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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