Wenhan Dai

Lie Algebra and Representations

Lie algebras form a fundamental class of mathematical objects that are not only important for the study of algebra, but also have numerous applications in mathematical fields such as differential geometry and even in physics. This course introduces the basics of finite-dimensional complex Lie algebras, with emphasis on the structure and classification of complex semisimple Lie algebras, and will also briefly discuss the basic properties of the representations.

Here are prepared notes for a mini-course given by Professor Jinpeng An (Peking University) at the invitation of Tianyuan Mathematical Center in Southwest China in February 2022.

The course closely follows [Hum12] and [Car05].

Contents

  1. Introduction.
    Basic notions, the main classification theorem of simple lie algebras.
  2. Abelian, nilpotent, and solvable Lie algebras.
    Ad-semisimple and ad-nilpotent elements, a characterization of abelian Lie algebras, Engel’s theorem for nilpotent Lie algebras, Lie’s theorem for linear solvable Lie algebras.
  3. Invariant bilinear forms and applications.
    An application of the trace form, Jordan decomposition, Cartan’s criterions, structure of semisimple Lie algebras, abstract Jordan decomposition.
  4. Root spaces and root systems.
    Root space decompositions, root systems, conjugacy of Cartan subalgebras, simple Lie algebras and irreducible root systems.
  5. Classification of root systems.
    Basic types: A, B, C, D, and others.
  6. Representations.
    Basic notions, Weyl’s theorem on complete reducibility, application of Weyl’s theorem: Jordan decomposition, representations of sl(2,C).

References

  1. [Car05] Roger W. Carter. Lie algebras of finite and affine type. Number 96. Cambridge University Press, 2005.
  2. [Hum12] James E. Humphreys. Introduction to Lie algebras and representation theory, volume 9. Springer Science & Business Media, 2012.