The University of Massachusetts Boston

Department of Mathematics

MA 480 The Geometry of Lie Group Actions I, Fall 2009

Instructor: Prof. Alfred Noël

Office : 3-175 Science Building

Phone : (617)-287-6458

Email : anoel@math.umb.edu

Url : http://www.math.umb.edu/~anoel

Room: S2-62.

Class hours : Friday 4:00 PM – 6:30 PM (May Change).

Office hours: Tuesday, Thursday 3:45 PM – 5:15 PM.

Text : Symmetry, Representations, and Invariants (Graduate Texts in Mathematics) by Roe Goodman and Nolan R. Wallach published by Springer Verlag.

Course Description: This course will introduce Lie groups and algebraic groups using examples from the Classical groups. We hope to cover the first three chapters that consist of the following:

· The classical linear groups (real and complex forms)

· Closed subgroups of GL(n) as Lie groups

· Linear algebraic groups and rational representations

· Structure of complex classical groups: maximal torus, roots, adjoint representation, Weyl group

· Highest weight theory for representations of semisimple Lie algebras

· Complete reducibility of representations of semisimple Lie algebras and classical groups

This is a very ambitious project. We may not achieve the goal stated above. However, we shall try our best. I will lecture most of the time and will go over assigned exercises in class. It would be nice to start producing a solution manual for the book also.

This will be an opportunity for those of you who are interested in working with Professor Jackson or me to have an idea of the type of research that we are currently pursuing. The course should be of interest to those who intend to take on graduate studies in Geometry, Algebraic Combinatorics, Representation Theory etc … In fact the material is fundamental to a genuine understanding of Modern Mathematics and Physics.

Swami Iyer will create a wiki-type page where we could post interesting findings and notes.