Talbot 2008: Affine Lie algebras and chiral structures

Mentored by Dennis Gaitsgory.

Notes

The notes are still in very rough form, but are available upon request from Sheel Ganatra. (E-mail ganatra AT math DOT mit). There are, however, notes from the chiral algebras seminar led by Mike Hopkins, which also has a nice list of references.

Talk Schedule

 Monday Tuesday Wednesday Thursday Friday Introduction/Overview Affine grassmannian and factorization D-modules Reps of quantum groups = FS Constructing the functor Intro to quantum groups Factorization algebras and $E_2$ algebras Twistings and twisted D-modules More affine grassmannian Proof of equivalence Rep theory for quantum groups $E_2$ modules and factorization modules Chiral categories discussion session TBD Drinfeld doubles Koszul duality, $E_2$ algebras, and Drinfeld doubles FS category The twisted Whittaker category discussion session

-- Monday --

1. Introduction/Overview. [Dennis Gaitsgory]

2. Overview of quantum groups. [Ian]
Definition of quantum universal enveloping algebras $U_q(g)$; the structure of its category of representations -- braided and ribbon tensor structure and R-matrix; the big and small quantum group, at and away from roots of unity.

3. Representation theory for quantum groups. [Travis]

4. Drinfeld doubles. [Nick]
Show the equivalence $DD(U_q(n_+)) = \operatorname{Rep} U_q(g)$ away from roots of unity, and fully faithful embedding at roots of unity.

-- Tuesday --

5. Affine grassmannian and factorization structures. [Owen]
Definition of the affine grassmannian. How it gives a factorization space.

6. Factorization algebras and E2 algebras. [John or Jacob]

7. E2 modules and factorization modules. [John or Jacob]

8. Koszul duality, E2 algebras and Drinfeld doubles. [Jacob]

-- Wednesday --

9. D-modules. [Zhiwei]

10. Twistings and twisted D-modules. [Reimundo]

11. Chiral categories. [Jacob or John]

12. Factorizable sheaves. [Carl]

-- Thursday --

13. Why FS is the same as chiral modules for the partial Koszul dual of Uq(n+). [Dennis Gaitsgory]

14. More affine grassmannian. [Vivek]

15. Discussion.

16. The twisted Whittaker category. [Xinwen]
Definition of the twisted Whittaker category $\mathrm{Whit}^c$ of the affine Grassmannian.

-- Friday --

17. Construction of the functor $\mathbf{Whit}^c$ → FS. [Scott]
What it says.

18. Proof that the functor is an equivalence. [Richard]
Ditto.

19. TBD.
...