18.755 | Fall 2004 | Graduate

Introduction to Lie Groups


In addition to the table of contents of the required textbook, given below is a list of additional readings for the course.

Required Textbook

Helgason, Sigurdur. Differential Geometry, Lie Groups, and Symmetric Spaces. Providence, R.I.: American Mathematical Society, 2001. ISBN 0821828487.

Table of Contents

Chapter I: Elementary Differential Geometry

  1. Manifolds
  2. Tensor Fields
    1. Vector Fields and 1-Forms
    2. Tensor Algebra
    3. The Grassman Algebra
    4. Exterior Differentiation
  3. Mappings
    1. The Interpretation of the Jacobian
    2. Transformation of Vector Fields
    3. Effect on Differential Forms
  4. Affine Connections
  5. Parallelism
  6. The Exponential Mapping
  7. Covariant Differentiation
  8. The Structural Equations
  9. The Riemannian Connection
  10. Complete Riemannian Manifolds
  11. Isometries
  12. Sectional Curvature
  13. Riemannian Manifolds of Negative Curvature
  14. Totally Geodesic Submanifolds
  15. Appendix 1. Topology 2. Mappings of Constant Rank

Chapter II: Lie Groups and Lie Algebras

  1. The Exponential Mapping
    1. The Lie Algebra of a Lie Group
    2. The Universal Enveloping Algebra
    3. Left Invariant Affine Connections
    4. Taylor’s Formula and the Differential of the Exponential Mapping
  2. Lie Subgroups and Subalgebras
  3. Lie Transformation Groups
  4. Coset Spaces and Homogeneous Spaces
  5. The Adjoint Group
  6. Semisimple Lie Groups
  7. Invariant Differential Forms
  8. Perspectives

Chapter III: Structure of Semisimple Lie Algebras

  1. Preliminaries
  2. Theorems of Lie and Engel
  3. Cartan Subalgebras
  4. Root Space Decomposition
  5. Significance of the Root Pattern
  6. Real Forms
  7. Cartan Decompositions
  8. Examples. The Complex Classical Lie Algebras

Chapter IV: Symmetric Spaces

  1. Affine Locally Symmetric Spaces
  2. Groups of Isometries
  3. Riemannian Globally Symmetric Spaces
  4. The Exponential Mapping and the Curvature
  5. Locally and Globally Symmetric Spaces
  6. Compact Lie Groups
  7. Totally Geodesic Submanifolds. Lie Triple Systems

Chapter V: Decomposition of Symmetric Spaces

  1. Orthogonal Symmetric Lie Algebras
  2. The Duality
  3. Sectional Curvature of Symmetric Spaces
  4. Symmetric Spaces with Semisimple Groups of Isometries
  5. Notational Conventions
  6. Rank of Symmetric Spaces

Chapter VI: Symmetric Spaces of the Noncompact Type

  1. Decomposition of a Semisimple Lie Group
  2. Maximal Compact Subgroups and Their Conjugacy
  3. The Iwasawa Decomposition
  4. Nilpotent Lie Groups
  5. Global Decompositions
  6. The Complex Case

Chapter VII: Symmetric Spaces of the Compact Type

  1. The Contrast between the Compact Type and the Noncompact Type
  2. The Weyl Group and the Restricted Roots
  3. Conjugate Points. Singular Points. The Diagram
  4. Applications to Compact Groups
  5. Control over the Singular Set
  6. The Fundamental Group and the Center
  7. The Affine Weyl Group
  8. Application to the Symmetric Space U/K
  9. Classification of Locally Isometric Spaces
  10. Geometry of U/K. Symmetric Spaces of Rank One
  11. Shortest Geodesics and Minimal Totally Geodesic Spheres
  12. Appendix. Results from Dimension Theory

Chapter VIII: Hermitian Symmetric Spaces

  1. Almost Complex Manifolds
  2. Complex Tensor Fields. The Ricci Curvature
  3. Bounded Domains. The Kernel Function
  4. Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type
  5. Irreducible Orthogonal Symmetric Lie Algebras
  6. Irreducible Hermitian Symmetric Spaces
  7. Bounded Symmetric Domains

Chapter IX: Structure of Semisimple Lie Groups

  1. Cartan, Iwasawa, and Bruhat Decompositions
  2. The Rank-One Reduction
  3. The SU (2,1) Reduction
  4. Cartan Subalgebras
  5. Automorphisms
  6. The Multiplicities
  7. Jordan Decompositions

Chapter X: The Classification of Simple Lie Algebras and of Symmetric Spaces

  1. Reduction of the Problem
  2. The Classical Groups and Their Cartan Involutions
    1. Some Matrix Groups and Their Lie Algebras
    2. Connectivity Properties
    3. The Involutive Automorphisms of the Classical Compact Lie Algebras
  3. Root Systems
    1. Generalities
    2. Reduced Root Systems
    3. Classification of Reduced Root Systems. Coxeter Graphs and Dynkin Diagrams
    4. The Nonreduced Root Systems
    5. The Highest Root
    6. Outer Automorphisms and the Covering Index
  4. The Classification of Simple Lie Algebras over C
  5. Automorphisms of Finite Order of Semisimple Lie Algebras
  6. The Classifications
    1. The Simple Lie Algebras over C and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV
    2. The Real Forms of Simple Lie Algebras over C. Irreducible Riemannian Globally Symmetric Spaces of Type I and Type IV
    3. Irreducible Hermitian Symmetric Spaces
    4. Coincidences between Different Classes. Special Isomorphisms

Additional Readings

The first two papers below are quite elementary and non-technical and are passed out at the very beginning of the course. They serve as motivation. The third one is more technical and is passed out near the end of the course.

  1. Helgason, Sigurdur. “Sophus Lie, the mathematician” (PDF - 1.2 MB). O.A. Laudal and B. Jahren (eds.) The Sophus Lie Memorial Conference. Oslo 1992 Proceedings , Oslo: Universitetsforlaget (Scandinavian University Press) 1994.
  2. Sophus Lie and the Role of Lie Groups in Mathematics (PDF). Opening lecture by Sigurdur Helgason at a Nordic Teachers Conference in Reykjavik 1990.
  3. Reprinted with permission from Springer.  Helgason, Sigurdur. “A Centennial: Wilhelm Killing and the Exceptional Groups.” (PDF - 1.2MB) The Mathematical Intelligencer 12, no. 3 (1990).

For a thorough treatment of the history of the subject, see

  1. Hawkins, Thomas. The Emergence of the Theory of Lie Groups. New York: Springer, 2000. ISBN: 0387989633.
  2. Borel, Armand. Essays in the History of Lie Groups and Algebraic Groups. Providence, R.I.: American Mathematical Society, 2000. ISBN: 0821802887.

Course Info

As Taught In
Fall 2004
Learning Resource Types
Lecture Notes
Problem Sets with Solutions