### Course material from previous semesters

Spring 2005

### Course material for current semester (Fall, 2009)

**Lecturer:**Prof. Roe Goodman

**Office Hours:**

MTh 2:00-3:00 (and usually W 2:00-3:00)

**Text:** *Symmetry, Representations, and Invariants* by Roe
Goodman and Nolan R. Wallach (Springer Graduate Texts in Mathematics
255)

The required text and the supplementary books by Rossmann (*Lie Groups: An Introduction
Through Linear Groups*) and Hawkins (*Emergence of the Theory of
Lie Groups*) are available on reserve in the math
library.

A presentation of Lie groups from the applied mathematician's viewpoint with applications to numerical analysis are surveyed in the paper Lie-group methods by Iserles et al.

**Description and Prerequisites:**This course will be an introduction to Lie groups and algebraic groups. The prerequisites are real analysis, linear algebra, and elementary topology. No prior knowledge of Lie algebras, Lie groups, or representation theory will be assumed.

**Course Outline:**

- 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

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

**Lectures:** Here is a detailed
syllabus

Roe Goodman / goodman "at" math "dot" rutgers "dot" edu / Revised September 2, 2009

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Information posted prior to the beginning of the semester is frequently tentative, or based on previous semesters. Textbooks should not be purchased until confirmed with the instructor. For generally reliable textbook information—with the exception of sections with an alphabetic code like H1 or T1, and topics courses (197,395,495)—see the textbook list.