~
math261A Home Page
#
Lie groups

**Instructor:**
Vera Serganova

**Email
address:** serganov@math
**webpage:**/~serganov
**Phone Number:** 642-2150
**Office hours:** MW 5:00-6:00 in 709 Evans
**Prerequisites:** To understand this course you need basic knowledge of Algebra and Differential
Geometry. In other words you have to know what is a group and what is
a differentiable manifold. However, I will give a short introduction
to differential geometry.
**Homework:** Each Monday I will post on my web page a problem
assignment on the material of
the previous week lectures. The homework will be collected on
Wednesday the following week.
**Grading policy:** The grade will be computed according to the following
proportions: 60% for your homework and 40% for final project.
**Recommended Texts:** Fulton-Harris: Representation theory (a
first course), Humphreys: Introduction to
Lie algebras and Representation theory.
**Berkeley lecture notes for the course.
****
**

# Course outline

Lie Groups. Definitions and examples. Closed linear groups.
Lie algebras and exponential map. Relation between subgroups and
subalgebras. Campbell-Hausdorff formula.
Fundamental group of a Lie group. Coverings, isogeny.
Structure theory of Lie algebras. Solvable and nilpotent algebras.
Engel's and Lie's theorems. Semisimple Lie algebras. The Killing Form
and Cartan's criterion. Jordan decomposition.
Universal enveloping algebras. PBW Theorem.
Representations of semisimple Lie Algebras. Casimir operator and
complete reducibility.
Ado's and Levi's theorems.
Lie algebras cohomology.
Representations of sl(2) and sl(3). Spinor
representations of simply-connected cover of an orthogonal group.
Classification of complex semisimple Lie algebras. Root systems,
Dynkin diagrams and Weyl groups. Exceptional Lie algebras.
Highest weight modules. Weyl character formula.
Compact Lie groups and their Representations. Peter-Weyl theorem.
# Problem sets

**Problem set 1 **
**Problem set 2 **
**Problem set 3 **
**Problem set 4 **
**Problem set 5 **
**Problem set 6 **