N/N0 is a pro-Lie group as a closed subgroup of G/N0 by the Closed Subgroup Theorem 3.35. Since N/N0 is a totally disconnected normal subgroup of a connected group, it is central by Lemma 12.55. Thus N/N0 is a totally disconnected abelian proLie group and so is prodiscrete by Proposition 4.23. Hence there is an open subgroup M of N such that M/N0 , being central in G/N0 , is normal in G/N0 , whence M is normal in G. The quotient morphism q : G/M → G/N has the discrete kernel N/M and therefore is a covering morphism. Since G/N is simply connected, q is an

Lie’s Third Fundamental Theorem for Pro-Lie Groups

251

isomorphism, that is M = N. Therefore, the prodiscrete group N/N0 has no open normal subgroups and thus is singleton. Hence N = N0 which is what we had to show. (ii) ⇒ (i): Assume that G is prosimply connected and that N is connected. Since N S(G) is cofinal in N (G), there is an M such that M ⊆ N. Thus there is a quotient morphism q : G/M → G/N. Now assume that ϕ : (G/N ) → G/N is the universal covering morphism of connected Lie groups. Since G/M is simply connected, by the lifting property, there is a morphism f : G/M → (G/N ) such that ϕ  f = q. Let qM : G → G/M denote the quotient morphism. Since ϕ is a covering morphism and f is the lifting of a quotient morphism, f and thus f  qM are quotient morphism. Also (G/N), being the universal covering of the Lie group G/N , is a Lie group. Thus def N1 = ker(f  qN ) ∈ N (G) and N1 ⊆ N. Also, N/N1 ∼ = ker ϕ, and the kernel ker ϕ of the universal covering of G/N is discrete. Hence N1 is open in the connected group N, whence N1 = N. Thus ϕ is an isomorphism and so G/N is a simply connected Lie group, that is, N ∈ N S(G). (b) Assume the existence of C as stated in (b). We have to show that G has arbitrarily small connected identity neighborhoods. Indeed G has arbitrarily small identity neighborhoods V containing an N ∈ C such that V N = V and that V /N is an open cell neighborhood of the identity in the Lie group G/N . Let W be any nonempty open closed subset of V . Since N is connected, for all w ∈ W we get wN ⊆ W . Hence W N = W . Then W/N is nonempty open closed in the open cell V /N and thus W = V . So V is connected, and (b) is proved. (c) is now an immediate consequence of (a) and (b). If G = SO(3)N , then the members of N (G) are precisely the cofinite partial products of G, and all of these are connected. But N S(G) = {G}. One notices directly, that G is locally connected. At any rate, for any prosimply connected pro-Lie group G, by Proposition 6.3, all members of N S(G) are connected. The occurrence of prosimply connected pro-Lie groups is not a rare event. Indeed, Theorem 6.4 below asserts the existence of a plethora of prosimply connected pro-Lie groups. Subsequent results will place the construction into a more functorial frame. But first we recall a basic result of Lie group theory utilizing the universal property of simple connectivity: Lemma L. Let S be a simply connected Lie group and L a Lie group, and assume that there is a morphism α : L(S) → L(L). Then there is a unique morphism of Lie groups a : S → L such that L(a) = α. The following diagram commutes: α

L(S) −−→ L(L) ⏐ ⏐ ⏐exp expS ⏐   L S −−→ L.

(SC)

a

Proof (Indication). The proof is standard Lie theory: Let B be a convex zero neighborhood of L(S) such that expS induces a homeomorphism expS |B : B → V onto an identity neighborhood of S. Then expL  α  (expS |B)−1 : B → L is a local morphism

252

6 Lie’s Third Fundamental Theorem

which by the simple connectivity of S extends uniquely to a morphism a : S → L such that the diagram (SC) commutes (see e.g. [102, Corollary A2.26]; note that U is assumed to be connected there). This completes the indication of proof of the lemma. Theorem 6.4 (Lie’s Third Fundamental Theorem for Pro-Lie Algebras). Let g be a pro-Lie algebra. Then there is a prosimply connected pro-Lie group (g) for which there is an isomorphism ηg : g → L((g)). Proof. For each i ∈  (g) there is (up to isomorphism) a unique simply connected Lie group Gi such that there is an isomorphism ϕi : g/i → L(Gi ). Since all Gi are simply connected, for i ⊇ j in  (g) by Lemma L, the quotient map quotij : g/j → g/i induces a unique surjective morphism fij : Gj → Gi such that the following diagram commutes: quotij

←−−−−− g/j g/i ⏐ ⏐ ⏐ ⏐ϕ ϕi   j L(G −−−− L(G ⏐ i ) ←− ⏐ j) L(fij ) ⏐expG expG ⏐  i j Gi ←−−−−− Gj . fij

Now {fij : Gj → Gi | (i, j) ∈  (g) ×  (g), j ⊇ i} def

is a projective system of finite-dimensional Lie groups. Let (g) = limi∈ (g) Gi be its limit and let fi : (g) → Gi be the limit maps. We claim that (g) is a prodef

Lie group. Indeed the filter basis N S(G) = {ker fi | i ∈  (g)} converges to 1 by Theorem 1.27 (ii), and (g) is complete as the limit of complete groups, whence γGN S(G) : (g) → GN S(G) is an isomorphism by 1.30 and 1.33, inverting the natural isomorphism η : GN S(G) → (g) of 1.27 (ii). For each i ∈  (g), the following diagram commutes: η / (g) GN S(G) MMM MMM MMMfi νker fi fi MMM MMM M &   / Gj , (g)/ ker fi  fi

where η = γG−1 . The filter basis N S(G) is cofinal in N ((g)). The fi are quotient N S(G) morphisms by 1.27 (iii). Therefore each fi : (g)/ ker fi → Gi is an isomorphism of finite-dimensional Lie groups. Thus all quotients (g)/ ker fi are simply connected. Now GN is a pro-Lie group by 1.29 and 1.40. Thus (g) is a pro-Lie group as claimed. By Definition 6.1 and what we showed above, it is simply connected. Since L is a continuous functor by 2.25 (ii) we have an isomorphism α : L((g)) → limi∈ (g) L(Gi ) and thus we obtain an isomorphism ηg as a composition

Lie’s Third Fundamental Theorem for Pro-Lie Groups γg

253

α −1

limi∈ (g) ϕi

g −−→ limi∈ (g) g/i −−−−−−→ limi∈ (g) L(Gi ) −−→ L(limi∈ (g) Gi ) = L((g)). This completes the proof. Exercise E6.1. Prove the following assertion: Let g be an abelian pro-Lie algebra, that is, a weakly complete topological vector space; then the additive group (g) of g is a simply connected pro-Lie group such that L((g)) = Hom(R, (g)) ∼ = g. This theorem has a functorial enhancement of considerable elegance. We have to refer to the concept of adjoint functors. (See e.g. [102, Appendix 2, Definition A3.29].) Lie’s Third Fundamental Theorem, Functorial Version Theorem 6.5. The construction  of Theorem 6.4 assigning to a pro-Lie algebra a prosimply connected pro-Lie group extends to a functor  : proLieAlg → proLieGr which is left adjoint to the Lie algebra functor L : proLieGr → proLieAlg. Proof. By TheoremA3.28 of [102] we have to verify the universal property best pursued by the following diagram. proLieAlg ηg

proLieGr

g ⏐ ⏐ ∀f 

−−−−→

L((g)) ⏐ ⏐  L(f )

L(H )

−−−−→

L(H )

idL(H )

(g) ⏐ ⏐  ∃!f

()

H

For each morphism of topological Lie algebras f : g → L(H ) for a pro-Lie group H we have to find a unique morphism of topological groups f  : (g) → H such that L(f  )  ηg = f . Thus assume we are given f : g → L(H ). For each N ∈ N (H ) let qN : H → H /N be the quotient map. Then incl

L(qN )

0 → L(N ) −−−−→ L(H ) −−−−→ L(H /N ) is exact since L preserves kernels (see e.g. 3.22). Since dim L(H /N ) < ∞ we have def

L(N) ∈  (L)(H ). Then i(N ) = f −1 (L(N )) ∈  (g) and f induces a morphism of topological Lie algebras fN : g/i(N ) → L(G/N ). By definition, (g) = limi∈ (g) Gi for a projective system of simply connected Lie groups Gi such that there is an isomorphism ϕi : g/i → L(Gi ). Since Gi(N ) is simply connected there is a unique morphism of topological groups gN : Gi(N ) → H /N such that the following diagram is commutative: −1 ϕi(N) fN L(G⏐i(N ) ) −−−−→ g/i(N ) −−−−→ L(H⏐/N ) ⏐ ⏐exp expG  H /N i(N)  Gi(N ) −−−−→ Gi(N ) −−−−→ H /N; idGi(N)

gN

254

6 Lie’s Third Fundamental Theorem

−1 consequently, L(gN ) = ϕi(N )  fN . Let

pi : αi : βi :

(g) −→ Gi , limj∈ (g) g/j −→ g/i, limj∈ (g) L(Gj ) −→ L(Gi )

be the limit morphisms. Then we have, in particular, a commutative diagram in which lim is short for limi∈ (g) : λg

limi ϕi

L(qN )

id

g −−−−→ lim −−−−→ lim L(G g ⏐ ⏐i ⏐ i ) ←−−− ⏐ ⏐β αi(N) ⏐ =   i(N) id g g − − − − → − − − − → L(G i(N ⏐ ⏐) ⏐i(N ) ) ←−−− ϕi(N) quot ⏐ ⏐ ⏐L(g ) f fN   i(N) id L(H ) −−−−→ L(H /N ) −−−−→ L(H /N ) ←−−− α

L((g)) ⏐ ⏐L(p (N ))  i L(G⏐i(N ) ) ⏐L(g )  i(N) L(H /N ).

def

For each N ∈ N (H ) we set fN = gN pi(N ) : (g) → H /N. The map α −1 (limi ϕi )λg from the top left corner of the diagram to the top right corner is ηg by the definition of ηg in the proof of 6.4. Thus the outside of the diagram yields L(fN )  ηg = L(qn )  f.

(1)

If N1 ⊇ N2 in N (H ) we straightforwardly see that fN 2 = qN1 N2 fN 1 with the obvious quotient morphism qN1 N2 : H /N2 → H /N1 . Then the universal property of the limit H = limN∈N (H ) H /N attaches to the cone {fN : L((g)) → L(H /N ) | N ∈ N (H )} a unique morphism f  : (g) → H such that qN  f  = fN for all N ∈ N (H ) and thus L(qN )  L(f  ) = L(fN ). (2) So from (1) and (2) we have L(qN )L(f  )ηg = L(qN )f for all N ∈ N (H ). Since L is limit preserving, the L(qN ) form a limit cone and thus separate the points. Therefore L(f  )  ηg = f. The question of the uniqueness of f  remains. Assume also that f  : (g) → H satisfies L(f  )  ηg = f = L(f  )  ηg . Since ηg is an epic (being even an isomorphism) this means L(f  ) = L(f  ). Now consider the two superimposed commutative diagrams L(f  )

L((g)) ⏐ ⏐ exp(g) ⏐  (g)

−−−−→ −−−−→  L(f ) f

−−−−→ −−−− →  f

L(H ⏐ ) ⏐ ⏐expH  H

and conclude f   exp(g) = f   exp(g) . Since the exponential function of a pro-Lie group is generating, we conclude f  = f  , and this completes the proof.

