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Math 607 WINTER 2014, List of lectures
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On this page I will post content of all lectures with reference to the
book. All handouts also will be posted here.
Monday, January 6: The Witt algebra. (One dimensional) central extensions
of Lie algebras. Here is homework!
Wednesday, January 8: Theorem: H^2(W)=C. The Virasoro algebra.
Other examples: Heisenberg (or oscillator) and affine Lie algebras.
Friday, January 10: Positive energy representations.
Unitary representations.
Monday, January 13: Fock spaces as representations of Heisenberg algebra.
Wednesday, January 15: Virasoro algebra action on the Fock spaces.
Friday, January 17: Highest weight representations of the Virasoro algebra.
Existence of Hermitian contravariant form.
New homework!
Wednesday, January 22: Unitary representations of Virasoro algebra
from deformed action on the Fock space and from the fermions.
Friday, January 24: Kac determinant. Some non-unitary representations.
Monday, January 27: Divisibility properties of Kac determinant.
Wednesday, January 29: Degree of Kac determinant. Relations in loop algebra
of sl(2).
Friday, January 31: Contragredient Lie algebras and Kac-Moody Lie algebras.
Monday, February 3: Kac-Moody Lie algebras are integrable.
Wednesday, February 5: Affine Lie
algebras as Kac-Moody algebras.
Friday, February 7: Class was cancelled.
Monday, February 10: Highest weight representations. Integrable representations and Weyl group.
Wednesday, February 12: Weyl-Kac character formula and denominator identity.
Friday, February 14: Invariant bilinear form.
Monday, February 17: Casimir operator.
Wednesday, February 19: Proof of Weyl-Kac formula.
New (incomplete) homework!
Friday, February 21: Characters for affine sl(2) and theta functions.
Heisenberg subalgebra and basic representation.
Monday, February 24: Vertex operator construction of basic module
for affine sl(2).
Wednesday, February 26: Unitarity of integrable highest weight
representations of affine sl(2). Updated homework!
Friday, February 28: Affine and Virasoro Lie algebras:
Sugawara construction.
Monday, March 3: Coset construction.
Wednesday, March 5: Singular vectors for Verma modules over
Virasoro algebra.
Friday, March 7: Quantum fields and locality. Vertex algebras.
Monday, March 10: More of vertex algebras
Wednesday, March 12: Conformal blocks.
Friday, March 14: Tensor categories.