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Math 607 WINTER 2018, 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 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.