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Math 682 WINTER 2009, 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 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