Locally Compact GroupsEuropean Mathematical Society, 2006 - 302 pages Locally compact groups play an important role in many areas of mathematics as well as in physics. The class of locally compact groups admits a strong structure theory, which allows to reduce many problems to groups constructed in various ways from the additive group of real numbers, the classical linear groups and from finite groups. The book gives a systematic and detailed introduction to the highlights of that theory. In the beginning, a review of fundamental tools from topology and the elementary theory of topological groups and transformation groups is presented. Completions, Haar integral, applications to linear representations culminating in the Peter-Weyl Theorem are treated. Pontryagin duality for locally compact Abelian groups forms a central topic of the book. Applications are given, including results about the structure of locally compact Abelian groups, and a structure theory for locally compact rings leading to the classification of locally compact fields. Topological semigroups are discussed in a separate chapter, with special attention to their relations to groups. The last chapter reviews results related to Hilbert's Fifth Problem, with the focus on structural results for non-Abelian connected locally compact groups that can be derived using approximation by Lie groups. The book is self-contained and is addressed to advanced undergraduate or graduate students in mathematics or physics. It can be used for one-semester courses on topological groups, on locally compact Abelian groups, or on topological algebra. Suggestions on course design are given in the preface. Each chapter is accompanied by a set of exercises that have been tested in classes. |
Table des matières
Contents | 1 |
B Topological Groups | 26 |
Linear Groups over Topological Rings | 50 |
Topological Transformation Groups | 91 |
The Haar Integral | 113 |
Applications to Linear Representations | 123 |
15 | 133 |
Categories | 143 |
Embedding Cancellative Directed Semigroups into Groups | 250 |
Compact Semigroups | 254 |
Groups with Continuous Multiplication | 259 |
H Hilberts Fifth Problem | 261 |
Dimension of Locally Compact Groups | 264 |
The Rough Structure | 268 |
Notions of Simplicity | 272 |
Compact Groups | 276 |
17 | 151 |
18 | 158 |
Compact Groups | 169 |
Pontryagins Duality Theorem | 191 |
24 | 201 |
Automorphism Groups of Locally Compact Abelian Groups | 207 |
Locally Compact Rings and Fields | 212 |
Homogeneous Locally Compact Groups | 230 |
G Locally Compact Semigroups | 242 |
Countable Bases Metrizability | 279 |
NonLie Groups of Finite Dimension | 280 |
Arcwise Connected Subgroups | 281 |
Algebraic Groups | 285 |
287 | |
291 | |
297 | |
Expressions et termes fréquents
Abelian group additive group algebraic assume automorphism bijection Cauchy filterbasis CGAL closed subgroup closure compact Hausdorff group compact neighborhood compact-open topology compactly connected component consider contains continuous homomorphism continuous map converges Corollary countable defined Definition denote embedding Exercise exists fact finite dimension finite subset following hold GL(n group G Haar integral Hausdorff space implies inverse isomorphic kernel Lemma Let G Lie group linear locally compact Abelian locally compact group locally compact Hausdorff maximal compact subgroup metric monic Mor(A morphism multiplication natural map natural number neighborhood basis nonempty normal subgroup obtain open sets open subgroup product topology projective limit Proof proves assertion quotient map real number semigroup sequence subgroup of G subspace surjective Theorem topological group topological ring topological space topology induced torsion-free totally disconnected vector space vector subgroup XjEJ yields