Math 682 WINTER 2011, List of lectures

  • On this page I will post content of all lectures. All handouts also will be posted here.
  • Monday, January 3: Universal enveloping algebra. PBW theorem.
  • Wednesday, January 5: Proof of PBW theorem, I.
  • Friday, January 7: Proof of PBW theorem, II.
  • Monday, January 10: Free Lie algebras; generators and relations. Serre's relations.
  • Wednesday, January 12: Lie algebra L_0 decomposes as a direct sum of L_-, H, L_+.
  • Friday, January 14: Proof of Serre's theorem.
  • Wednesday, January 19: End of proof of Serre's theorem.
  • Friday, January 21: Standard cyclic modules.
  • Monday, January 24: Verma modules. Possible highest weights of finite dimensional irreducible representations.
  • Wednesday, January 26: Existense theorem for finite dimensional irreducible representations.
  • Friday, January 28: Set of weights of irreducible finite dimensional representation. Characters.
  • Monday, January 31: Examples of representations of sl_3.
  • Wednesday, February 2: W-invariant functions on H.
  • Friday, February 4: Invariant functions on L.
  • Monday, February 7: Chevalley restriction theorem.
  • Wednesday, February 9: Harish-Chandra theorem.
  • Friday, February 11: Proof of Harish-Chandra theorem.
  • Monday, February 14: Composition series of Verma modules.
  • Wednesday, February 16: Weyl character formula.
  • Friday, February 18: Dimension formula. Examples in rank 2.
  • Monday, February 21: Examples in type A. Vandermonde determinant and Schur functions.
  • Wednesday, February 23: Branching Law gl(n)->gl(n-1). Gelfand-Cetlin basis. Semistandard tableaux.
  • Friday, February 25: Kostant multiplicity formula for branching. Proof of branching law gl(n)->gl(n-1).
  • Monday, February 28: Tensor products and Steinberg's formula.
  • Wednesday, March 2: How to decompose tensor products: example in type G_2.
  • Friday, March 4: Some results on tensor product decompositions.
  • Monday, March 7: Schur-Weyl duality.
  • Wednesday, March 9: Glimpse at Invariant Theory.
  • Friday, March 11: Chevalley basis and Chevalley groups of adjoint type.
  • Please find Final Exam.
  • THE END