Math 682 WINTER 2013, List of lectures

  • On this page I will post content of all lectures. All handouts also will be posted here.
  • Monday, January 7: Universal enveloping algebra. PBW theorem.
  • Wednesday, January 9: Proof of PBW theorem, I.
  • Friday, January 11: Proof of PBW theorem, II.
  • Monday, January 14: Free Lie algebras; generators and relations. Serre's relations.
  • Wednesday, January 16: Lie algebra L_0 decomposes as a direct sum of L_-, H, L_+.
  • Friday, January 18: Proof of Serre's theorem.
  • Monday, January 21: Martin Luther King Jr day.
  • Wednesday, January 23: Standard cyclic modules.
  • Friday, January 25: Existence of standard cyclic modules: Verma modules.
  • Monday, January 28: Possible highest weights of finite dimensional irreducible representations.
  • Wednesday, January 30: Existense theorem for finite dimensional irreducible representations.
  • Friday, February 1: Set of weights of irreducible finite dimensional representation. Characters.
  • Monday, February 4: W-invariant functions on H.
  • Wednesday, February 6: Invariant functions on L.
  • Friday, February 8: Chevalley restriction theorem.
  • Monday, February 11: Harish-Chandra theorem.
  • Wednesday, February 13: Proof of Harish-Chandra theorem.
  • Friday, February 15: Composition series of Verma modules.
  • Monday, February 18: Weyl character formula.
  • Wednesday, February 20: Dimension formula. Examples in rank 2.
  • Friday, February 22: Examples in type A. Vandermonde determinant and Schur functions.
  • Monday, February 25: Fundamental representations of classical Lie algebras. Spinors.
  • Wednesday, February 27: Branching Law gl(n)->gl(n-1). Gelfand-Cetlin basis. Semistandard tableaux.
  • Friday, March 1: Kostant multiplicity formula for branching. Proof of branching law gl(n)->gl(n-1).
  • Monday, March 4: Tensor products and Steinberg's formula.
  • Wednesday, March 6: How to decompose tensor products: example in type G_2.
  • Friday, March 8: Some results on tensor product decompositions.
  • Monday, March 11: Schur-Weyl duality.
  • Wednesday, March 13: Glimpse at Invariant Theory.
  • Friday, March 15: Chevalley basis and Chevalley groups of adjoint type.
  • Please find Final Exam.
  • THE END