p-Adic Lie Groups (eBook)

Peter Schneider (Autor)

eBook Download: PDF

2011 | 2011
XII, 256 Seiten
Springer Berlin (Verlag)
978-3-642-21147-8 (ISBN)

Infos

Manifolds over complete nonarchimedean fields together with notions like tangent spaces and vector fields form a convenient geometric language to express the basic formalism of p-adic analysis. The volume starts with a self-contained and detailed introduction to this language. This includes the discussion of spaces of locally analytic functions as topological vector spaces, important for applications in representation theory. The author then sets up the analytic foundations of the theory of p-adic Lie groups and develops the relation between p-adic Lie groups and their Lie algebras. The second part of the book contains, for the first time in a textbook, a detailed exposition of Lazard's algebraic approach to compact p-adic Lie groups, via his notion of a p-valuation, together with its application to the structure of completed group rings.

Inhalt

Introduction 6
Contents 9
Part A p-Adic Analysis and Lie Groups 12
I Foundations 13
1 Ultrametric Spaces 13
2 Nonarchimedean Fields 18
3 Convergent Series 24
4 Differentiability 27
5 Power Series 35
6 Locally Analytic Functions 48
II Manifolds 54
7 Charts and Atlases 54
8 Manifolds 56
9 The Tangent Space 65
10 The Topological Vector Space Can (M, E), Part 1 83
11 Locally Convex K-Vector Spaces 88
12 The Topological Vector Space Can (M, E), Part 2 93
III Lie Groups 98
13 Definitions and Foundations 98
14 The Universal Enveloping Algebra 110
15 The Concept of Free Algebras 115
16 The Campbell-Hausdorff Formula 120
17 The Convergence of the Hausdorff Series 133
18 Formal Group Laws 141
Part B The Algebraic Theory of p-Adic Lie Groups 163
IV Preliminaries 164
19 Completed Group Rings 164
20 The Example of the Group Zpd 170
21 Continuous Distributions 171
22 Appendix: Pseudocompact Rings 172
V p-Valued Pro-p-Groups 175
23 p-Valuations 175
24 The Free Group on Two Generators 181
25 The Operator P 184
26 Finite Rank Pro-p-Groups 187
27 Compact p-Adic Lie Groups 198
VI Completed Group Rings of p-Valued Groups 201
28 The Ring Filtration 201
29 Analyticity 207
30 Saturation 214
VII The Lie Algebra 224
31 A Normed Lie Algebra 224
32 The Hausdorff Series 237
33 Rational p-Valuations and Applications 248
34 Coordinates of the First and of the Second Kind 252
References 256
Index 258

Produkt-Details

• Erscheint lt. Verlag: 11.6.2011
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Reihe/Serie: Grundlehren der mathematischen Wissenschaften
• Zusatzinfo: XII, 256 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: 22E20, 16S34 • completed group ring • Lie group • locally analytic manifold • p-adic • p-valuation
• ISBN-10: 3-642-21147-X / 364221147X
• ISBN-13: 978-3-642-21147-8 / 9783642211478