# Kazhdan-Lusztig Equivalences

### at the University of Sydney

Seminar meeting: Wednesdays 9-11am in SMRI Seminar room

Exercise sessions: Wednesdays 1:30-3pm in SMRI Seminar room

First meeting: August 14

## Seminar description:

In a series of four papers, Kazhdan and Lusztig constructed braided monoidal equivalences between (smooth) representations of affine Lie algebras at level 𝜅 , where 𝜅 lies in C-Q⩾0, and representations of quantum groups at parameter q=exp(i𝜋/𝜅 ). These equivalences generalized earlier work of Drinfeld, and are intimately connected to Knizhnik-Zamolodchikov (KZ) equations, the Virasoro algebra, and conformal field theory. Kazhdan-Lusztig equivalences were later extended by Finkelberg to positive integral levels of the affine Lie algebra, and their extension to other levels are the subject of active research.

This seminar will focus on learning about the main actors which appear in this story: affine Lie algebras, quantum groups, KZ equations, and fusion products. After everything is defined and established, we will try to make sense of the equivalences.

Schedule

The seminar will meet on Wednesdays from 9-11am in the SMRI seminar room.

There will be an exercise session on Wednesdays 1:30-3pm, again in the SMRI Seminar room.

The following is a tentative schedule. The rough plan: first half of the course we will try to understand Drinfeld's equivalence which describes the associators and braiding on quantum group representations in terms of asymptotics of the KZ equation. In the second half we will attempt to explain the fusion product on representations of affine Lie algebras and how Kazhdan and Lusztig constructed an equivalence.

The focus will always be on examples, especially sl(2).

August 14: Representations of quantum groups (Elijah Bodish, University of Oregon)

August 21: Affine Lie algebras (Finn Klein)

August 28: Knizhnik-Zamolodchikov (KZ) equations I (Kevin Coulembier)

September 4: Knizhnik-Zamolodchikov (KZ) equations II: solutions (Pieter Roffelsen)

September 11: Knizhnik-Zamolodchikov (KZ) equations III (Kevin Coulembier)

September 18: Vertex operators and free field realization (Alex Sherman)

September 25: Local systems, monodromy, and flat connections (Chris Hone)

October 2: No seminar, mid-semester break.

October 9: Asymptotics of KZ equations and the Drinfeld-Kohno theorem (Alex Sherman)

October 16: Coinvariants and fusion products of affine Lie algebra representations, Part I (Sam Jeralds)

October 23: Coinvariants and fusion products of affine Lie algebra representations, Part II (Sam Jeralds)

October 30: Fusion products and Finkelberg's theorem for integer levels (Tao Qin)

November 6: The Kazhdan-Lusztig equivalence: how it works (general picture), Joe Newton

November 13: No meeting---seminar is finished!

Exercises:

Resources:

Affine Lie algebras and quantum groups (survey paper by Kazhdan and Lusztig)

Tensor structures arising from affine Lie algebras I-IV (original papers of Kazhdan and Lusztig):

Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, by Etingof, Frenkel, and Kirillov (book): if you are having trouble finding it, please email the organizers for assistance

Contact the organisers Sam Jeralds or Alexander Sherman with any questions or to join the mailing list.