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math261A Home Page
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Lie groups

**Instructor:**
Vera Serganova

**Email
address:** serganov@math
**webpage:**http://math.berkeley.edu/~serganov
**Phone Number:** 642-2150
**Office hours:** MW 3:30-5:00 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 Friday I will post on my web page a problem assignment (3-5
problems) on the material of
the week lectures. The homework will be collected the next Friday.
**Grading policy:** The grade will be computed according to the following
proportions: 60% for your homework and 40% for the take home final.
**Recommended Texts:** Fulton-Harris: Representation theory (a
first course), Humphreys: Introduction to
Lie algebras and Representation theory.
**Lecture notes for the course.
****
**** Attention: there will be lectures on January 23,25,27 (the
previous cancellation is cancelled) **

# Course outline

Lie Groups. Definitions and examples. Closed lienar 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.
Representations of semisimple Lie Algebras. Casimir operator and
complete reducibility.
Ado's and Levi's theorems.
Representations of classical Lie groups. Representations of
$sl_2$ and $sl_3$. Spinor
representations of simply-connected cover of an orthogonal group.
Universal enveloping algebras. PBW Theorem.
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**
**Problem set 7**
**Problem set 8**
**Problem set 9**
**Problem set 10**
**Problem set 11**
**Problem set 12**
**Final exam **