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Math 682 WINTER 2013, List of lectures
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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