Math 681 FALL 2008, List of lectures
On this page I will post content of all lectures. All handouts also will be posted here.
Monday, September 29: Definition of a Lie algebra. Examples.
Wednesday, October 1: Linear Lie algebras of type BCD.
Friday, October 3: Ideals, homomorphisms, simple Lie algebras. Representations
and modules. Adjoint representation.
Monday, October 6: Derivations and automorphisms. Solvable Lie algebras.
Wednesday, October 8: Nilpotent Lie algebras. Statement of Engel's theorem.
Friday, October 10: Proof of Engel's theorem. Lie's theorem.
Monday, October 13: Proof of Lie's theorem.
Wednesday, October 15: Jordan decomposition.
Friday, October 17: Cartan's criterion.
Monday, October 20: Killing form and applications.
Wednesday, October 22: Complete reducibility, I: reminder about modules.
Friday, October 24: Complete reducibility, II: Casimir operator.
Monday, October 27: Complete reducibility, III: Proof of Weyl theorem.
Please find homework here.
Wednesday, October 29: Representations of sl(2), I.
Friday, October 31: Representations of sl(2), II. Toral subalgebras are
Monday, November 3: Root decomposition: Centralizer of a maximal toral
Wednesday, November 5: Root decomposition: dimension of root spaces and
multiples of roots.
Friday, November 7: Root decomposition: Cartan integers and strings of roots. Roots form a root system.
Monday, November 10: Root systems: 2-dimensional examples.
Wednesday, November 12: Root systems: simple roots.
Friday, November 14: Root systems: Weyl group.
Monday, November 17: Length function on Weyl group. Fundamental domain
for the action of Weyl group. Decomposition into a direct sum of irreducible
Wednesday, November 19: Properties of irreducible root systems:
maximal root; long and short roots.
Friday, November 21: Cartan matrix determines a root system uniquely.
Classification of Dynkin diagrams, I.
Monday, November 24: Classification of Dynkin diagrams, II.
Wednesday, November 26: Construction of root systems.
Friday, November 28: Thanksgiving.
Monday, December 1: Isomorphism theorem: reduction to the case of simple
Wednesday, June 4: End of proof of isomorphism theorem.
Please find final exam here.
Friday, June 6: Root system of sl(n,F).