Math 8710 Lie Algebra

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General Information:
Time and Venue: Every Tue, Thu 11:00-12:15, 317 Kerchoff.
Office Hour: Mon/Wed 10:30-11:30am or by email appointments.
Course Description:
This course is aimed at giving a gentle introduction to Lie algebras. Our main goal is to have a basic understanding of the fundamental subject of complex semisimple Lie algebras, their origins and their representation theory. We will cover:
  • Basics of topological groups, Lie group. Why Lie algebras?
  • Nilpotent and Solvable Lie algebras
  • Represention theory of sl(2)
  • Semisimple Lie algebras
  • Root systems, Dykin diagrams, classification of semisimple Lie algebras
  • Representations of semisimple Lie algebras
  • Weyl character formula and applications
    • Prerequisites:
    Advanced linear algebra (Math 4651 level) and basic familiarity with notions of groups, rings and modules.
    Referencess:
    [1] J. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, Springer, The online version of the book can be downloaded on Grounds through the link.
    [2] W. Fulton and J. Harris, Representation Theory, A First Course, GTM 129, Springer. The online version of the book can be downloaded on Grounds through the link.
    Homeworks:
    Homework1.
    Some Notes:
    Here are some notes on the ADE phenomena.
    The first is about quiver representations and applications to spectral sequences: Quiver Representation and Spectral Sequences.
    The second instance appears in the algebraic McKay correspondence: McKay Correspondence.
    Last Updated 01/23/22
    by You Qi.