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:

*Representation Theory: A First Course*by William Fulton and Joe Harris*Lie Groups*by J.J. Duistermaat and J.A.C. Kolk*Complex Semisimple Lie Algebras*by Jean-Pierre Serre*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*by Brian C. Hall

**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.

**Syllabus:**

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:
http://www.www.ehs.stonybrook.edu/fire/disabilities.shtml