Basic Lie Theory

The following sections are included:

• Contents

• Preface and Acknowledgments

• Notations

The following sections are included:

• Topological Groups

• Lie Groups

• Covering Maps and Groups

• Group Actions and Homogeneous Spaces

• Lie Algebras

The following sections are included:

• Elementary Properties of a Lie Group

• Taylor's Theorem and the Coefficients of expX expY

• Correspondence between Lie Subgroups and Subalgebras

• The Functorial Relationship

• The Topology of Compact Classical Groups

• The Iwasawa Decompositions for GL(n, ℝ) and GL(n, ℂ)

• The Baker-Campbell-Hausdorff Formula

The following sections are included:

• Haar Measure on a Locally Compact Group

• Properties of the Modular Function

• Invariant Measures on Homogeneous Spaces

• Compact or Finite Volume Quotients

• Applications

• Compact Linear Groups and Hilbert's 14^th Problem

The following sections are included:

• Basics of Lie Algebras
  • Ideals and Related Concepts
  • Semisimple Lie Algebras
  • Complete Lie Algebras
  • Lie Algebra Representations
  • The Irreducible Representations of sl(2, k)
  • Invariant Forms
  • Complex, Real and Rational Lie Algebras

• Engel and Lie's Theorems
  • Engel's Theorem
  • Lie's Theorem

• Cartan's Criterion and Semisimple Lie Algebras
  • Some Algebra
  • Cartan's Solvability Criterion
  • Explicit Computations of Killing Form
  • Further Results on Jordan Decomposition

• Weyl's Theorem on Complete Reducibility

• Levi-Malcev Decomposition

• Reductive Lie Algebras

• The Jacobson-Morozov Theorem

• Low Dimensional Lie Algebras over ℝ and ℂ

• Real Lie Algebras of Compact Type

The following sections are included:

• Introduction

• Maximal Tori in Compact Lie Groups

• Maximal Tori in Compact Connected Lie Groups

• The Weyl Group

• What Goes Wrong If G is Not Compact

The following sections are included:

• Introduction

• The Schur Orthogonality Relations

• Compact Integral Operators on a Hilbert Space

• The Peter-Weyl Theorem and its Consequences

• Characters and Central Functions

• Induced Representations

• Some Consequences of Frobenius Reciprocity

The following sections are included:

• Introduction

• The Polar Decomposition

• The Cartan Decomposition

• The Case of Hyperbolic Space and the Lorentz Group

• The G-invariant Metric Geometry of P

• The Conjugacy of Maximal Compact Subgroups

• The Rank and Two-Point Homogeneous Spaces

• The Disk Model for Spaces of Rank 1

• Exponentiality of Certain Rank 1 Groups

The following sections are included:

• Root and Weight Space Decompositions

• Cartan Subalgebras

• Roots of Complex Semisimple Lie Algebras

• Real Forms of Complex Semisimple Lie Algebras

• The Iwasawa Decomposition

The following sections are included:

• Lattices in Euclidean Space

• GL(n, ℝ)/GL(n, ℤ) and SL(n, ℝ)/ SL(n, ℤ)

• Lattices in More General Groups

• Fundamental Domains

The following sections are included:

• Introduction

• A Density Theorem for Cofinite Volume Subgroups

• Consequences and Extensions of the Density Theorem

The following sections are included:

• Appendix A Vector Fields

• Appendix B The Kronecker Approximation Theorem

• Appendix C Properly Discontinuous Actions

• Appendix D The Analyticity of Smooth Lie Groups

• Bibliography

• Index