Lie Algebras, Math 7161 (class number 35371),
Autumn Semester 2024

Instructor: Sergei Chmutov

Classes: Monday, Wednesday, Friday 12:40--1:35 pm. The classes will be in-person in Caldwell Lab 115. All the classes will be recorded through ZOOM and the the link will be posted on Carmen.

Carmen: https://carmen.osu.edu

Main textbook:

  • Anthony Knapp, Lie Groups Beyond an Introduction. Birkhäuser, 2002. ISBN:0-8176-4259-5.

    Additional textbooks:

  • James Humphreys: Introduction to Lie Algebras and Representation Theory. Springer, 1972. ISBN:3540900527.
  • W. Fulton and J. Harris: Representation Theory – A first course. Springer, 1999. ISBN:0387974954.
  • J-P. Serre: Complex semisimple Lie algebras. Springer, 1987. ISBN:3540965696.

    Homework will be assigned weekly through Carmen. You are supposed to upload your solutions to Carmen.

    Exams: There will be one midterm and a final exams.

    TENTATIVE EXAMINATION SCHEDULE:
    MIDTERM Monday, October 7. FINAL EXAM Thursday, December 12, 2:00 - 3:45 PM

    Grading: Homework assignments average (60%), the midterm (20% each), and the final exam (20%).
    GRADING SCALE:
    A A- B+ B B- C+ C C- D+ D
    90 87 83 80 77 73 70 67 63 60

    Tentative weekly schedule of topics:
    Week 1 Introduction. Definitions and examples. Arnold's example. Adjoint representation.
    Week 2 Killing form. Nilpotent and solvable Lie algebras.
    Week 3 Cartan subalgebras. Semi-simple Lie algebras. Cartan's criterion.
    Week 4 Representations of sl2(C). Root space decomposition.
    Week 5 Root systems and Weyl groups.
    Week 6 Classification of root systems. Cartan matrices. Dynkin diagrams.
    Week 7 An, Bn, Cn, Dn, E6, E7, E8, F4, G2.
    Week 8 Finite-dimensional representations of semi-simple Lie algebras. Characters.
    Week 9 The Weyl character formula.
    Week 10 Kostant's multiplicity formula. Steinberg's formula.
    Week 11 The universal enveloping algebra. The Poincare-Birkhoff-Witt theorem.
    Week 12 Free Lie algebras and the Lyndon words.
    Week 13 Center of the universal enveloping algebra.
    Week 14 Symmetric polynomials. The Harish-Chandra isomorphism.
    Week 15 The Duflo isomorphism.

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