**Lecturer:** Prof.
Roe Goodman

**Text:** Wulf Rossmann, *Lie Groups: An Introduction
Through Linear Groups*,

ISBN 0-19-859683-9,
Oxford University Press, New York, 2002.

**Supplementary Text:** Roe Goodman and Nolan R. Wallach,
*Representations and Invariants of the Classical Groups*,

ISBN 0-521-66348-2,
Cambridge University Press, 2003.

**Grading:** There are graded
homework exercises that are due every two weeks during the term.

This course will be an introduction to Lie groups, beginning with the general linear group and the other classical groups (the unitary, orthogonal and symplectic groups in both compact and noncompact forms), and finishing with the Weyl character formula and the Borel-Weil theorem for the irreducible representations of a compact, connected Lie group such as the unitary group U(n) or the orthogonal group SO(n). We will follow Rossmann's book with additional material from the relevant chapters and appendices of the Goodman-Wallach book. The prerequisites are real analysis (differentiation and integration of functions of several real variables), linear algebra, and elementary ideas from topology (such as covering spaces). The point of view will be much more analytic than in the Fall 2004 Lie algebra course 640:550, and students with no prior knowledge of Lie algebras should not be at a disadvantage in the Lie groups course.

Topics will include the following:

- The exponential map and exponential coordinates for matrices; the Campbell-Baker-Hausdorff series.
- Linear Lie groups and their Lie algebras; correspondence between Lie subalgebras and Lie subgroups.
- The classical linear groups as Lie groups.
- Homogeneous spaces for Lie groups.
- Integration on manifolds, Lie groups, and homogeneous spaces; Weyl integral formula for compact Lie groups.
- Representations of compact Lie groups; Peter-Weyl theorem; Weyl character formula; Borel-Weil theorem

Here is a lecture-by-lecture syllabus.

Roe Goodman / goodman "at" math "dot" rutgers "dot" edu / Revised October 25, 2007