p-Adic Lie Groups (eBook)

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