Math 681 FALL 2010, List of lectures

  • On this page I will post content of all lectures. All handouts also will be posted here.
  • Monday, September 27: Definition of a Lie algebra. Examples.
  • Wednesday, September 29: Linear Lie algebras of type BCD. Derivations and automorphisms.
  • Friday, October 1: Ideals, homomorphisms, simple Lie algebras. Lie algebra sl_2(F) is simple.
  • Monday, October 4: Representations and modules. Adjoint representation. Solvable Lie algebras.
  • Wednesday, October 6: Nilpotent Lie algebras. Statement of Engel's theorem.
  • Friday, October 8: Proof of Engel's theorem. Lie's theorem.
  • Monday, October 11: Proof of Lie's theorem.
  • Wednesday, October 13: Jordan decomposition.
  • Friday, October 15: Cartan's criterion.
  • Monday, October 18: Killing form and applications.
  • Wednesday, October 20: Derivations of semisimple Lie algebras. Abstract Jordan decomposition.
  • Friday, October 22: Finite dimensional irreducible sl(2)-modules.
  • Monday, October 25: Complete reducibility for finite dimensional sl(2)-modules.
  • Wednesday, October 27: Casimir operator and complete reducibility for finite dimensional representations of semisimple Lie algebras (Weyl's theorem).
  • Friday, October 29: Toral subalgebras of semisimple Lie algebras.
  • Monday, November 1: Root decomposition: Centralizer of a maximal toral subalgebra. Please find homework here.
  • Wednesday, November 3: Root decomposition: dimension of root spaces and multiples of roots.
  • Friday, November 5: Root decomposition: Cartan integers and strings of roots. Roots form a root system.
  • Monday, November 8: Root systems: 2-dimensional examples. Simple roots.
  • Wednesday, November 10: Root systems: Weyl group.
  • Friday, November 12: Root systems: More on Weyl group. Decomposition of a root system into the union of irreducible ones.
  • Monday, November 15: Long and short roots. Cartan matrix determines a root system.
  • Wednesday, November 17: Classification of Dynkin diagrams, I.
  • Friday, November 19: Classification of Dynkin diagrams, II.
  • Monday, November 22: Simplicity of a Lie algebra and irreducibility of its root system. Reductive Lie algebras and criterion for reductivity/semisimplicity.
  • Wednesday, November 24: Lie algebra sl(V) is simple. Isomorphism Theorem.
  • Friday, November 26: Thanksgiving.
  • Monday, November 29: Proof of isomorphism theorem.
  • Wednesday, December 1: Maximal toral subalgebras are conjugated. Please find Final Exam here.
  • Friday, December 3: Maximal toral subalgebras = Cartan subalegbras.
  • THE END