Math 552: Introduction to Lie Groups and Lie Algebras

Fall 2008
Department of Mathematics
SUNY at Stony Brook

This course will offer an introduction to the theory of Lie groups, Lie algebras, and their representations. This material is frequently used by mathematicians in a wide variety of fields. A tentative syllabus/schedule is given below, but the course may differ slightly depending on the interests of the students taking the course.

Time/Location: MWF 9:35am-10:30am, Physics P122

Instructor: Corbett Redden
Math Tower 3-114. Phone: 632-8261. email: redden at math dot sunysb dot edu
Office Hours: MWF 10:40 - 11: 40a, or drop-in, or by appointment.

Textbook: Introduction to Lie groups and Lie algebras, by Alexander Kirillov. Cambridge Studies in Advanced Mathematics (No. 113), 2008.

There are a number of good books on the subject, including:

Prerequisites: Students are expected to be familiar with most of the material of Math 530-531 (Geometry/Topology I-II) and Math 534-535 (Algebra I-II). Manifolds and group/algebra/module theory will be frequently used.

Requirements: There will be regular homework assignments (less than a core course, but more than a topics course) and a final exam/project. Possible topics for a final project are listed here.

Week Notes Section Description
9/1 - 9/5 No class 9/1 §2.1-§2.6 Definition of Lie groups, subgroups, cosets, group actions on manifolds, homogeneous spaces.
9/8 - 9/12 No class 9/12 §2.7-§3.6 Classical groups, exponential and logarithmic maps, Lie bracket, Lie algebras, subalgebras, ideals, stabilizers, center
9/15 - 9/19 Homework 1
§2: 6-8. §3: 1,3,9
Due Friday 9/26
§3.7-§3.10 Baker-Campbell-Hausdorff formula, Lie's Theorems (above all done in slightly different order)
9/22 - 9/26 §4.1-§4.4 Representations, operations on representations, irreducible representations, Schur's lemma
9/29 - 10/3 No class (9/30,) 10/1 §4.5-§4.7 Unitary representations and complete reducibility, representations of finite groups, Haar measure on compact Lie groups, characters
10/6 - 10/10 (No class 10/9) §4.7, §5.1-§5.3 Peter-Weyl theorem, universal enveloping algebra and Poincare-Birkoff-Witt, commutants
10/13 - 10/17 Homework 2 Due 10/17 §5.4-§5.8 Solvable and nilpotent Lie algebras (with Lie/Engel theorems), semisimple and reductive algebras, invariant bilinear forms, Killing form, Cartan criteria
10/20 - 10/24 §5.9-§6.3 Jordan decomposition, complex semisimple Lie algebras, compact groups/algebras, complete reducibility of representations
10/27 - 10/31 §4.8, §6.4-§6.6 Toral subalgebras, Cartan subalgebras, root systems
11/3 - 11/7 Homework 3 Due 11/7
Graph Paper
§6.7-§7.3 Regular elements and Cartan subalgebras, abstract root systems, Weyl group, rank 2 root systems
11/10 - 11/14 §7.4-§7.7 Positive roots, simple roots, weight lattice, root lattice, Weyl chambers, simple reflections
11/17 - 11/21 §7.8-§7.10 Dynkin diagrams, classification of root systems, classification of semisimple Lie algebras
11/24 - 11/28 No class (11/27,) 11/28 §8.1-§8.2 Representations of semisimple Lie algebras, weight decomposition, characters, highest weight representations, Verma modules
12/1 - 12/5 §8.3-§8.6 Classification of irreducible finite-dimensional representations, BGG resolution, Weyl character formula
12/8 - 12/12 Class projects/assorted topics, end of §8
12/15 - 12/19 Last class 12/15

Disabilities: If you have a physical, psychological, medical or learning disability that may impact your course work, please contact Disability Support Services, ECC (Educational Communications Center) Building, room 128, (631) 632-6748. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential. Students requiring emergency evacuation are encouraged to discuss their needs with their professors and Disability Support Services. For procedures and information, go to the following web site: