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p-Adic Lie Groups

By Peter Schneider

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In addition to providing a self-contained introduction to p-adic lie groups, this volume discusses spaces of locally analytic functions as topological vector spaces, important to applications in representation theory.

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  • ISBN13: 978-3-6422-1146-1
  • 265 Pages
  • Publication Date: June 11, 2011
  • Available eBook Formats: PDF

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Full Description
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.
Table of Contents

Table of Contents

  1. Introduction.
  2. Part A: p
  3. Adic Analysis and Lie Groups.
  4. I.Foundations.
  5. I.1.Ultrametric Spaces.
  6. I.2.Nonarchimedean Fields.
  7. I.3.Convergent Series.
  8. I.4.Differentiability.
  9. I.5.Power Series.
  10. I.6.Locally Analytic Functions.
  11.   II.Manifolds.
  12. II.7.Charts and Atlases.
  13. II.8.Manifolds.
  14. II.9.The Tangent Space.
  15. II.10.The Topological Vector Space C^an(M,E), part 1.
  16. II.11 Locally Convex K
  17. Vector Spaces.
  18. II.12 The Topological Vector Space C^an(M,E), part 2.
  19. III.Lie Groups.
  20. III.13.Definitions and Foundations.
  21. III.14.The Universal Enveloping Algebra.
  22. III.15.The Concept of Free Algebras.
  23. III.16.The Campbell
  24. Hausdorff Formula.
  25. III.17.The Convergence of the Hausdorff Series.
  26. III.18.Formal Group Laws.
  27. Part B:The Algebraic Theory of p
  28. Adic Lie Groups.
  29. IV.Preliminaries.
  30. IV.19.Completed Group Rings.
  31. IV.20.The Example of the Group Z^d_p.
  32. IV.21.Continuous Distributions.
  33. IV.22.Appendix: Pseudocompact Rings.
  34. V.p
  35. Valued Pro
  36. p
  37. Groups.
  38. V.23.p
  39. Valuations.
  40. V.24.The free Group on two Generators.
  41. V.25.The Operator P.
  42. V.26.Finite Rank Pro
  43. p
  44. Groups.
  45. V.27.Compact p
  46. Adic Lie Groups.
  47. VI.Completed Group Rings of p
  48. Valued Groups.
  49. VI.28.The Ring Filtration.
  50. VI.29.Analyticity.
  51. VI.30.Saturation.
  52. VII.The Lie Algebra.
  53. VII.31.A Normed Lie Algebra.
  54. VII.32.The Hausdorff Series.
  55. VII.33.Rational p
  56. Valuations and Applications.
  57. VII.34.Coordinates of the First and of the Second Kind.
  58. References.
  59. Index.

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