Math 681 FALL 2008, List of lectures

  • On this page I will post content of all lectures. All handouts also will be posted here.
  • Monday, September 29: Definition of a Lie algebra. Examples.
  • Wednesday, October 1: Linear Lie algebras of type BCD.
  • Friday, October 3: Ideals, homomorphisms, simple Lie algebras. Representations and modules. Adjoint representation.
  • Monday, October 6: Derivations and automorphisms. Solvable Lie algebras.
  • Wednesday, October 8: Nilpotent Lie algebras. Statement of Engel's theorem.
  • Friday, October 10: Proof of Engel's theorem. Lie's theorem.
  • Monday, October 13: Proof of Lie's theorem.
  • Wednesday, October 15: Jordan decomposition.
  • Friday, October 17: Cartan's criterion.
  • Monday, October 20: Killing form and applications.
  • Wednesday, October 22: Complete reducibility, I: reminder about modules.
  • Friday, October 24: Complete reducibility, II: Casimir operator.
  • Monday, October 27: Complete reducibility, III: Proof of Weyl theorem. Please find homework here.
  • Wednesday, October 29: Representations of sl(2), I.
  • Friday, October 31: Representations of sl(2), II. Toral subalgebras are commutative.
  • Monday, November 3: Root decomposition: Centralizer of a maximal toral subalgebra.
  • Wednesday, November 5: Root decomposition: dimension of root spaces and multiples of roots.
  • Friday, November 7: Root decomposition: Cartan integers and strings of roots. Roots form a root system.
  • Monday, November 10: Root systems: 2-dimensional examples.
  • Wednesday, November 12: Root systems: simple roots.
  • Friday, November 14: Root systems: Weyl group.
  • Monday, November 17: Length function on Weyl group. Fundamental domain for the action of Weyl group. Decomposition into a direct sum of irreducible root systems.
  • Wednesday, November 19: Properties of irreducible root systems: maximal root; long and short roots.
  • Friday, November 21: Cartan matrix determines a root system uniquely. Classification of Dynkin diagrams, I.
  • Monday, November 24: Classification of Dynkin diagrams, II.
  • Wednesday, November 26: Construction of root systems.
  • Friday, November 28: Thanksgiving.
  • Monday, December 1: Isomorphism theorem: reduction to the case of simple Lie algebras.
  • Wednesday, June 4: End of proof of isomorphism theorem. Please find final exam here.
  • Friday, June 6: Root system of sl(n,F).
  • THE END