# 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.
• 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