Lie’s Third Fundamental Theorem for Pro-Lie Groups

255

From purely category theoretical considerations, we obtain the following version of the universal properties of the adjoint functors L and  (see e.g. [102, Appendix 3, Proposition A3.36]). There is a natural morphism πG : (L(G)) → G of pro-Lie groups with the following universal property: Given a pro-Lie group G and any morphism f : (h) → G for some pro-Lie algebra h, there is a unique morphism f  : h → L(G) of pro-Lie algebras such that f = πG  (f  ). proLieAlg

proLieGr πG

L(G)  ⏐ ∃!f  ⏐

(L(G))  ⏐  ⏐(f )

−−−−→

G  ⏐∀f ⏐

h

(h)

−−−−→

(h).

id(h)

(⊥)

However, with the information from Chapter 4 that L preserves quotient morphisms it serves a useful purpose if we elaborate on this situation in the concrete case of pro-Lie groups. The Theorem on the Reflection into the Category of Prosimply Connected Pro-Lie Groups Theorem 6.6. (i) Let G be a pro-Lie group. Then there is a prosimply connected  and a pro-open morphism pro-Lie group G  −−−→ G πG : G which induces an isomorphism of pro-Lie algebras ∼ =

 −−−→ L(G). L(πG ) : L(G)  extends to a functor proLieGr → proLieGr onto the (ii) The assignment G  → G full subcategory proSimpConLieGr of prosimply connected pro-Lie groups, which is right adjoint to the inclusion functor proSimpConLieGr → proLieGr. In particular,  → G is a natural transformation with a prodiscrete kernel P (G) def = ker πG . πG : G (iii) The following statements (a) and (b) are equivalent:  → G is an isomorphism. (a) πG : G (b) G is prosimply connected. (iv) The image of πG is dense in G0 . (v) Let f : G → H be a morphism of connected pro-Lie groups such that L(f ) is an isomorphism and H is prosimply connected. Then f is an isomorphism.

256

6 Lie’s Third Fundamental Theorem

(vi) By a slight abuse of notation denote the corestriction of the functor  to the category of prosimply connected pro-Lie groups by  : proLieAlg → proSimpConLieGr and the restriction of the functor L to the category of prosimply connected pro-Lie groups by L : proSimpConLieGr → proLieAlg. Then the two functors  : proLieAlg → proSimpConLieGr,

L : proSimpConLieGr → proLieAlg

implement an equivalence of categories. (vii) If f : G → H is a bijective morphism of connected pro-Lie groups, then  → H  is L(f ) : L(G) → L(H ) is an isomorphism of pro-Lie algebras and f: G an isomorphism of prosimply connected pro-Lie algebras. In particular, a bijective morphism of prosimply connected pro-Lie groups is an isomorphism.  = (g) be the pro-Lie Proof. (i) Set g = L(G); then g is a pro-Lie algebra. We let G  group constructed in Theorem 6.1; then there is an isomorphism ηg : L(G) → L(G).  We construct πG : G → G. Let M ∈ N (G) and note that the quotient morphism qM : G → G/M induces an exact sequence L(qM )

incl

0 → L(M) −−−−→ L(G) = g −−−−→ L(G/M) → 0 ∼ limi∈ (g) g/i and L(qM ) induces is exact by Corollary 4.16. Hence L(M) ∈  (g) = †  = limi∈ (g) Gi for a family of an isomorphism qM : g/L(M) → L(G/M). Recall G simply connected finite-dimensional Lie groups Gi with an isomorphism ϕi : g/i → L(Gi ). In particular, there is an isomorphism ϕL(M) : g/L(M) → L(GL(M) ). Then † −1 qM  ϕL(M) : L(GL(M) ) → L(G/M) is an isomorphism. Hence the two Lie groups GL(M) and G/M are locally isomorphic (see e.g. [102, Theorem 5.42]; this theorem does not depend on the linearity of the Lie groups involved); thus there is an open connected identity neighborhood U of GL(M) , an open identity neighborhood V of G/M, and a homeomorphism μ0 : U → V satisfying μ0 (uu ) = μ0 (u)μ0 (u ) for u, u , uu ∈ U . Since GL(M) is a finite-dimensional simply connected Lie group, there is a unique morphism μM : GL(M) → G/M extending μ0 and implementing a local isomorphism (see e.g. [102, Corollary A2.26], as in the proof of Lemma L used in the proof of Theorem 6.4); the following diagram is commutative: L(G ⏐ i) ⏐ expG L(M)  GL(M)

L(μM )

−−−−→ L(G/M) ⏐ ⏐exp  G/M −−−−→ G/M, μM

 → Gi be a limit morphism. Then for each and L(μM ) =  Let pi : G  → G/N , αM = μM  pL(M) . The cone M ∈ N (G) we have a morphism αM : G property qM1 M2  αM2 = αM1 for M1 ⊇ M2 in N (G) is straightforwardly verified. Since G is a pro-Lie group, λG : G → limN ∈N (G) G/N is an isomorphism, and by  → G such that the universal property of the limit, there is a unique morphism πG : G qM  πG = αM , i.e., that the following diagram commutes for all M ∈ N (G): † qM

−1 ϕL(M) .

Lie’s Third Fundamental Theorem for Pro-Lie Groups

 G ⏐ ⏐ pM  GL(M)

πG

−−−−→ −−−−→ μM

G ⏐ ⏐q M G/M.

257

(3)

Then πG is pro-open and for each M there is a commutative diagram L(πG )

 −−−−→ L(⏐G) ⏐ L(pM ) L(GL(M) ) −−−−→ L(μM )

L(G) ⏐ ⏐L(q )  M L(G/M).

(4)

Recall that L(μM ) is an isomorphism of Lie algebras. The functor L preserves limits. Thus L(γG )

g = L(G) −−−−→

lim

M∈N (G)

† −1 limM∈N (G) (qM )

L(G/M) −−−−−−−−−−→

lim

M∈N (G)

L(G)/L(M)

is an isomorphism. This implies that the filter basis {L(M) ∈  (g) | M ∈ N (G)} converges to 0 in g. From Proposition 1.40 it follows that it is cofinal in  (g) and that,  may be identified with limM∈N (G) GL(M) by the Cofinality Lemma 1.21, therefore G in such a way that the pL(M) are the limit maps. As a consequence, in diagram (4), the vertical maps are the respective limit maps, and since the L(μM ) are isomorphisms it follows that L(πG ) is an isomorphism. This concludes the construction of πG and finishes the proof of (i).  = (L(G)) with the back adjunction πG (see e.g. [102, We have reconstructed G Definition A3.37]) in (⊥).  extends to a right adjoint functor to the inclusion (ii) In order to prove that G  → G  functor and that πG : G → G is a natural transformation, it again suffices to verify the universal property in a fashion analogous to that in the proof of 6.5: proSimpConLieGr

 G  ⏐  f ⏐ H

proLieGr πG  −−− G −→  ⏐  ⏐f

H

−−−−→ idH

G  ⏐f ⏐ H

We have to show that for each morphism of pro-Lie groups f : H → G from a simply  of topological connected pro-Lie group H there is a unique morphism f  : H → G groups such that π  f  = f . So let f : H → G be given and let M ∈ N (G). Since L(μM ) is an isomorphism, μM : GL(M) → G/M implements a covering morphism of the identity component (G/M)0 . Since H is simply connected, by Lemma L, qM f

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6 Lie’s Third Fundamental Theorem

has a unique lifting νM : H → GL(M) such that H ⏐ ⏐ f G

νM

−−−−→ GL(M) ⏐ ⏐μ  M −−−−→ G/M

(5)

qM

is commutative for all M ∈ N (G). We recall that {L(M) | M ∈ N (G)} is cofinal in  (g) and that therefore  = lim GL(M) G M∈N (G)

up to a natural identification, where the limit maps are denoted pM . If M1 ⊆ M2 then straightforwardly pM1 M2 νM2 = νM1 . Thus by the universal property of the limit, there  such that pM f  = νM for all M ∈ N (G). Then is a unique morphism f  : H → G    qM πG f = μM pM f = μM fM = qM f for all M ∈ N (G). Since the qM separate points, πG f  = f . Thus we have the existence of f  ; it remains to show that it is unique. If also πG f  = f = πG f  , then κ : H → ker πG , κ(h) = f  (h)(f  (h)−1 , is a continuous function from a connected space to ker πG . But L(ker πG ) = ker L(πG ) = {0} since L(πG ) is an isomorphism. Hence ker πG is totally disconnected, equivalently, prodiscrete by 3.30 (iv). Thus κ is constant and this proves f  = f  and completes the proof of (ii).  is prosimply connected. Let us prove (b) ⇒ (a): (iii) Trivially, (a) ⇒ (b) since G  such that We apply (i) with H = G and f = idG . This gives us a ν : G → G   being the πG ν idG . Thus G is a retract of G in proLieGr, the semidirect factor of G totally disconnected group ker πG . As a topological direct factor of a connected space, it must be singleton. Thus πG is a monic retraction and therefore is an isomorphism. (See [102, Remark A3.13 (ii)].)  → L(G) is an isomorphism. Then (iv) From (i) we know that L(πG ) : L(G) 0 ) by 4.22 (ii). Since G 0 = G  as G  is prosimply connected, the assertion G0 = πG (G is proved. The following commutative diagram may be helpful to visualize the situation: L(πG )

 −−−−→ L(G) L(⏐G) ⏐ ⏐ ⏐exp expG˜   G  G. G −−−−→ πG

(v) Let g = L(G); since H is prosimply connected we may write H = (h) and identify h and L((h)) via the isomorphism ηh in () in the proof of 6.5. Then by the universal property expressed in (), there is a unique morphism f ∗ : H = (h) → G such that L(f ∗ ) = L(f )−1 . Now L(f ∗  f ) = L(f ∗ )  L(f ) = idL(G) = L(idG ), and by the uniqueness in the universal property, we have f ∗  f = idG . Similarly, f  f ∗ = idH . Hence f ∗ = f −1 and f is an isomorphism. (vi) Let g be a pro-Lie algebra. Then ηG : g → L((g)) is an isomorphism for each pro-Lie algebra g by 6.4, and πG : (L(G)) → G is an isomorphism for each

