Talbot 2008: Affine Lie algebras and chiral structures
Mentored by Dennis Gaitsgory.
March 30 to April 5, 2008.
Plymouth, Massachusetts.
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 E_{2} algebras. [John or Jacob]
7. E_{2} modules and factorization modules. [John or Jacob]
8. Koszul duality, E_{2} 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 U_{q}(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.
...