18.755 | Fall 2004 | Graduate

Introduction to Lie Groups

Lecture Notes

These lecture notes were created using material from Prof. Helgason’s books Differential Geometry, Lie Groups, and Symmetric Spaces and Groups and Geometric Analysis, intermixed with new content created for the class. The notes are self-contained except for some details about topological groups for which we refer to Chevalley’s Theory of Lie Groups I and Pontryagin’s Topological Groups. Documenting the material from the course, the text has a fairly large bibliography up to 1978. Since then, a huge number of books on Lie groups has appeared.

Source Textbooks

All excerpts courtesy of the American Mathematical Society. Used with permission.

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

Helgason, Sigurdur. Groups and Geometric Analysis. Providence, R.I.: American Mathematical Society, 2000. ISBN: 0821826735.

Referenced Textbooks

Chevalley, Claude. Theory of Lie groups, I. Princeton, Princeton University Press, 1946.

Pontryagin, L. S. Topological Groups. Translated from the Russian by Arlen Brown, with additional material translated by P. S. V. Naidu. 3rd ed. New York: Gordon and Breach Science Publishers, 1986. ISBN: 2881241336 (Switzerland).

Preface (PDF)

Chapter I: Elementary Differential Geometry (PDF)

1. Manifolds    
2. Mappings    
3. Affine Connections    
4. Parallelism    
5. The Exponential Mapping    
6. Covariant Differentiation

Chapter II: Lie Groups and Lie Algebras (PDF 1 of 2 - 1.9 MB) (PDF 2 of 2 - 1.8 MB)

1. 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. The Universal Covering Group    
8. General Lie Groups    
9. Differential Forms    
10. Integration of Forms    
11. Invariant Differential Forms    
12. Invariant Measures on Coset Spaces    
13. Real Forms of Complex Lie Algebras    
14. The Classical Groups and their Cartan Involutions

Chapter I: Exercises and Further Results (PDF)

A. Manifolds    
B. The Lie Derivative and the Interior Product    
C. Affine Connections    
D. Submanifolds    
E. The Hyperbolic Plane

Chapter II: Exercises and Further Results (PDF)

A. On the Geometry of Lie Groups    
B. The Exponential Mapping    
C. Subgroups and Transformation Groups    
D. Closed Subgroups    
E. Invariant Differential Forms    
F. Invariant Measures    
G. Compact Real Forms and Complete Reducibility

Solutions to Exercises (PDF)

Course Info

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