Math 731 — Seminar in Algebra

Classical Groups

Fall 2002

Instructor: R. Keith Dennis

Time: MWF 3:35-4:25

Room: Malott 230

Prerequisite: Linear algebra (e.g., Math 433) and basic group theory (e.g., Math 434 or Math 631).

This course will be accessible to beginning graduate students and should be of interest to those interested in algebra, representation theory, K-theory, algebraic topology, among others.

Linear groups over fields play a fundamental role in many parts of mathematics such as: The Classification Theorem for Finite Nonabelian Simple Groups asserts that essentially all such finite simple groups have such an origin. Generalization to coefficients in arbitrary rings provides the foundation for algebraic K-theory.

The basic definitions and properties of many of the interesting groups will be developed in the course.

Probable topics include the following:

• The general linear group
• Bilinear forms
• Symplectic groups
• Orthogonal geometry & the orthogonal group
• Clifford algebras
• Hermitian forms, unitary spaces, & unitary groups

There will be no text for the course. Some suggested references:

• E. Artin, Geometric Algebra, Wiley Interscience, 1957.
• A. J. Hahn and O. T. O'Meara, The Classical Groups and K-Theory, Springer, 1989.
• Larry Grove, Classical Groups and Geometric Algebra, AMS, 2001.
• L. C. Grove and C. T. Benson, Finite Reflection Groups, 2nd Ed., Springer, New York, 1985.
• John B. Sullivan, Groups and Geometry, Brown, 1994.
• Donald E. Taylor, The Geometry of the Classical Groups, Heldermann, 1992.