Math 681 FALL 2012, List of lectures
On this page I will post content of all lectures. All handouts also will be posted here.
Monday, September 24: Definition of a Lie algebra. Examples.
Wednesday, September 26: Linear Lie algebras of type BCD. Derivations and automorphisms.
Friday, September 28: Ideals, homomorphisms, simple Lie algebras.
Lie algebra sl_2(F) is simple. Representations and modules.
Monday, October 1: Solvable Lie algebras and nilpotent Lie algebras.
Wednesday, October 3: Engel's theorem.
Friday, October 5: Lie's theorem.
Monday, October 8: Jordan decomposition.
Wednesday, October 10: Cartan's criterion.
Friday, October 12: Killing form.
Monday, October 15: Derivations of semisimple Lie algebras.
Abstract Jordan decomposition.
Wednesday, October 17: Finite dimensional irreducible sl(2)-modules.
Friday, October 19: Complete reducibility for finite dimensional
Monday, October 22: More on complete reducibility for sl(2).
Casimir operator for finite dimensional representation of semisimple Lie
Wednesday, October 24: Complete reducibility
for finite dimensional representations of semisimple Lie algebras (Weyl's
Friday, October 26: Abstract Jordan decomposition and representations.
Toral subalgebras of semisimple Lie algebras.
Please find take-home midterm here.
Monday, October 29: Root decomposition: Centralizer of a maximal toral
Wednesday, October 31: Root decomposition: dimension of root spaces and
multiples of roots.
Friday, November 2: Root decomposition: Cartan integers and strings of roots. Roots form a root system.
Monday, November 5: Root systems: 2-dimensional examples. Simple roots.
Wednesday, November 7: Root systems: existence of a base and Weyl chambers.
Friday, November 9: Root systems: Weyl group is generated by simple
reflections. Length function.
Monday, November 12: Decomposition
of a root system into the union of irreducible ones.
Long and short roots. Cartan matrix determines a root
Wednesday, November 14: Classification of Dynkin diagrams, I.
Friday, November 16: Classification of Dynkin diagrams, II.
Monday, November 19: Simplicity of a Lie algebra and irreducibility of
its root system. Isomorphism Theorem.
Wednesday, November 21: Proof of isomorphism theorem.
Friday, November 23: Thanksgiving.
Monday, November 26: Classical Lie algebras are semisimple.
Wednesday, November 28: Maximal toral subalgebras are conjugated.
Friday, November 30: Maximal toral subalgebras = Cartan subalgebras.
Please find Final Exam here.