Lie’s Third Fundamental Theorem for Pro-Lie Groups

259

prosimply connected pro-Lie group G by (iii) above. These statements together prove Claim (vi). (vii) By Theorem 2.25, the functor L preserves kernels; hence the injectivity of f implies the injectivity of L(f ). By the connectivity of G and by Theorem 4.22 (ii)(b), the morphism L(f ) is a surjective continuous and open morphism of pro-Lie algebras and is therefore an isomorphism. As does any functor, the functor  : proLieAlg → →H  proSimpConLieGr preserves isomorphisms. Since = L we obtain that f: G is an isomorphism. The last assertion then follows from this and (iii) above. We shall call the kernel P (G) of G the Poincaré group of G. In the case that G is a Lie group, this notation agrees with the classical one and thus correctly extends this  → G as the universal morphism of G; in the concept. We also shall refer to πG : G  will be called case of Lie groups this is the universal covering morphism. The group G the universal group. (Cf. Theorem 8.21 below.)  is isomorphic to the additive group of L(G) If G is a compact abelian group, then G and πG is equivalent to expG : L(G) → G which is a morphism of topological groups  and G will be quite different in almost all aspects; in this case. Therefore, in general, G this is a fact one knows from Chapter 8 in [102], on compact abelian groups. We note that 6.6 (iv) says that the categories proLieAlg of pro-Lie algebras and continuous Lie algebra morphisms and the full subcategory of the category of topological groups and continuous group homomorphisms consisting of prosimply connected proLie groups are faithful images of each other; whatever is true in one of them, grosso modo, holds in the other as well, and the two functors  and L achieve the translation. Since  : proLieAlg → proLieGr is a left adjoint, it preserves colimits, it preserves cokernels. Before we exploit this useful fact, let us quickly review the concept of a cokernel. In a pointed category (that is, a category with a terminal and initial object and thus with zero morphisms; see e.g. [102, Appendix 3]) the kernel of a morphism f : X → Y is morphism e : K → X which is the equalizer of f and the zero morphism X → Y (in the category of groups, for instance, the constant morphism). As an equalizer, a kernel K is a limit. Dually, the cokernel of a morphism f : X → Y is a morphism c : Y → C which is the coequalizer of f and the zero morphism. In the category of Hausdorff topological groups, the cokernel c is the quotient morphism c : Y → Y /N where N is the smallest closed normal subgroup containing f (X). In fact, whenever ϕ : Y → Z is a morphism of topological groups such that ϕ f : X → Z is the constant morphism, then f (X) ⊆ ker ϕ and thus N ⊆ ker ϕ. So ϕ factors uniquely through the quotient morphism c : Y → Y /N. The Strict Exactness Theorem for  Theorem 6.7. Assume that

e

f

n −−−→ g −−−→ h is a strict exact sequence of morphisms of pro-Lie algebras; that is, assume that im e = ker f . Then (f ) (e) (n) −−−−→ (g) −−−−→ (h)

260

6 Lie’s Third Fundamental Theorem

is a strict exact sequence of morphisms of pro-Lie groups, that is, the image of (e) : (n) → (g) is the kernel of (f ) : (g) → (h) and this kernel is prosimply connected. Proof. The morphism f factors through g/ ker f and ker f is also the kernel of the quotient morphism g → g/ ker f . We may therefore replace f by the quotient morphism g → g/ ker f and will therefore assume, without losing generality, that f is itself a quotient morphism. It is then the cokernel of e and since  preserves cokernels, (q) is the cokernel of (e) in proLieGr. We let K = ker (f ). Then the quotient morphism p : (g) → (g)/K is the cokernel of (e) in TopGr. Let Q ⊇ (g)/K be the completion and j : (g)/K → Q the inclusion. Then the coextension j  p : (g) → Q is the cokernel of (e) in proLieGr. By the uniqueness of cokernels, there is an isomorphism ι : (h) → Q such that ι  (f ) = j  p: (f ) (g) −−−−→ ((h) ⏐ ⏐ ⏐ ⏐ι p (∗)  (g)/K −−−−→ Q. j

By the Strict Exactness Theorem for L 4.20, the map L(j ) : L((g)/K) → L(Q) is an isomorphism. Trivially, L(ι) : L((g)) → L(Q) is an isomorphism. Since L preserves kernels, we have L(K) = ker L((f )) ⊆ L((g)). In the commutative diagram e

f

L((e))

L((f ))

n −−−−→ g −−−−→ h ⏐ ⏐ ⏐ ⏐ ⏐ ηh ηg ⏐ ηn    L((n)) −−−−→ L((g)) −−−−→ L((h)) the vertical arrows are isomorphisms by 6.5 and thus, if we identify pro-Lie algebras def

via the natural automorphism η, we see that k = L(K) equals e(n). This means that the corestriction of (e) to K is the unique morphism e : (n) → K such that L(e ) : n = L((n)) → k = e(n) is the corestriction of e to k. The situation is illustrated by the following commutative diagram L(e )

−−−−→ ⏐k n ⏐ exp(n) ⏐ expK ⏐   (n) −−−− → K  e

−−−−→

L(i)

f

i

(f )

g −−−−→ ⏐ ⏐exp  (g) −−−−→ (g) −−−−→

h ⏐ ⏐exp  (h) (h),

where i : K → (g) denotes the inclusion morphism. Thus if we identify L((g)/K) with g/k (via 4.20), and η with the identity then L(p) identifies with the quotient map g → g/k and we obtain from (∗) upon applying the functor L the commuting diagram f

g −−−−→ ⏐ ⏐ quot g/k −−−−→ L(j )

h ⏐ ⏐L(ι)  L(Q).

Lie’s Third Fundamental Theorem for Pro-Lie Groups

261

We claim that K is connected, that is K0 = K. Let j ∗ : (g)/K0 → Q∗ denote completion. Then L(j ∗ ) : L((g)/K0 ) = g/k → L(Q∗ ) is an isomorphism and the surjective morphism s : (g)/K0 → (g)/K, s(xK0 ) = xK induces a unique extension s ∗ : Q∗ → Q which in turn induces an isomorphism L(s) : L(Q∗ ) → L(Q). But Q∗ is a connected pro-Lie group since (g)/N0 is connected, and Q is prosimply connected since Q ∼ = (h). As a consequence s ∗ is an isomorphism of topological groups by 6.6 (v). Since the morphism s is a restriction of s ∗ , it is injective. As it is also surjective it is bijective. But this means K = K0 as asserted. Now we apply the Closed Subgroup Theorem for Projective Limits 1.34 for (g) in place of G, for N S((g)) in place of N and K in place of H . Then N ∼ = limM∈N S((g)) K/(K ∩ M), where K/(K ∩ M) ∼ = KM/M. Now for each M ∈ N S((g)), the quotient group K/(K ∩ M) is a quotient of a connected pro-Lie group, and is therefore a connected proto-Lie group by 4.1; on the other hand, KM/M is a subgroup of the Lie group (g)/M and therefore has an identity neighborhood, in which the singleton subgroup is the only subgroup. Thus KM/M ∼ = K/(K ∩ M) is a normal Lie subgroup of the simply connected Lie group (g)/M and is therefore simply connected (see e.g. [17, Chap. 3, §6, no 6, Proposition 14] or [85, p. 224, III.3.17]). So K is a projective limit of simply connected Lie groups and is therefore prosimply connected by 6.1. Hence πK : (k) → K is an isomorphism of topological groups by 6.6 (iii). If k : k → g is the inclusion, then e = k  L(e ) and we note a commutative diagram (L(e ))

(i)

e

incl

−−−−→ (k) −−−−→ (n) ⏐ ⏐ ⏐ ⏐ π K0  id (n) −−−− → K0 −−−−→ 

(g) ⏐ ⏐ id (g).

This means that im (e) = K. Thus im (e) = ker (f ) and this is exactly the claim of the theorem. Theorem on the Quotient Preservation in Lie’s Third Theorem Corollary 6.8. (i) Assume that f : g → h is a surjective morphism of pro-Lie algebras. Then the morphism of topological groups (f ) : (g) → (h) is a quotient morphism. In particular, it is surjective. Moreover, ker (f ) is connected. (ii) Assume that G is a pro-Lie group and N a closed normal subgroup. Let F : G → def : G →H  H = G/N be the quotient morphism and j : N → G the inclusion. Then F is a quotient morphism of prosimply connected pro-Lie groups whose kernel is the  in G,  that is, there is a strict exact sequence image of N j˜

 F

−−−−→G −−−−→H  → 1. 1→N If H is a pro-Lie group, then there is a commutative diagram whose rows are strict exact sequences

262

6 Lie’s Third Fundamental Theorem ˜



j

F

j F   −−− →1 1→N −→ H ⏐ −−−−→ G ⏐ ⏐ ⏐ ⏐ ⏐ pN  pG  pH  1 → N −−−−→ G −−−−→ H → 1.

(iii) Let G a prosimply connected pro-Lie group and N a closed normal subgroup. Then the following two statements are equivalent: (a) G/N is prosimply connected. (b) N is prosimply connected. (c) N is connected. (iv) Assume that F : G → H is a surjective morphism of pro-Lie groups and that : G →H  is a quotient morphism. G is connected. Then F Proof. (i) The surjectivity of f implies the bijectivity of the induced map f  : g/n → h. In the category of weakly complete topological vector spaces, however, a bijective morphism, as the adjoint of a bijective morphism of vector spaces, is an isomorphism. (Compare this with [102, Theorem 7.30 (iv)], cited below as Theorem 7.7 (iv).) Hence f : g → h is a quotient morphism and if n denotes the kernel of f , then f is in fact the cokernel of the inclusion morphism e : n → g. Applying  to the exact sequence f

e

{0} → n −−−→ g −−−→ h → {0}, by the Strict Exactness Theorem for , 6.7, we get an exact sequence of morphisms of pro-Lie groups (f )

(e)

const

{1} → (n) −−−−→ (g) −−−−→ (h) −−−−→ {1}. In particular im (e) = ker (f ), which is a closed connected normal subgroup N of def

G = (g), and im (g) = ker const = (h). Thus (f ) is a surjective morphism of connected pro-Lie groups with connected kernel. It is also the cokernel of (e) in proLieGr since  preserves cokernels. In TopGr, the quotient morphism q : G → G/N is the cokernel of (e). By Theorem 4.1 the factor group G/N is a proto-Lie group and its completion Q ⊇ G/N is a pro-Lie group. Let j : G/N → Q be the inclusion. Then j  q : G → Q is the cokernel of (e) in proLieGr. By the uniqueness of cokernels, there is an isomorphism ι : (h) → Q such that G ⏐ ⏐ q G/N

(f )

−−−−→ (h) ⏐ ⏐ι  −−−−→ Q j

commutes. Since (f ) is surjective and j is injective, it follows that j is bijective. That is, G/N = Q is complete and therefore is a pro-Lie group. Then (f ) and q are equivalent, and thus (f ) is in fact a quotient morphism.

