# Spring 2016 MATH 261B 001 LEC

Section | Days/Time | Location | Instructor | CCN |
---|---|---|---|---|

001 LEC | MWF 10-11A | 2 EVANS | HAIMAN, M | 54509 |

Units/Credit | Final Exam Group | Enrollment |
---|---|---|

4 | NONE | Limit:28 Enrolled:19 Waitlist:0 Avail Seats:9 [on 02/29/16] |

**Prerequisites:** 261A, or other previous introduction to Lie groups and Lie algebras

**Description:** The course will cover several topics in Lie theory, including some areas with open research problems. It should be of interest to a variety of students with some previous knowledge of Lie groups and Lie algebras, not limited to those continuing from 261A this past fall.

We will begin with the description of the semisimple Lie algebras and their representations, and Lie algebra cohomology (this is where we left off in the fall and may serve as useful review for those joining in the spring). Next we will discuss Hopf algebras and algebraic groups, and see how to recover a linar algebraic group from its tensor category of representations; this will lead to the remarkable fact that compact real Lie groups, reductive complex Lie groups, and reductive linear algebraic groups over fields of any characteristic are classified by the same 'Cartan data' of a root system and weight lattice.

The next topic will be flag varieties G/P of a reductive group G and the Borel-Weil-Bott geometric construction of the representations of G. I will discuss some open problems in this area, of a very concrete and combinatorial nature.

Finally we will discuss 'quantum groups,' which are non-commutative q-deformations of the Hopf algebra of functions on a reductive group, and dually, of the enveloping algebra of its Lie algebra. This leads to the theory of canonical bases, which provide a beautiful combinatorial structure underlying the representations of reductive groups. This structure is present even in the classical representations, even though the only known way to see it is through the q-deformation. There are plenty of open problems in this area also.

Depending on time, possible additional topics are Kac-Moody algebras, groups, and quantum groups (especially the affine case); and Hecke algebras and Kazhdan-Lusztig theory.

**Office:** 855 Evans

**Office Hours:** TBA

**Required Text:** V.S. Varadarajan,
Lie Groups, Lie Algebras, and Their
Representations. This book is
available
online from campus computers or through the library
proxy server.

**Recommended Reading:** see course web page

**Grading:** Based on homework - probably a smaller quantity than in 261A due to the more topics course flavor of the spring semester. As an alternative, you can earn an A by working on any open research problem. For this you don't have to solve the problem, but should make a serious attempt to understand it and work out some special cases.

**Course Webpage:** https://math.berkeley.edu/~mhaiman/math261-fall15-spring16/