MATH 650: Lie Groups (Fall 2005)

Instructor: Dan Barbasch

Meeting Time & Room

A Lie group is a group with the additional structure of a differentiable manifold for which the group operation is differentiable. The name Lie group comes from the Norwegian mathematician M. Sophus Lie (1842-1899) who was the first to study these groups systematically in the context of symmetries of partial differential equations.

The theory of Lie groups and their representations plays a vital role in the description of symmetries in Physics (quantum physics, elementary particles), Geometry and Topology, and Number Theory (automorphic forms).

In this course we will begin by studying the basic properties of Lie groups. Much of the structure of a connected (real) Lie group and its Lie subgroups is captured by its Lie algebra, which may be defined as the algebra of left invariant vector fields.

After this introduction we will study nilpotent, solvable and compact Lie groups. Examples are the groups of strictly upper triangular, upper triangular groups, and the unitary, ortogonal and symplectic groups. We will study the representation theory of such groups, and some examples from harmonic analysis.

The final part of the course will be devoted to algebraic groups, and their representation theory. In particular we will focus on real and p-adic reductive groups, and finite Chevalley groups.

Prerequisites

A knowledge of algebra, analysis and differential geometry at an advanced undergraduate level. Lie algebras 649 is also very useful.

Bibliography

[B] N. Bourbaki, Groupes et Algebres de Lie, Hermann, Paris, 1971.

[H] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978.

[OV] A. L. Onishchik, E. B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, Heidelberg, 1990.

[V] V. S. Varadarajan, Lie Groups, Lie Algebras, and their Representations, Prentice-Hall, Engelwood Cliffs, NJ, 1974.

[W] F. Warner, Foundations of Differentiable Manifolds and Lie Groups}, Scott, Foresman and Co., Glenview, IL, 1971.

[R] W. Rossmann, Lie groups: An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Number 5, Oxford University Press, 2002; ISBN 0198596839

[DK] J.J. Duistermaat & J.A.C. Kolk, Lie Groups, Universitext serie, Springer-Verlag, New York, 2000. ISBN 3-540-15293-8, cat prijs DM 79.

[BD] T. Bröcker & T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York, 1985.

Last modified: April 5, 2005