Lie’s Third Fundamental Theorem for Pro-Lie Groups

263

(ii) By the Strict Exactness Theorem 4.20 for L, we have a strict exact sequence L(F )

incl

0 → n −−−−→ g −−−−→ h → 0 with a quotient morphism L(F ) and with n = ker L(F ) ∼ = L(N ) by the kernel preservation of L. Now by the Strict Exactness Theorem 6.7 for  we obtain a strict exact sequence (incl)

L(F )

1 → (n) −−−−→ (g) −−−−→ (h) → 1, where L(F ) is a quotient morphism by (i) above. Now we recall from Theorem 6.6 and its proof that  · =   L and that πG :  g → G is that natural morphism inducing an isomorphism on the Lie algebra level. Thus (ii) follows. (iii) We let F : G → H be the quotient morphisms as in (ii). The group G is prosimply connected iff pG is an isomorphism by Theorem 6.6 (iii). If, in the commuting diagram in (ii), two of the three vertical morphisms pN , pG , and pH are isomorphisms, so is the third one. Hence by 6.6 (iii) again, (a) and (b) are equivalent.  with G by the isomorphism Trivially (b) implies (c). Finally assume (c). We identify G  pG and consider both N and N as subgroups of G via the inclusions j˜ and j . Then pN  ⊆ N . (Note that this is consistent with F = pH  F  is an inclusion map, that is N     and N = ker F ⊆ ker(pH  F ) = ker F = N.) Since L(pN ) : L(N ) → L(N ) is an  = N0 . Our assumption, however, isomorphism, from Corollary 4.22 we conclude N is N0 = N, and so pN is the identity map, whence (b) holds. (iv) By the connectivity of G and by Theorem 4.22 (ii)(b), the morphism L(F ) is a  = (L(F )) is a quotient quotient morphism of pro-Lie algebras; then by (i) above, F morphism of pro-Lie groups. Theorem on the Preservation of Embeddings in Lie’s Third Theorem Corollary 6.9. (i) If j is any closed ideal of a pro-Lie algebra g, then the inclusion i : j → g induces an embedding (i) : (j) → (g). That is, (j) may be considered as a closed normal subgroup of (g). (ii) If N is the kernel of a morphism F : G → H of pro-Lie groups and j : N → G →G  is an embedding so that N  may be identified the inclusion morphism, then j˜ : N  with the kernel of F . i

q

Proof. (i) The exact sequence 0 → j −−→ g −−→ g/j → 0 by the Exactness Theorem 6.7 gives an exact sequence i

(q)

{1} → (j) −−−→ (g) −−−→ (g/j) → {1}. If K = ker (q), then K is a closed normal subgroup and thus a pro-Lie group by the Closed Subgroup Theorem 3.35. Since L preserves kernels, L(K) = j. By the Exactness Theorem 6.7, the corestriction ε : (j) → K of (j ) is a bijective morphism inducing an isomorphism L(ε) : ((j) → L(K) = j) which becomes the identity when L((j)) is identified with j via ηj . It follows from the uniqueness in

264

6 Lie’s Third Fundamental Theorem

the universal property of  (see 6.6) that ε is none other than the canonical morphism  → K. In 6.8, however, we saw that K was prosimply connected. πK : (L(K)) = K But then by 6.6 (iii), ε = πK is an isomorphism. But this means that (i) is an embedding algebraically and topologically, as asserted. (ii) Since the functor L preserves kernels, the Lie algebra n = L(N ) is (up to natural isomorphism) the kernel of L(F ) : L(G) → L(H ). Now we apply (i) above to n in place of j and recall  · =   L again to obtain the assertion.

Semidirect Products We discussed in Chapter 1 the concept of a semidirect product of topological groups in Exercise E1.5. The concept of a semidirect product in the category of topological groups is very fruitful, both traditionally in classical Lie group theory and in the theory of pro-Lie groups – as we still have to see. Let us review the concept with particular emphasis on the functorial aspects of it. Remember that a self map f of a set is called idempotent if f 2 = f . Also recall that for topological groups N and H and a morphism α : H → Aut(N ) such that (h, n)  → α(h)(n) : H × N → N is continuous, the product N × H becomes a topological group N α H with the multiplication (n, h)(n , h ) = (nα(h)(n ), hh ), called the semidirect product of N by H (with respect to α). If n and h are topological Lie algebras and δ : h → Der(n) is a morphism of Lie algebras such that (Y, X)  → δ(Y )(X) : h × n → n is continuous, then n × h becomes a Lie algebra n ⊕δ h with respect to componentwise scalar multiplication and addition and the bracket multiplication [(X, Y ), (X , Y  )] = (δ(Y )(X  ) − δ(Y  )(X), [Y, Y  ]), called the semidirect sum of n with h (with respect to δ). Exercise E6.2. Verify that the bracket multiplication of the semidirect sum is antisymmetric and satisfies the Jacobi identity (see Definition A1.1 (ii), p. 624). Proposition 6.10. (a) Let G be a topological group and N a closed normal subgroup. Then the following conditions are equivalent: (i) There is an idempotent endomorphism ϕ of G such that ker ϕ = N . (ii) There are morphisms q : G → K and σ : K → G such that q  σ = idK and ker q = N. (iii) There is a closed subgroup H such that the function (n, h)  → nh : N × H → G is an isomorphism of the semidirect product N I H of N by H with respect to the morphism I defined by inner automorphisms I (h)(n) = hnh−1 onto G. If these conditions are satisfied, H = ϕ(G) = σ (K).

Semidirect Products

265

(b) Let g be a topological Lie algebra and n a closed ideal. Then the following conditions are equivalent: (i) There is an idempotent endomorphism  of g such that ker  = n. (ii) There are morphisms p : g → k and ρ : k → g such that p  ρ = idk and ker p = n. (iii) There is a closed subalgebra h such that the function (X, Y )  → X+Y : n×h → g is an isomorphism of the semidirect sum n ⊕ad h of n by h with respect to the morphism ad defined by the adjoint representation ad(Y )(X) = [Y, X] onto g. If these conditions are satisfied, h = (g) = ρ(k). Proof. We prove part (a): (i) ⇒ (ii): We let K = ϕ(G) and let q be the corestriction of ϕ to its image. Define σ : K → G be the inclusion map. Then ker q = ker ϕ = N, and if k ∈ K, then k = ϕ(g) for some g ∈ G and (q σ )(k) = q(ϕ(g)) = ϕ 2 (g) = ϕ(g) = k. (ii) ⇒ (iii): Define H = σ (K). Let μ : N I H → G be defined by μ(n, h) = nh. Then μ is easily seen to be a morphism of topological groups. The function g  → (gσ (q(g)), q(g)) is readily verified to be the inverse map μ−1 of μ. (iii) ⇒ (i): Assume that μ : N I H → G is an isomorphism of topological groups, μ(n, h) = nh. Let i : H → G be the inclusion morphism and pr H : N I H → H the projection onto H . Define ϕ : G → G by ϕ = i  pr 2 μ−1 . Assume g = nh. Then μ−1 (g) = (n, h), and μ−1 (h) = (1, h), whence ϕ 2 (g) = i pr 2 μ−1 i pr 2 μ−1 (g) = h = i pr 2 μ−1 (g) = ϕ(g). Part (b) is proved in a completely analogous fashion and is left as an exercise. Exercise E6.3. Spell out the details of the proof of Part (b) of Proposition 6.10. Theorem 6.11 (Preservation of Semidirect Products). Let g be the semidirect sum of def the closed ideal n by the closed subalgebra h. Then G = (g) is the semidirect product of a closed normal subgroup N naturally isomorphic to (n) and a subgroup H isomorphic to (h). Proof. By Proposition 6.10 (b), the hypothesis is equivalent to the existence of an idempotent endomorphism  of g with kernel n and image h. We apply the functor  def

and obtain and idempotent endomorphism ϕ = () : G → G. Let i : n → g denote the inclusion morphism. The exact sequence i



0 → n −−→ g −−→ g gives rise to an exact sequence (i)

(ϕ)

1 → (n)−−−−→(g)−−−−→(g) by the Exactness Theorem for  6.7, and (j ) is an embedding by Corollary 6.6. Set N = (j)((n)). Then (n) ∼ = N = ker () = ker ϕ. It now follows from

266

6 Lie’s Third Fundamental Theorem def

Proposition 6.10 that G is the semidirect product of N by H = ϕ(G). It remains to identify im ϕ. def

Now let p : g → h = (g) be the corestriction of  to its image. Then p is a quotient morphism inducing an isomorphism g/n → h. Then the exact sequence p

g −−→ h → 0 induces an exact sequence (p)

G = (g) −−−−→ (h) → 0 def

by the Exactness Theorem for  6.7, and q = (p) : G → (h) is a quotient morphism by Corollary 6.8. If we let ρ : h → g denote the inclusion, we have pρ = idh , whence def

def

q  σ = id(h) for σ = (ρ). Then H = σ ((h)) is isomorphic to (h). Also  = ρ  p and thus ϕ = () = (ρ)  (p) = σ  q whence im ϕ = im σ = H .

Postscript From classical Lie group theory we know that if G is a finite-dimensional Lie group, then there exists a unique (up to isomorphism) simply connected finite-dimensional Lie  such that their Lie algebras L(G) and L(G)  are isomorphic; further, there is group G,  onto G implementing a local isomorphism. an open continuous homomorphism of G When one moves out of the world of finite-dimensional Lie groups or rather out of the world of manifolds, the first task is to be clear about what “simply connected” should mean. For a connected topological space X with a base point x0 several definitions of simple connectivity are in use. For instance, • X is called simply connected if it has the following universal property: For any covering map p : E → B between topological spaces, any point e0 ∈ E and any continuous function f : X → B with p(e0 ) = f (x0 ) for some x0 ∈ X there is a continuous map f: X → E such that p  f = f and f(x0 ) = e0 . f

X ⏐ −−−−→ ⏐ idX 

E ⏐ ⏐p 

X −−−−→

B.

f

Alternatively, • X is called simply connected if π1 (X, x0 ) = {0}, that is if every loop attached to x0 is contractible.

Postscript

267

No nondegenerate connected compact abelian group is simply connected with respect to the first definition. Any connected compact abelian group G such that  Z) = {0} (such as e.g. G = Q ) is simply connected with respect to the second Hom(G, definition. For arcwise connected, locally arcwise connected, and locally arcwise simply connected spaces, in particular for manifolds, the two definitions agree (see e.g. [102, Proposition A2.10]). In [102], we opted for the first definition because of its universal properties which are practically built into the definition. Taking these complications into consideration, we ask what should be the right concept of simple connectivity for pro-Lie groups? We say that • a pro-Lie group G is prosimply connected if N (G) contains a cofinal subset N S(G) such that G/N is a simply connected Lie group for each N ∈ N S(G). This definition takes the characteristic property G ∼ = limN ∈N (G) G/N into account. But is it useful for the structure theory of pro-Lie groups? With the aid of the structure theory of pro-Lie groups that we will develop in later chapters, we will show that a pro-Lie group G is prosimply connected if and only it is simply connected, and that it satisfies π1 (G) = {0} whenever it is prosimply connected. That is a reassuring fact. But recall that no compact connected abelian group is prosimply connected. That may make us a bit nervous. However, within the full category of pro-Lie groups in the category of topological groups, the full subcategory of prosimply connected pro-Lie groups has precisely the right universal properties. Firstly, for every profinite Lie algebra g there is a unique prosimply connected pro-Lie group G (up to isomorphism) such that L(G) = g. (Lie’s Third Fundamental Theorem). Thus there is essentially a “bijection” between the possible Lie algebras of pro-Lie groups and the prosimply connected specimens in proLieGr. If g is abelian, then G is the additive group of g; this is reasonable and has no analog in such categories as the category of locally compact groups. Secondly, for each connected pro-Lie group G there is a (functorially attached)  and a pro-open natural morphism πG : G →G prosimply connected pro-Lie group G  inducing an isomorphism L(πG ) : L(G) → L(G) of Lie algebras, and every morphism  of G into a prosimply connected pro-Lie group uniquely lifts to a morphism from G, factoring through πG . All of this very satisfactorily generalizes familiar facts from classical Lie group theory. Yet none of these facts is exactly trivial. For a compact  is naturally isomorphic to the additive group of L(G) and πG : G →G abelian group, G identifies with expG : L(G) → G. With our theorems, among other things we are saying: the category proLieAlg of profinite Lie algebras has “a precise copy”, namely, proSimpConLieGr, the full subcategory of the category proLieGr of pro-Lie groups containing precisely the prosimply

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connected pro-Lie groups. Moreover, the category of connected Lie groups allows a left-reflection into this category, that is a left adjoint functor which behaves like a retraction onto proSimpConLieGr. We have seen that Lie’s Third Fundamental Theorem in reality complements the Lie algebra functor to an adjoint situation. This is of course true in the finite-dimensional situation, but as a rule, less emphasis is placed on this fact in elementary Lie theory. Among other reasons, the most compelling one is that in the finite-dimensional case we are not dealing with complete categories. Adjoint situations are most appropriate where limits are present, because right adjoint functors (such as L in our case) preserve all limits. The functor  : proLieAlg → proLieGr that is left adjoint to the Lie algebra functor L : proLieGr → proLieAlg has excellent exactness properties, as we have seen in Theorem 6.7 and its two corollaries. This is a parallel to the Exactness Theorem for L 4.20 in Chapter 4. The theorems on the preservation of quotients 6.8 and of embeddings 6.9 are particularly relevant since quotients are somewhat problematic as we have seen in Chapter 4, and since we cannot have an Open Mapping Theorem for pro-Lie groups in general. The general covering theory for topological groups introduced by Berestovskii and Plaut fits very well with our theory for pro-Lie groups (see [6], [7], [8]). For a  connected pro-Lie group G, the Berestovskii–Plaut-cover agrees with G.

