Lie Groups

Lie Groups by J.J. Duistermaat, Johan A.C. Kolk.

This book is devoted to an exposition of the theory of finite-dimensional Lie groups and Lie algebras, which is a beautiful and central topic in modern mathematics. At the end of the nineteenth century this theory came to life in the works of Sophus Lie. It had its origins in Lie's idea of appl...

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Bibliographic Details
Main Authors: Duistermaat, J.J (Author), Kolk, Johan A.C (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2000.
Edition:1st ed. 2000.
Series:Universitext,
Springer eBook Collection.
Subjects:
Group theory.
Topological groups.
Lie groups.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1. Lie Groups and Lie Algebras
  • 1.1 Lie Groups and their Lie Algebras
  • 1.2 Examples
  • 1.3 The Exponential Map
  • 1.4 The Exponential Map for a Vector Space
  • 1.5 The Tangent Map of Exp
  • 1.6 The Product in Logarithmic Coordinates
  • 1.7 Dynkin’s Formula
  • 1.8 Lie’s Fundamental Theorems
  • 1.9 The Component of the Identity
  • 1.10 Lie Subgroups and Homomorphisms
  • 1.11 Quotients
  • 1.12 Connected Commutative Lie Groups
  • 1.13 Simply Connected Lie Groups
  • 1.14 Lie’s Third Fundamental Theorem in Global Form
  • 1.15 Exercises
  • 1.16 Notes
  • 2. Proper Actions
  • 2.1 Review
  • 2.2 Bochner’s Linearization Theorem
  • 2.3 Slices
  • 2.4 Associated Fiber Bundles
  • 2.5 Smooth Functions on the Orbit Space
  • 2.6 Orbit Types and Local Action Types
  • 2.7 The Stratification by Orbit Types
  • 2.8 Principal and Regular Orbits
  • 2.9 Blowing Up
  • 2.10 Exercises
  • 2.11 Notes
  • 3. Compact Lie Groups
  • 3.0 Introduction
  • 3.1 Centralizers
  • 3.2 The Adjoint Action
  • 3.3 Connectedness of Centralizers
  • 3.4 The Group of Rotations and its Covering Group
  • 3.5 Roots and Root Spaces
  • 3.6 Compact Lie Algebras
  • 3.7 Maximal Tori
  • 3.8 Orbit Structure in the Lie Algebra
  • 3.9 The Fundamental Group
  • 3.10 The Weyl Group as a Reflection Group
  • 3.11 The Stiefel Diagram
  • 3.12 Unitary Groups
  • 3.13 Integration
  • 3.14 The Weyl Integration Theorem
  • 3.15 Nonconnected Groups
  • 3.16 Exercises
  • 3.17 Notes
  • 4. Representations of Compact Groups
  • 4.0 Introduction
  • 4.1 Schur’s Lemma
  • 4.2 Averaging
  • 4.3 Matrix Coefficients and Characters
  • 4.4 G-types
  • 4.5 Finite Groups
  • 4.6 The Peter-Weyl Theorem
  • 4.7 Induced Representations
  • 4.8 Reality
  • 4.9 Weyl's Character Formula
  • 4.10 Weight Exercises
  • 4.11 Highest Weight Vectors
  • 4.12 The Borel-Weil Theorem
  • 4.13 The Nonconnected Case
  • 4.14 Exercises
  • 4.15 Notes
  • References for Chapter Four
  • Appendices and Index
  • A Appendix: Some Notions from Differential Geometry
  • B Appendix: Ordinary Differential Equations
  • References for Appendix.

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