*Everyone is welcome to attend. Refreshments will be served in the Math Lounge before the exam.*

Wednesday, December 5, 2018

11:10 a.m.

BA1170

PhD Candidate: Chia-Cheng Liu

Co-Supervisors: Joel Kamnitzer/Alexander Braverman

Thesis title: Semi-innite Cohomology, Quantum Group Cohomology, and the Kazhdan-Lusztig

Equivalence

The Kazhdan-Lusztig tensor equivalence is a monoidal functor which sends modules over ane

Lie algebras at a negative level to modules over quantum groups at a root of unity. A positive

level Kazhdan-Lusztig functor is dened using Arkhipov-Gaitsgory’s duality between ane Lie

algebras of positive and negative levels. We prove that the semi-innite cohomology functor

for positive level modules factors through the positive level Kazhdan-Lusztig functor and the

quantum group cohomology functor with respect to the positive part of Lusztig’s quantum

group. This is the main result of the thesis.

Monoidal structure of a category can be interpreted as factorization data on the associated

global category. We describe a conjectural reformulation of the Kazhdan-Lusztig tensor equivalence

in factorization terms. In this reformulation, the semi-innite cohomology functor at

positive level is naturally factorizable, and it is conjectured that the factorizable semi-innite

cohomology functor is essentially the positive level Kazhdan-Lusztig tensor functor modulo the

Riemann-Hilbert correspondence. Our main result provides an important technical tool in a

proposed approach to a proof of this conjecture.

A copy of the thesis can be found here: thesis_chiachengliu-1