Chapter 7

Profinite-Dimensional Modules and Lie Algebras

In the previous chapters we recognized the significance of the category of topological groups with profinite-dimensional Lie algebras. This category can be understood only if one first comprehends the nature of pro-Lie algebras, that is, profinite-dimensional Lie algebras themselves. In this chapter we develop the structure theory of pro-Lie algebras by utilizing the duality between the category of weakly complete topological vector spaces and the category of vector spaces we review completely in Appendix 2. The use of this duality is made possible through representation theory or, equivalently, module theory of a fixed Lie algebra L. In fact we shall consider profinite-dimensional modules which emulate the definition of pro-Lie algebras in such a fashion that the adjoint module of a pro-Lie algebra is exactly a profinite-dimensional module. While pro-Lie algebras themselves do not dualize in an obvious way, profinite-dimensional modules do, and we shall make ample use of this fact. Prerequisites. In this chapter we shall make use of the duality theory of weakly complete topological vector spaces as it is presented in Appendix 2. We shall collect the essential features in Theorem 7.7. The representation theory of Lie algebras used here is simple. But we do resort to the basic structure theory of finite-dimensional Lie algebras presented in source books that are widely used such as [16]. In this chapter we do not use any information from Chapters 3, 4 and 5.

Modules over a Lie Algebra We begin with some elementary facts on the linear algebra of modules over a Lie algebra. Definition 7.1. (i) Let L be a Lie algebra and E a vector space. Then E is an L-module if there is a bilinear map (x, v)  → x · v : L × E → E

satisfying

[x, y] · v = x · (y · v) − y · (x · v)

for all x, y ∈ L and v ∈ E. A function f : E1 → E2 between L-modules is said to be a morphism of L-modules if it is linear and satisfies (∀x ∈ L, v ∈ E1 )

f (x · v) = x · f (v).

(1)

(ii) If L is a Lie algebra and V is a topological vector space, then V is said to be a continuous L-module if the underlying vector space is an L-module in the sense of (i) above and the functions v  → x · v : V → V are continuous for all x ∈ L.

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(iii) If L is a topological Lie algebra, then a topological vector space V is said to be a topological L-module if it is a continuous L-module and the maps x  → x · v : L → V are continuous for all v ∈ V ; in other words, if (x, v)  → x · v : L × V → V is continuous in each variable separately. Let us consider some prominent examples. Example 7.2. For any vector space E, let gl(E) denote the Lie algebra of all endomorphisms of E with the Lie bracket [ϕ, ψ] = ϕψ − ψϕ. If L is any Lie algebra, let |L| denote the underlying vector space. If we define ad x : L → L by (ad x)(y) = [x, y], then the Jacobi identity shows that ad : L → gl(|L|) is a morphism of Lie algebras, called the adjoint representation. If for x ∈ L and v ∈ |L| we set x · v = (ad x)v, then |L| becomes an L-module called the adjoint module. If L is a topological Lie algebra then the underlying topological vector space |L| becomes a topological L-module, again called the adjoint module. In particular, if g is a pro-Lie algebra and |g| is the underlying weakly complete topological vector space, then |g| is a topological g-module via the adjoint operation. If E is an L-module, for x ∈ L we define xE : E → E by xE (v) = x · v. Then x  → xE : L → gl(E) is a morphism of Lie algebras. Conversely, if π : L → gl(E) is a representation, i.e. a morphism of Lie algebras, then E becomes an L-module via x · v = π(x)(v). Thus the set of representations of L on V is in bijective correspondence with the L-module structures on E. If V is a topological vector space and an L-module, then each xV is an endomorphism of the topological vector space V , and if gl(V ) denotes the Lie algebra of all endomorphisms of the topological vector space V with the Lie bracket [ϕ, ψ] = ϕψ − ψϕ, then x  → xV : L → gl(V ) is a representation. Conversely, every representation of L into gl(V ) yields on the topological vector space V the structure of an L-module. If L is a topological Lie algebra and V a topological L-module, then x  → xV : L → gl(V ) is a morphism of topological Lie algebras if gl(V ) is given the topology of pointwise convergence, sometimes called the strong operator topology on V . We shall now make use of the duality of topological vector spaces. If V is a topological vector space, its topological dual is the vector space V  = Hom(V , R) of all continuous linear functionals. In accordance with this definition, V  is a vector space without additional structure, but it may be equipped with many relevant vector space topologies; we shall consider two of them here. The coarser of the two is the weak *-topology, also called the topology of pointwise convergence, that is the topology induced from the product RV . If V  is equipped with this topology, we denote it by V  . The second topology we consider is the topology of uniform convergence on compact subsets of V also called the compact open topology. (See [102, Proposition 7.1ff].) If the topological dual V  is endowed with the compact open topology, we denote  have the same underlying vector . The topological vector spaces V  and V it by V . For a space V  , but the topology of V  may be properly coarser than that of V  is defined to be TopGr(G, T) = Hom(G, T) topological group G the character group G with the compact open topology; of course, this works well only when G is abelian. If V

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271

is a topological vector space, then the natural morphism Hom(V , R) → Hom(V , R/Z) is an isomorphism of topological vector spaces when both hom-groups are given the compact open topology (see [102, Proposition 7.5 (iii)]). This justifies our notation of  for V  with the compact open topology. V Now let L be a Lie-algebra and V a topological vector space. If ω ∈ V  , then ω  xV : V → R is a continuous linear functional. Thus if we define x · ω by x · ω = −ω  xV ∈ V  we have (∀x ∈ L, v ∈ V )

(x · ω)(v) = −ω(x · v).

(2)

Lemma 7.3. (i) Let L be a Lie algebra and V a topological vector space and L→V , module. With respect to the bilinear map (x, ω) → x · ω = −ω  xV : L × V  the topological vector space V is an L-module. (ii) If L is a topological Lie algebra and V a topological L-module, then V  is a topological L-module. Proof. (i) First we compute ([x, y] · ω)(v) = −ω([x, y] · v) = −ω(x · (y · v)) + ω(y · (x · v)) = −y · ω(x · v) + x · ω(y · v) = x · (y · ω)(v) − y · (x · ω)(v). Thus V  is an L-module. Now we observe the required continuity property, namely, that xV  , defined by xV  (ω) = x · ω = −ω  xV , is continuous for all x ∈ L. But that is immediate from the continuity of all xV and ω. (ii) Let ω ∈ V  and v ∈ V . All functions x  → x · v : L → V are continuous. Hence all functions x  → (x · ω)(v) = −ω(x · v) : L → R are continuous and that means that the linear function x  → x · ω : L → V  is continuous with respect to the topology of pointwise convergence on V  . Lemma 7.4. (i) Let L be a Lie algebra and V a topological vector space and L-mod→V , the ule. With respect to the bilinear map (x, ω)  → x · ω = −ω  xV : L × V  is an L-module. topological vector space V (ii) If L is a topological Lie algebra and V a topological L-module such that  is a topological L-module. (x, v)  → x · v : L × V → V is continuous, then V Proof. (i) After 7.1 (ii) we have to verify only that for all x ∈ L the linear maps →V  are continuous. For a proof we assume that a compact subset C of V and xV : V a zero neighborhood U of R are given and thus determine a basic zero neighborhood W (C, U ) = {ω ∈ V  | ω(C) ⊆ U }. Since xV is continuous, −x · C = −xV (C) is compact, and every functional ω satisfying ω(−x · C) ⊆ U also satisfies ω(−x · C) = (x · ω)(C) ⊆ U . Thus xV maps W (−x · C, U ) into W (C, U ) →V , (ii) For a proof of the continuity of the functions x → x · ω = −ω  xV : V we again assume that a compact subset C of V and an open zero neighborhood U of R are given. We must find a zero neighborhood W of L such that x ∈ W implies (x · ω)(C) ⊆ U i.e., ω(−x · C) ⊆ U . Now ω−1 (U ) is an open identity neighborhood

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of V . If the function (x, v) → x · v : L × V → V is assumed to be continuous, then an open symmetric zero neighborhood W of L exists such that W · C ⊆ ω−1 (U ), and this proves the assertion. Definition 7.5. 7.5. The L-module V  is called the dual module of V . Example 7.6. Let L be a topological Lie algebra and let |L| denote the underlying topological vector space. Define L to be the topological dual |L| of |L| in the weak *-topology. Then the dual module of the adjoint module |L| on L is a topological  x)(ω) = L-module which is called the coadjoint module, written Lcoad . If we set (ad   x · ω = ω  ad x, then ad : L → gl(L) is a continuous morphism of Lie algebras if one endows gl(|L|) with the topology of pointwise convergence.  denotes the topological dual of |L| with the compact open topology, then L  is If L a topological L-module with respect to the coadjoint action.

Duality of Modules We consider two categories of topological vector spaces which are dual to each other under the passage to the topological dual with the compact open topology. (See [102, Proposition 7.1ff].) Namely, we consider firstly the category Vect of vector spaces and linear maps between them, and secondly, the category WCVect of weakly complete topological vector spaces and continuous linear maps between them. Any real vector space can be considered as a topological vector space with respect to the finest locally convex vector space topology (see Appendix 2, Proposition A2.3 and the passage that precedes it), and any linear map between vector spaces is automatically continuous with respect to this topology on the domain and the range space. An abelian topological group A is called reflexive if the natural evaluation morphism  η: A →  A into the double dual is an isomorphism of topological groups. (See [102, Definition 7.8ff].) We denote by RefTopGr the full subcategory of TopGr whose objects are all reflexive abelian topological groups. Note that a morphism f : V → W of topological groups between the additive groups of topological vector spaces is always linear. To see this note that if r = m n with m, n ∈ N, then n · f (r · v) = f (m · v) = m · f (v), whence f (r · v) = r · f (v), and this relation extends by continuity to all r ∈ R. the set of all ω ∈ V  If S is a subset of a topological vector space V , then S ⊥ denotes   ⊥ such that ω(S) = {0}. If S is a subset of V , then S = ω∈S ker ω. This will not cause confusion if we always keep track of the vector space in which S is located. The vector subspace S ⊥ is called the annihilator of S. For easy reference, we review the duality of vector spaces and weakly complete topological vector spaces.

