Mathematics 534, section 101. Lie Theory I

Term 1, 2017, TTh 11:00 - 12:30, Buchanan D216

  • Instructor :
    Zinovy Reichstein
    Office: 1105 Math Annex
    Phone: 2-3929
    E-mail: reichst at math dot ubc dot ca

  • Textbook:
    James E. Humphreys, Introduction to Lie algebras and representation theory, Springer, 1972.
  • Course description:
    Lie theory is the study of continuous group of tranformations. These groups play an important role in various areas of mathematics, from PDEs to number theory, as well as in physics. Their structure is most easily understood by in studying their ``linear approximations", otherwise known as Lie algebras. This course we will focus on the study of finite-dimenional Lie algebras and their representations by algebraic methods. We will discuss nilpotent, solvable, and semisimple Lie algebras, root systems, weights, highest weight modules, and (if time permits) universal enveloping algebras. Our ultimate goal will be the classification of complex semisimple Lie algebras. This material is foundational for many areas of pure mathematics. Our textbook is concise and beautifully written. I plan to follow it closely through Chapters I-III, occasionally supplementing the lectures with additional material, such as the Levi Decomposition Theorem.
  • Prerequisites:
    High comfort level with linear algebra. Familiarity with abstract algebra will also be helpful.
  • Homework:
    The course mark will be based entirely on homework assignemnts. I plan to assign 6-8 problem sets during the term.

    Problem Set 1. Due Tuesday, September 26.

    Solution to Problem 6.

    Problem Set 2. Due Thursday, October 5.

    Solution to Problem 4.

    Notes on the proof of Cartan's Criterion.

    Problem Set 3. Due Thursday, October 19.

    Problem Set 4. Due Thursday, November 2.

    Problem Set 5. Due Thursday, November 16.

    Problem Set 6. Due Thursday, November 30.