Math 607 WINTER 2018, List of lectures

  • On this page I will post content of all lectures with reference to the book. All handouts also will be posted here.
  • Monday, January 8: Overview of Lie groups and Lie algebras.
  • Wednesday, January 10: Overview of semisimple Lie algebras: Killing form, Jordan decomposition, Cartan subalgebras.
  • Friday, January 12: Overview of semisimple Lie algebras: root space decomposition; Weyl theorem on complete reducibility.
  • Wednesday, January 17: Integrable representations of sl(2). Quantum sl(2).
  • Friday, January 19: Quantum groups and Hopf aalgebras.
  • Monday, January 22: The Witt algebra. (One dimensional) central extensions of Lie algebras.
  • Wednesday, January 24: Theorem: H^2(W)=C. The Virasoro algebra. Other examples: Heisenberg (or oscillator) and affine Lie algebras.
  • Friday, January 26: Positive energy representations. Unitary representations.
  • Monday, January 29: Fock spaces as representations of Heisenberg algebra. Here is homework!
  • Wednesday, January 31: Virasoro algebra action on the Fock spaces.
  • Friday, February 2: Highest weight representations of the Virasoro algebra. Existence of Hermitian contravariant form.
  • Monday, February 5: Unitary representations of Virasoro algebra from deformed action on the Fock space and from the fermions.
  • Wednesday, February 7: Class was cancelled.
  • Friday, February 9: Class was cancelled.
  • Monday, February 12: Kac determinant. Some non-unitary representations.
  • Wednesday, February 14: Divisibility properties of Kac determinant.
  • Friday, February 16: Degree of Kac determinant.
  • Monday, February 19: Relations in loop algebra of sl(2). Contragredient Lie algebras.
  • Wednesday, February 21: Generalized Cartan matrices and Kac-Moody Lie algebras.
  • Friday, February 23: Kac-Moody Lie algebras are integrable.
  • Monday, February 26: Affine Lie algebras as Kac-Moody algebras.
  • Wednesday, February 28: Highest weight representations. Integrable representations and Weyl group.
  • Friday, March 2: Weyl-Kac character formula and denominator identity.
  • Monday, March 5: Casimir operator.
  • Wednesday, March 7: Characters of Verma modules.
  • Friday, March 9: End of proof of Weyl-Kac formula.
  • Monday, March 12: Vertex operator construction of basic modules over affine sl(2).
  • Wednesday, March 14: Sugawara construction.
  • Friday, March 16: Coset construction.
  • Extra lectures:
  • Monday, March 19: Vertex algebras.
  • Wednesday, March 21: Conformal blocks.
  • THE END.