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Theorem 7.7 (Duality of Real Vector Spaces). Let E be a real vector space and endow it with its finest locally convex vector space topology, and let V be a weakly complete real topological vector space. Then   is an isomorphism of topological vector spaces. (i) E is reflexive; i.e. ηE : E → E  and E  is a weakly complete Thus E belongs to RefTopGr. Moreover, E  = E, topological vector space.   is an isomorphism of topological vector spaces, (ii) V is reflexive; i.e. ηV : V → V  has the finest locally convex topology, and thus V belongs to RefTopGr. The dual V   ∼   and V = V = V . (iii) Vect and WCVect are subcategories of RefTopGr, and the contravariant functor  · : RefTopGr → RefTopGr exchanges the categories Vect and WCVect. (iv) Every closed vector subspace V1 of V is algebraically and topologically a direct summand; that is there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces. Every surjective morphism of weakly complete topological vector spaces f : V → W splits; that is, there is a morphism σ : W → V such that f  σ = idW . (v) For every closed vector subspace H of E, the relation H ⊥⊥ = H ∼ = (E  /H ⊥ )  ⊥ ⊥  holds and E /H is isomorphic to H . The map F  → F is an antiisomorphism of the complete lattice of vector subspaces of E onto the lattice of closed vector subspaces of E  . Proof. See Appendix 2, notably Theorems A2.8, A2.11, and A2.12. A subset M of an L-module V is called a submodule if L · M ⊆ M. If M is a closed submodule then V /M is an L-module with respect to the bilinear map (x, v + M)  → x ∗ (v + M) = x · v + M. Definition 7.8. Let L be a Lie algebra, E a vector space and V a topological vector space such that E and V are L-modules. (i) V is called a profinite-dimensional L-module if it is complete as a topological vector space and the filter basis M of closed submodules M ⊆ V such that dim V /M < ∞ converges to 0. (ii) E is called a locally finite-dimensional L-module if for each finite subset S of E there is a finite-dimensional submodule F of E containing S. We specifically draw the reader’s attention to the facts that according to our definitions, a profinite-dimensional L-module is a topological vector space with a module structure, while a locally finite-dimensional L-module is merely a vector space with an L-module structure. It is possible at all times to consider it as a topological vector space with a module structure by equipping it with the finest locally convex topology, but doing that is not likely to give us additional insights. Thus a locally finite-dimensional module is a purely algebraic object even if L is a topological Lie algebra. Let g be a finitely generated topological Lie algebra on an infinite-dimensional vector space. Then the underlying vector space |g| with the adjoint module structure cannot be locally finite-dimensional, since a finite set of generators of g is not contained

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in a finite-dimensional subalgebra, let alone an ideal, and thus it is not contained in a finite-dimensional submodule. A simple example is as follows: Let s = so(3) and ej ∈ s, j ∈ Z(3) = Z/3Z, a basis of s such ' that [e & j , ej +1 ] = ej +2 for j = 0, 1, 2 mod 3. Then [s, e0 ] contains e1 and e2 and s, [s, e0 ] = s. Now set g = sN and x = (xn )n∈N with xn = e0 for all n. In ' &N the adjoint module we have g·x = [s, e0 ]N and g·(g·x) = s, [s, e0 ] = sN = g. Thus the adjoint module is singly generated algebraically and yields a profinite-dimensional g-module whose underlying vector space does not yield a locally finite-dimensional g-module, since {x} is not contained in a finite-dimensional submodule. Proposition 7.9. For an L-module on a topological vector space V , the following statements are equivalent: (i) V is a profinite-dimensional L-module.  = limM∈M V /M (see 1.29) is an isomorphism. (ii) The morphism γV : V → V Every profinite-dimensional L-module is weakly complete. Proof. The assertion follows from Theorems 1.29 and 1.30. Proposition 7.10. For an L-module E the following conditions are equivalent: (i) (ii) (iii) (iv)

E is locally finite-dimensional. Each element of E is contained in a finite-dimensional submodule. E is the union of its finite-dimensional submodules. E is a sum of a family of finite-dimensional submodules.

Proof. Exercise E7.1. Exercise E7.1. Prove Proposition 7.10. [Hint. (i) ⇒ (ii) ⇒ (iii) ⇒ (i) and (iii) ⇒ (iv) are trivial. If (iv) holds, then E = 5 . Let F denote the family j ∈J Vj for a family of finite-dimensional submodules Vj of all finite sums of the submodules Vj . Then (iv) says E = F and this implies (iii).] The Duality Theorem for Profinite-Dimensional and Locally Finite-Dimensional L-Modules Theorem 7.11. Let L be a Lie algebra, E a vector space and V a topological vector space such that E and V are L-modules. Then the following conclusions hold.  = E  is weakly complete. (ia) The underlying vector space of the dual module E   of E  is E (up to natural isomorphism). If E is a locally finiteThe dual module E  is a profinite-dimensional L-module. dimensional L-module, then E (ib) If L is a topological Lie algebra and E is a topological L-module then the dual  = E  is a topological L-module. module E   has the finest locally convex topology. Its dual (V ) = V  is (ii) The dual module V  V (up to natural isomorphism). If V is a profinite-dimensional L-module, then V is a locally finite-dimensional L-module.

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(iii) The functor  : Vect → WCVect maps the category of locally finite-dimensional L-modules to the category of profinite-dimensional topological L-modules. The functor  : WCVect → Vect maps the category of profinite-dimensional topological L-modules to the category of locally finite-dimensional L-modules. (iv) Every closed L-submodule V1 of V is a direct topological vector space summand; that is, there is a closed vector subspace V2 of V such that (x, y)  → x + y : V1 × V2 → V is an isomorphism of topological vector spaces. Every surjective morphism of profinite-dimensional topological L-modules f : V → W splits in terms of topological vector spaces; i.e. there is a morphism of topological vector spaces σ : W → V such that f  σ = idW . (v) A vector subspace H of E is an L-submodule iff its annihilator H ⊥ in E  is an L-submodule of E  . A closed vector subspace K of V is an L-submodule of V iff its annihilator in V  is an L-submodule of V  . For every L-submodule H of E, the . The map relation H ⊥⊥ = H ∼ = (E  /H ⊥ ) holds and E  /H ⊥ is isomorphic to H F  → F ⊥ is an antiisomorphism of the complete lattice of L-submodules of E onto the lattice of closed L-submodules of E  , and F ⊥⊥ = F for each submodule F of E and each closed submodule F of V .  and E  is a weakly complete topological vector Proof. (i) By 7.7 (i) and (iii), E  = E  space. By 7.3, E is an L-module, and if L is a topological Lie algebra and E is a  ∼ topological L-module, then E  is a topological L-module as well. The relation E =E follows from 7.7 (i). Now assume that E is a locally finite-dimensional L-module.  is complete and that the filter basis of closed vector Since we know that E  = E  subspaces N with dim E/N < ∞ converges to 0, in order to verify condition 7.8 (i)  we have to show that every closed vector subspace N of E  such that E/N  is for V = E  finite-dimensional contains an L-submodule M such that E/M is finite-dimensional. By the Annihilator Mechanism 7.7 (v), the annihilator N ⊥ of N in E is finite-dimensional. Since E is a locally finite-dimensional L-module, there is a finite-dimensional L-module F of E containing N ⊥ . Then we let M be the annihilator F ⊥ of F in E  .  we have   ⊥∼ Since annihilators of submodules are submodules and E/M = E/F =F  dim E/M < ∞. Thus the proof of (i) is complete.  has the finest locally convex topology and by (i) above and 7.7 (i) (ii) By 7.7 (iii), V  ) = V ∼ and (ii), we have (V = V . Now we assume that V is a profinite-dimensional L-module. We claim that V  is a locally finite-dimensional L-module. So let S be a finite subset of V  . Let S ⊥ be the annihilator of this set in V . Then by 7.7 (v), the vector space V /S ⊥ ∼ = (span S) is finite-dimensional. Since V is a profinite-dimensional L-module, there is a submodule M such that M ⊆ S ⊥ and that dim V /M < ∞. def ∼ Then F = M ⊥ , the annihilator of M in V  is a submodule and since F = V /M we ⊥ ⊥⊥ know that it is finite-dimensional. Also S ⊆ M implies S ⊆ S ⊆ M⊥ = F . Statement (iii) is now a consequence of (i) and (ii), and conditions (iv) follows directly from 7.7 (iv). (v) Let H be a vector subspace of E. Then x · H ⊆ H for x ∈ L iff ω(H ) = {0} implies x · ω(H ) = −ω(x · H ) = {0} for all ω ∈ E  since H = H ⊥⊥ by 7.7 (v). Thus x · H ⊆ H iff x · H ⊥ ⊆ H ⊥ . For a closed vector subspace K of V the same argument

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applies since for a closed vector subspace K of V one has V = V ⊥⊥ by 7.7 (v). The remainder of (v) is an immediate consequence of 7.7 (v). Corollary 7.12. Let L be a topological Lie algebra and V a weakly complete topological L-module. Then the following statements are equivalent: (i) V is a profinite-dimensional L-module. (ii) V  is a locally finite-dimensional L-module. Proof. Since E = V  with the finest locally convex topology and V are duals of each other by 7.7, the corollary is an immediate consequence of Theorem 7.11. The theory of profinite-dimensional modules is primarily developed to deal with profinite-dimensional Lie algebras. This is why the following example is crucial. Exercise E7.2. Prove the following proposition: Let g be a weakly complete topological Lie algebra. Then the following statements are equivalent. (i) g is a pro-Lie algebra. (ii) The coadjoint module gcoad of g is locally finite-dimensional. [Hint. With g = L and V = |g| (the underlying topological vector space), this is but a special case of Corollary 7.12.] As we progress further into duality theory we have reached the point to recall that categorical concepts such as limits and their various manifestations such as products, pullbacks, equalizers projective limits come in pairs, as each concept has its dual concept. Technically, if D : J → C is a diagram its colimit is the limit of the diagram D : J → C op in the opposite category. However, in order to have things self-contained in this book as much as possible, we explicitly formulate the definitions of a colimit and colimit cone. Definition 7.13. Let J and C be categories. A diagram D : J → C is said to have a colimit colim D ∈ ob C if there is a cone λ : D → const(colim D), called the colimit cone1 such that for each cone α : D → const A there is a unique morphism α  : colim D → A such that α = const(α  )  λ. Exercise E7.3. Formulate the definitions of coproduct, coequalizer, pushout, and injective limit. [Hint. Dualize the concepts introduced in 1.4.] of vector Lemma 7.14. (i) Let E be a vector space and {Fj | j ∈ J } a directed family  subspaces of E; i.e., J is a directed set and j ≤ k implies Jj ⊆ Jk . Then E = j ∈J Fj and E = colimj ∈J Fj are equivalent statements. 1 The

“cones” in this definition are “upside down” and should be called “cocones”. However, there are limits to the grammatical and etymological pliability of language. For instance, it is not true that a coconut is the dual of a nut. In fact, it is the bidual of a nut as the conut is its dual. For further references on philological analysis see the footnote in [102] regarding the terminology of Definitions 9.30.

