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Math 681 FALL 2010, 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, September 27: Definition of a Lie algebra. Examples.
Wednesday, September 29: Linear Lie algebras of type BCD. Derivations and automorphisms.
Friday, October 1: Ideals, homomorphisms, simple Lie algebras.
Lie algebra sl_2(F) is simple.
Monday, October 4: Representations and modules. Adjoint representation.
Solvable Lie algebras.
Wednesday, October 6: Nilpotent Lie algebras. Statement of Engel's theorem.
Friday, October 8: Proof of Engel's theorem. Lie's theorem.
Monday, October 11: Proof of Lie's theorem.
Wednesday, October 13: Jordan decomposition.
Friday, October 15: Cartan's criterion.
Monday, October 18: Killing form and applications.
Wednesday, October 20: Derivations of semisimple Lie algebras.
Abstract Jordan decomposition.
Friday, October 22: Finite dimensional irreducible sl(2)-modules.
Monday, October 25: Complete reducibility for finite dimensional
sl(2)-modules.
Wednesday, October 27: Casimir operator and complete reducibility
for finite dimensional representations of semisimple Lie algebras (Weyl's
theorem).
Friday, October 29: Toral subalgebras of semisimple Lie algebras.
Monday, November 1: Root decomposition: Centralizer of a maximal toral
subalgebra. Please find homework here.
Wednesday, November 3: Root decomposition: dimension of root spaces and
multiples of roots.
Friday, November 5: Root decomposition: Cartan integers and strings of roots. Roots form a root system.
Monday, November 8: Root systems: 2-dimensional examples. Simple roots.
Wednesday, November 10: Root systems: Weyl group.
Friday, November 12: Root systems: More on Weyl group. Decomposition
of a root system into the union of irreducible ones.
Monday, November 15: Long and short roots. Cartan matrix determines a root
system.
Wednesday, November 17: Classification of Dynkin diagrams, I.
Friday, November 19: Classification of Dynkin diagrams, II.
Monday, November 22: Simplicity of a Lie algebra and irreducibility of
its root system. Reductive Lie algebras and criterion for
reductivity/semisimplicity.
Wednesday, November 24: Lie algebra sl(V) is simple. Isomorphism Theorem.
Friday, November 26: Thanksgiving.
Monday, November 29: Proof of isomorphism theorem.
Wednesday, December 1: Maximal toral subalgebras are conjugated.
Please find Final Exam here.
Friday, December 3: Maximal toral subalgebras = Cartan subalegbras.
THE END