Mathematics Department - Math 549 - Lie Groups

Math 549 - Lie Groups

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:

  1. The classical linear groups (real and complex forms)
  2. Closed subgroups of GL(n) as Lie groups
  3. Linear algebraic groups and rational representations
  4. Structure of complex classical groups: maximal torus, roots, adjoint representation, Weyl group
  5. Highest weight theory for representations of semisimple Lie algebras
  6. 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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