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(ii) Let V be a profinite-dimensional L-module and M any filter basis of submodules M such that dim V /M < ∞ for all M ∈ M and that M converges to 0. Then the dual module V  is the union of the directed set of finite-dimensional submodules M ⊥ , M ∈ M. Proof. Exercise E7.4. Exercise E7.4. Verify Lemma 7.14. [Hint. (i) E = colimj ∈J Fj means that for every family of linear maps fj : Fj → F into a vector space F such that j ≤ k implies fk |Fj = fj there is a unique linear map f : E → F such that f |Fj = fj . Prove that this implies and is implied by E = colimj ∈J Fj . (ii) follows from (i) and the Duality Theorem 7.11.]

Semisimple and Reductive Modules Definition 7.15. (i) An L-module E is said to be simple if {0} and E = {0} are its only submodules. (ii) An L-module E is called semisimple if every submodule is a direct module summand. (iii) Let V be a profinite-dimensional topological vector space and an L-module. Then the module is said to be reductive if its dual module is semisimple. (iv) A pro-Lie algebra g is called reductive if its adjoint module is reductive. Note that a simple abelian module is isomorphic to R with the zero module operation x · r = 0 for all x ∈ L and r ∈ R. Recall that the direct sum  j ∈J Ej of a family Ej , j ∈ J of vector spaces is the set of all families (xj )j ∈J ∈ j ∈J Ej of elements xj ∈ Ej vanishing for all but a finite number of exceptions.5If each Ej is a vector subspace of a vector space E, then we shall say that the sum j ∈J Ej ⊆ E is direct if the morphism 6 8 6 xj : Ej → Ej (xj )j ∈J  → j ∈J

j ∈J

j ∈J

is an isomorphism of vector spaces. The Structure Theorem of Semisimple Locally Finite-Dimensional Modules Theorem 7.16. Let L be a Lie algebra and E a locally finite-dimensional L-module. Then the following statements are equivalent: (i) (i ) (ii) (iii)

E is a semisimple L-module. Every submodule is a direct module summand. Every finite-dimensional submodule of E is semisimple. E is the union of finite-dimensional semisimple submodules.

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(iv) E is (isomorphic to) a direct sum simple submodules Ej .

j ∈J

Ej of a family of finite-dimensional

Proof. (i) ⇔ (i ): Definition 7.15(ii). (i) ⇒ (ii): Let F be a finite-dimensional submodule of E and F1 a submodule of F . We must show that F1 has a complementary module summand in F . Since E is semisimple, there is a submodule E2 of E such that E = F1 ⊕ E2 . Set F2 = F ∩ E2 . Then F = F1 ⊕ F2 is readily checked (“Law of Modularity”). (ii) ⇒ (iii): Since E is a locally finite-dimensional module, every element is contained in a finite-dimensional submodule F . Hence this implication is trivial. (iii) ⇒ (iv): We consider sets S of finite-dimensional simple 5 submodules of E such that the morphism ε : F → E sending (x ) to F F ∈S F ∈S F ∈S xF ∈ E is injective. The set F of all sets S is partially ordered under inclusion. We claim that F is inductive. def  S and assert that For a proof let S be a totally ordered subset of F . We set5S = S ∈ F . Indeed we claim that the morphism (xF )F ∈S  → 5 x F ∈S F : F ∈S F → E is injective. Indeed let (xF )F ∈S be in its kernel. Then F ∈S xF = 0. The set T = {F ∈ S : xF = 0} is finite, hence is contained in some Sx ∈ S since S is the union of all S in S. But by definition of F the function 6 8 yF : F →E (yF )F ∈Sx  → F ∈Sx

F ∈Sx

is injective. Now (xF )F ∈ Sx is in the kernel of this map, and this implies xF = 0 for all F ∈ Sx . But xF = 0 for F ∈ S \ T and thus (xF )F ∈S is zero. This proves the claim and shows the assertion that S ∈ F 5. Hence F is indeed inductive. Now let S be a maximal element in F and set E0 = F ∈S F . If E0 = E we are finished. Suppose that E0 = E. Then by (ii) there is a finite-dimensional semisimple submodule W of E which is not contained in E0 . The submodule E0 ∩ W has a direct complement E1 in W , and thus E0 + E1 is a direct sum. Since E is locally finite-dimensional, so is E1 and thus there is a finite-dimensional nonzero submodule S of minimal dimension. Clearly S is a simple module. Since S ⊆ E1 , the sum E0 + S is direct. Hence the family S ∪ {S} is a member of F and is properly larger than S. This is a contradiction to the choice of S. Hence E0 = {0} and the proof is5 complete. (iv) ⇒ (i): We assume that E is a direct sum j ∈J Ej of submodules. Let F be a submodule of E. We must find a module complement of F5in E. For a fixed j ∈ J we either have Ej ⊆ F or Ej ∩ F = {0}. Let F1 = j ∈J, Ej ⊆F Ej and 5 E . Then E = F ⊕ F , and by the modular law we have F2 = 1 2 j ∈J, Ej ∩F ={0} j F = F1 ⊕ (F ∩ F2 ). Thus if we find a module complement of F ∩ F2 in F2 we are done. Thus in order to simplify notation, we might just as well 5 assume that F ∩Ej = {0} for all j ∈ J . The set  of all subsets I ⊆ J such that F ∩ 5j ∈I Ej = {0} is seen to be inductive again due to the fact that for every element x = j ∈J xj only finitely 5 many xj are nonzero. By Zorn’s Lemma let us pick a maximal subset I and set F = j ∈I Ej . Then clearly F ∩ F =5 {0}. We claim that E = F + F . Suppose that this is not the case. Then there is an x = j ∈J xj ∈ E \ (F + F ). In particular, there must be an i ∈ J \ I

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such that xi ∈ / F + F . We may say x = xi ∈ Ei . It follows that (F + F5) ∩ Ei = {0} which in turn entails F ∩ (F + Ei ) = {0}5because otherwise 0 = y = j ∈I ∪{i} yj ∈ F ∩ (F + Ej ) would imply yi = y − j ∈I yj ∈ Ei ∩ (F + F ) = {0} and thus 0 = y ∈ F ∩ F which is not possible. But now I ∪ {i} ∈  contradicting the assumed maximality of I . This contradiction proves the claim and finishes the proof. Exercise E7.5. Prove the following result. Proposition. In the class of L-modules, the following statements hold: (i) A submodule of a semisimple module is semisimple. (ii) A quotient of a semisimple module is semisimple. (iii) A direct sum of locally finite-dimensional semisimple modules is semisimple. [Hint. (i) Let F be a submodule of a semisimple module E and assume that F1 is a submodule of F . Since E is semisimple, there is a submodule E2 of E such that def

E = F1 ⊕ E2 . Define F2 = F ∩ E2 . Show that F = F1 ⊕ F2 . (ii) Let f : E → F be a surjective module morphism. Since E is semisimple, we find a submodule F1 of E such that E = ker f ⊕ F1 . Show that f |F1 : F1 → F is an isomorphism of modules and invoke (i) above to show that F1 is semisimple. (iii) By Theorem 7.16 every summand is a direct sum of simple submodules and thus the direct sum is a direct sum of simple submodules. Hence by Theorem 7.16 again the direct sum is semisimple.] Corollary 7.17. Assume that E is a locally finite-dimensional L-module. Let Ess be the union of all finite-dimensional semisimple submodules. Then: (i) Ess is the unique largest semisimple submodule. (ii) Ess is a direct sum of finite-dimensional simple modules. Proof. (i) By Theorem 7.16, the sum of two finite-dimensional semisimple submodules is a finite-dimensional semisimple submodule. Hence the set of all finite-dimensional semisimple submodules is directed. Therefore its union is a submodule. By Theorem 7.16 again it is semisimple, and it is obviously the largest semisimple submodule. (ii) This is an immediate consequence of (i) and Theorem 7.16. Theorem 7.18. (a) Let V be a profinite-dimensional L-module for a Lie algebra L. Then the following statements are equivalent: (i) V is reductive. (i ) Every closed submodule is a direct module summand algebraically and topologically. (ii) Every finite-dimensional quotient module of V is reductive. (iii) V is the projective limit of finite-dimensional reductive module quotients. (iv) V is isomorphic to a product of finite-dimensional simple modules. (b) Every profinite-dimensional L-module has a unique smallest submodule V ss such that V /V ss is reductive.

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Proof. Via duality 7.11, (a) follows from 7.16 and (b) from 7.17. In any L-module E we have y · (x · v) = yx · v − xy · v + x · (y · v) − x · v + x · v = [x, y] · v + x · (y · v − v) + x · v; hence the linear span of all x · v as (x, v) ranges through L × E is a submodule Eeff . Definition 7.19. For an L-module E define E0 = {v ∈ E | (∀x ∈ L) x · v = 0}; this set is clearly a submodule, called the maximal zero submodule. The submodule Eeff is called the effective submodule. In a locally finite-dimensional L-module E the submodule Ess is called the semisimple radical. Clearly, in E0 , every vector subspace is a submodule. If E is a locally finite-dimensional L-module, then E0 is just a vector space and thus is isomorphic to a direct sum R(I ) for some set I ; if V is a profinite-dimensional L-module, then V0 is just a weakly complete topological vector space and thus is isomorphic to RI for some set I (Corollary A2.9). In a locally finite-dimensional L-module, E0 ⊆ Ess . Proposition 7.20. (i) Assume that E is a simple L-module. Then E agrees with E0 if and only if it is zero or one-dimensional; if this is not the case, then E agrees with Eeff . In the latter case, if 0 = v ∈ E, then vL = E. (ii) 5If E is a semisimple locally finite-dimensional L-module, then E is the direct sum j ∈J Ej of a family of simple submodules Ej of E, and 6 E0 = Ej , (3) j ∈J dim Ej =1

Eeff =

6

Ej .

(4)

j ∈J dim Ej >1

In particular, E = E0 ⊕ Eeff .

(5)

Proof. (i) Let E be a simple L-module. Then E0 is either {0} or E. In the latter case, every vector subspace being a submodule, its dimension is either 0 or 1. In the former case, E = Eeff . Let 0 = v ∈ E. Then L · v is a submodule. Since E0 = {0} it is nonzero. Hence L · v = E 5 (ii)5 From 7.16 we deduce the direct 5 sum representation E = j ∈J Ej . Define F0 = j ∈J, dim Ej =1 Ej , and F1 = j ∈J, dim Ej >1 Ej . Let 0 = vj ∈ Ej . Then " {0} if dim Ej = 1, L · xj = Ej if dim Ej > 1 by 5 (i) above and thus F0 ⊆ E0 , F1 ⊆ Eeff . Now let v ∈ E0 . Then 0 = x · v = j ∈J x · vj for all x ∈ L, and thus x · vj = 0 for all x ∈ L and j ∈ J . Hence vj ∈ E0 for all j and thus v ∈ F0 . Therefore F0 = E0 .

