Math 682 WINTER 2009, List of lectures

  • On this page I will post content of all lectures. All handouts also will be posted here.
  • Monday, January 5: Engel subalgebras.
  • Wednesday, January 7: Cartan subalgebras; Cartan subalgebras are minimal Engel subalgebras; maximal toral subalgebras are Cartan subalgebras.
  • Friday, January 9: Cartan subalgebras of a solvable Lie algebra are conjugated.
  • Monday, January 12: Borel subalgebras.
  • Wednesday, January 14: Conjugacy of Borel subalgebras.
  • Friday, January 16: Universal enveloping algebra.
  • Wednesday, January 21: PBW theorem and its consequences.
  • Friday, January 23: Proof of PBW theorem.
  • Monday, January 26: Free Lie algebras; generators and relations.
  • Wednesday, January 28: Serre relations.
  • Friday, January 30: Lie algebra L_0 decomposes as a direct sum of L_-, H, L_+.
  • Monday, February 2: Serre's theorem.
  • Wednesday, February 4: Proof of Serre's theorem.
  • Friday, February 6: Standard cyclic modules. Verma modules.
  • Monday, February 9: Possible highest weights of finite dimensional irreducible representations.
  • Wednesday, February 11: Existense theorem for finite dimensional irreducible representations.
  • Friday, February 13: Proof of existense theorem for finite dimensional irreducible representations.
  • Monday, February 16: Representations of sl_3.
  • Wednesday, February 18: Formal characters.
  • Friday, February 20: Invariant functions on L.
  • Monday, February 23: Chevalley restriction theorem.
  • Wednesday, February 25: Center of universal enveloping algebra.
  • Friday, February 27: Harish-Chandra theorem.
  • Monday, March 2: Proof of Harish-Chandra theorem.
  • Wednesday, March 4: Composition series of Verma modules.
  • Friday, March 6: Weyl character formula.
  • Monday, March 9: Kostant multiplicity formula and dimension formula.
  • Wednesday, March 11: Tensor products.
  • Friday, March 13: Denominator formula, Vandermonde determinant, and Schur functions.
  • Final Exam
  • THE END