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But Eeff = (F0 ∩ Eeff ) ⊕ F1 . Now let 5 F0 ⊕ F1 = E and F1 ⊆ Eeff imply5 v = j ∈J,dim Ej =1 vj ∈ F0 ∩ Eeff . Then v = nm=1 xm · wm with xm ∈ L, wm ∈ E. 5 Now wm = j ∈J wmj with wmj ∈ Ej , and v=

6

xm · wmj =

m=1,...,n j ∈J

6

xm m=1,...,n j ∈J dim Ej >1

· wmj ∈ F1 .

But then v ∈ F0 ∩ F1 = {0}, and therefore v = 0. It follows that F0 ∩ Eeff = {0}, and thus that F1 = Eeff . Proposition 7.21. Let V be a profinite-dimensional L-module and V  its dual module.  and (V )⊥ = V  . Then (V0 )⊥ = Veff eff 0 Proof. Consider an ω ∈ V  ; then ω ∈ (Veff )⊥ iff ω(x · v) = 0, for all x ∈ L and v ∈ V iff x · ω = 0 for all x ∈ L iff ω ∈ V0 . Thus V0 = (Veff )⊥ . Similarly  )⊥ by duality. Applying ⊥, in view of the Duality Theorem 7.11 (v) we get V0 = (Veff ⊥  )⊥⊥ = V  = V  since every vector subspace of V  is closed. V0 = (Veff eff eff The Structure Theorem of Reductive Profinite-Dimensional L-Modules Theorem 7.22. Let V be a reductive profinite-dimensional L-module. Then  V is isomorphic to a product V0 × Veff , and V0 ∼ = RI for some set I and Veff ∼ = j ∈J Vj where each Vj is a simple submodule of V such that 1 < dim Vj < ∞. Proof. Since the module V is reductive and profinite-dimensional, its dual V  is semisimple and locally finite-dimensional. Thus by Proposition 7.20 we have V  =  . Then Theorem 7.11 (v) and Proposition 7.21 imply V = (V  )⊥ ⊕ (V  )⊥ = V0 ⊕ Veff eff 0 ⊥ = V  /V  ∼ V  , and by V0 ⊕ (V0 )⊥⊥ = V0 ⊕ Veff . The dual of Veff is V  /Veff 0 = eff 7.20 (ii), this module is a direct sum of simple modules with finite dimension greater 1. Thus Veff by duality is a product of simple modules of finite dimension greater than 1.

Reductive Pro-Lie Algebras Let us apply some of these results to pro-Lie algebras. def

Lemma 7.23. Let g be a pro-Lie algebra and i a closed ideal such that h = g/i is simple. Then dim h < ∞ and the annihilator i⊥ in the coadjoint module gcoad of g is a finite-dimensional simple submodule and is isomorphic to the coadjoint module of h.

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Proof. We consider the adjoint module of g and its dual, the coadjoint module. The vector space  h is isomorphic to i⊥ (see [102, Theorem 7.30 (v)]) and thus i⊥ is isomorphic to the coadjoint module hcoad of h which is a simple Lie algebra and therefore has a simple adjoint module, whence the coadjoint module i⊥ is simple. Proposition 7.24. Let g be a pro-Lie algebra and  (g) the filter basis of closed ideals i such that dim g/i  < ∞. Further let gcoad = Hom(|g|, R) denote the coadjoint module. Then gcoad = i∈ (g) i⊥ , and i⊥ is a finite-dimensional module which is the coadjoint module of the finite-dimensional Lie algebra g/i. Proof. From the Duality Theorem 7.11 we know that (i⊥ ) ∼ = g/i and (g/i) = i⊥ ⊥ for i ∈  =  (g). Since  is a filter basis, {i | i ∈  } is a directed family of finitedimensional vector subspaces of  E. As g = limi∈ g/i we have E = colimi∈ i⊥ , and by 7.15 this means exactly E = i∈ i⊥ . Definition 7.25. (a) Let g be a topological Lie algebra. Then its center z(g) is the set {x ∈ g | (∀y ∈ g) [x, y] = 0}. The span of all elements [x, y], x, y ∈ L is a subalgebra of L, called the commutator subalgebra [g, g]. The notation g = [g, g] is also used. The closure [g, g] is called the closed commutator algebra. If [g, g] = g, then g is called perfect. (b) A pro-Lie algebra g is called reductive if its adjoint module gad is a reductive g-module. It is called semisimple if it is reductive and its center z(g) is zero. In terms of the adjoint module gad of g, the center is the maximal zero submodule (gad )0 , the commutator subalgebra [g, g] is the effective submodule (gad )eff . In particular, the center and the commutator algebra are ideals. We shall see presently that the commutator subalgebra of a reductive pro-Lie algebra is closed. The following lemma uses a piece of information on finite-dimensional simple Lie algebras which we provide in Appendix 3. Indeed by Corollary A3.3 we know that Lemma BR. In any finite-dimensional semisimple real Lie algebra, each element is the sum of at most two brackets. This allows us to prove the following lemma. Lemma 7.26. Let {s j | j ∈ J } be any family of finite-dimensional real semisimple Lie algebras. Then s = j ∈J sj is its own algebraic commutator algebra s = [s, s], that is, the linear span of all elements [X, Y ] for X, Y ∈ s. In particular, every semisimple pro-Lie algebra is perfect. Proof. For subsets A and B of a Lie algebra g we define brack (A, B) = n

n 6

 [Xj , Yj ] | Xj ∈ A, Yj ∈ B, j = 1, . . . , n .

j =1

By Lemma BR above we have sj = brack 2 (sj , sj ). Since  brack n (s, s) = brack n (sj , sj ) for n ∈ N j ∈J

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we note in particular that brack 2 (s, s) =

 j ∈J

brack 2 (sj , sj ) =



sj = s.

j ∈J

Clearly, brack 2 (g, g) ⊆ [s, s]. Thus s = [s, s] follows, and this says that s is perfect. The Structure Theorem of Reductive Pro-Lie Algebras Theorem 7.27. (a) For a pro-Lie algebra g the following conditions are equivalent. (i) g is reductive. (i ) Every closed ideal of g is an ideal direct summand algebraically and topologically. (ii) g is the product of a family of finite-dimensional simple or one-dimensional ideals of g. (b) Let g be a reductive pro-Lie algebra. Then the commutator algebra [g, g] is closed and is a product of finite simple real Lie algebras. Further g ∼ = z(g) ⊕ [g, g] algebraically and topologically, and z(g) ∼ = RI for some set I . (c) Every pro-Lie algebra has a unique smallest ideal ncored (g) such that g/ncored (g) is reductive. Proof. (a) The Lie algebra g is reductive iff the adjoint module gad is reductive, and by Theorem 7.18, this is the case iff gad is a product of finite-dimensional simple gmodules. Now a vector subspace of g is a submodule of the adjoint module gad iff it is an ideal of g. This remark completes the proof of the equivalence of (i), (i ) and (ii).  By (a) above g ∼ = RI × j ∈J sj with a family of simple finite-dimensional real Lie algebras sj . By the preceding Lemma 7.26, the second factor s satisfies [s, s] = s, and the first factor is abelian. The assertion follows. (c) This is a consequence of Theorem 7.18 (b) applied to the adjoint module of g. We shall call ncored (g) the coreductive radical. We shall say more about ncored (g) later in Theorem 7.66 and Theorem 7.67. From Definition 7.25 (b) and Theorem 7.27 (b) we get at once the following conclusions. Corollary 7.28. Any semisimple pro-Lie algebra is perfect. Structure of Semisimple Pro-Lie Algebras Corollary 7.29. For a pro-Lie algebra, the following statements are equivalent. (I) g is semisimple. (II) g is the product of a family of finite-dimensional simple ideals of g.

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Lemma 7.30. Assume that E is a locally finite-dimensional L-module and S is a semisimple submodule such that S0 = {0} and E/S is a zero module. Let K denote the set {x ∈ L : (∀v ∈ E) x · v = 0}. Then S = Eeff and if L/K is semisimple, then E is semisimple. In particular, (6) E = S ⊕ E0 . Proof. By Proposition 7.20, S = Seff , and therefore Seff ⊆ Eeff . But Eeff /S ⊆ (E/S)eff = {0} implies Eeff ⊆ S, and so S = Eeff . It remains to show that E is semisimple. Let F be a finite-dimensional submodule. By Theorem 7.16 it suffices to show that F is semisimple. Now S ∩ F is contained in a def

finite sum D = Ej1 ⊕ · · · ⊕ Ejn of simple submodules of dimension > 1; by replacing F by F +D we may assume that F ∩S = D. Then F /(F ∩S) ∼ = (F +S)/S ⊆ (E/S)0 . We are reduced to the finite-dimensional case and may assume that dim E < ∞. Let F1 be a submodule of E containing S such that dim F1 /S = 1; if we can show that F1 is semisimple, then F1 = S ⊕ R · x1 with an element x1 such that L · x1 = {0}. Since this works for arbitrary one-dimensional submodules of E/S we will obtain E = S ⊕ R · x1 ⊕ · · · ⊕ R · xk , k = dim E/S, which will finish the proof. Hence we may as well assume that dim E/S = 1. We may assume that K = {0} because E is an L/K-module in the obvious way, and submodules and quotient modules are the same. But then, as we assume that L is semisimple, the assertion follows from [16, §6, no 3, Définition 3]. Proposition 7.31 (Levi–Mal’cev Theorem for Reductive Pro-Lie Algebras). Let g be a pro-Lie algebra such that g/z(g) is semisimple. Then g is reductive and g is an ideal direct sum, algebraically and topologically, g = z(g) ⊕ [g, g].

(7)

Proof. We apply Lemma 7.30 with L = g, E = gcoad , S = z(g)⊥ , K = z(g). Then, by duality, we have the module isomorphisms S ∼ = (g/z(g)) and E/S ∼ = z(g) . By hypothesis, g/z(g) is semisimple and thus, by Definitions 7.15 and 7.27 (i), S is a semisimple module. Also, since the g-submodule z(g) of the adjoint module g is the zero module, its dual E/S is a zero module. The module L/K = g/z(g) is semisimple. Thus Lemma 7.30 applies and shows that E is semisimple and this means by Definition 7.15 (iv), g is reductive. So 7.28 implies g = z(g) ⊕ [g, g]. Now [g, g] ∼ = g/z(g) is a semisimple pro-Lie algebra, and then g is a reductive one. Thus (7) follows from Theorem 7.27.

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Transfinitely Solvable Lie Algebras The familiar concept of solvability of a finite-dimensional Lie algebra requires some thought in the absence of dimensional restrictions. We are dealing with topological Lie algebras, and thus there are added complications due to the possible definition of “topological solvability” which again is not apparent in the situation of finite-dimensional Lie algebras. Similar comments apply to nilpotency which we shall treat in a way that is parallel to that of solvability. Definition 7.32. Let g be a Lie algebra. Set g(0) = g and define sequences of ideals g(α) indexed by the ordinals α, card α ≤ card g via transfinite induction. Assume that g(α) is defined for α < β.  (i) If β is a limit ordinal, set g(β) = α