MIT Infinite Dimensional Algebra Seminar (Fall 2008)

I will introduce nilpotent fusion categories, discuss some of their basic properties, and give examples (joint work with D. Nikshych and D. Naidu). In particular, fusion categories of prime power Frobenius-Perron dimension (= p-categories) are nilpotent. I will explain the Sylow decomposition of braided nilpotent fusion categories into a Deligne product of p-categories, and discuss the classification of p-categories (in particular of semisimple (quasi-)Hopf algebras of prime power dimension) (joint with V. Drinfeld, D. Nikshych and V. Ostrik).

A fusion category C is said to be nilpotent if it can be included in a chain of fusion categories C₀=Vec, C₁,...,C_n=C, such that for each i, C_i is graded by a finite group G_i, and the trivial component is C_i-1. If the group G_i can be chosen to be cyclic, then C is said to be cyclically nilpotent. A fusion category is weakly group theoretical if it is Morita equivalent to a nilpotent category, and solvable if it is Morita equivalent to a cyclically nilpotent category (the last definition is motivated by the fact that the category of representations of a finite group is solvable iff the group itself is solvable).

The goal of the talk is to discuss properties of weakly group-theoretical and solvable fusion categories. In particular, we will see that weakly group-theoretical categories are of Frobenius type, i.e. the dimension of every simple object divides the dimension of the category, and that any fusion category of dimension p^a q^b is solvable (a categorical analog of Burnside's theorem). At the end of the talk I will discuss some concrete applications of this theory (such as classification of semisimple Hopf algebras of dimension pq² and pqr for primes p,q,r). This is joint work with D. Nikshych and V. Ostrik.

Let X be a (smooth, complete) algebraic curve, G a reductive group, and κ a symmetric invariant form on g=Lie(G). Let KL^κ _G be the corresponding Kazhdan-Lusztig category, consisting of representions of the Kac-Moody extensions of g((t)) at level κ, that are G[[t]]-integrable. Let x₁,...,x_n be an n-tuple of points of X, and M₁,...,M_n be an n-tuple of objects of KL^κ_G. To this data, by taking coinvariants one can associate a twisted D-module on Bun_G.

(Moreover, by letting the points x₁,..., x_n move along X one obtains a twisted D-module on Xⁿ × Bun_G, which generalizes the KZ connection for X=P¹.)

On the other hand, to the data of ( x₁,..., x_n, M₁,..., M_n) one can associate a twisted D-module on a particular space of colored divisors on X. In the talk we will show how the two D-modules are connected, by the procedure of geometric Eisenstein series and Fourier-Mukai transform.

Let G a reductive group, and κ a symmetric invariant form on g=Lie(G). Let O_κ be the corresponding category O. On the one hand, we have the contragredient duality functor on O_κ. On the other hand, we shall show that semi-infinite cohomology identifies O_κ with the dual of O_κ' , where κ'= - κ+2κ_critical, at the level of derived categories. Combining the two functors, we will recover Arkhipov's equivalence between D(O_κ) and D(O_κ').

Suppose now that κ is negative. In this case, a localization result of Kashiwara-Tanisaki shows that (a direct summand of) O_κ is equivalent to the category of Iwahori-monodromic twisted D-modules on the thin affine flag scheme G((t))/I. On the other hand, when κ is positive, another result of Kashiwara-Tanisaki realizes O_κ via D-modules on the thick affine flag scheme.

In the second half of the talk, we will show how these two results can be obtained from one another via Arkhipov's functor.

To any differentiable manifold M one can associate a sheaf of vertex algebras called the chiral de Rham complex of M. To given geometric structures of M one can try to associate algebraic structures on this sheaf. In this talk we will describe the algebraic structures arising from generalized complex geometries on M.

Algebraic geometry over a field is based on the study of the category of commutative algebras (monoids) in the category of vector spaces. On the other hand, many times one has to enhance or replace vector spaces: for example, studying super or dg algebras, or (commutative) Lie groups (or monoids). Call this "categorical algebraic geometry".

I will define an embedding of associative (noncommutative) algebras into categorical algebraic geometry, Precisely, I embed every associative algebra A into a commutative algebra in the category of wheelspaces, called the "Fock space," F(A), of A. As an application, I will perform Grothendieck's construction of differential operators on F(A). Following Kontsevich's philosophy, these map to differential operators on representation varieties of A.

This also explains various constructions (due to Van den Bergh, Crawley-Boevey, Etingof, and Ginzburg) of Poisson, symplectic, etc., noncommutative geometry: given a Poisson or symplectic structure on F(A), the representation variety obtains the same structure.

If time allows, I can discuss "globalizations" of this where A is replaced by a nonaffine curve, or connections to topics such as necklace Lie algebras, Batalin-Vilkovisky structures, Kontsevich formality, pre-Lie algebras, and Yang-Baxter equations.

The main constructions are joint work with V. Ginzburg.

Nekrasov partition functions of certain gauge theories may be interpreted as traces of certain geometrically defined operators on homology or K-theory of moduli spaces of bundles. Remarkably, these operators may be described explicitly as certain vertex operators, which is what I would like to describe in this talk.

A special focus will be on the K-theory of Hilbert schemes of points on surfaces.

A W-algebra (of finite type) is a certain associative algebra constructed from a semisimple Lie algebra and its nilpotent element. Their study traces back to Kostant (late 70's) who considered the case of a principal nilpotent element. In the recent decade W-algberas were studied by Premet, Ginzburg, Brundan-Kleshchev, myself and others. The main reason why they are interesting is their relation to the representation theory of universal enveloping algebras. Also they are related to affine W-algebras, which are vertex algebras arising in QFT.

In these two talks I am going to explain my own work on W-algebras, arXiv:0707.3108, 0807.1023, and some work in progress. In the first talk I will explain how Fedosov deformation quantization applies to the study of W-algebras and then describe equivalences between certain categories of representations of W-algebras and of universal enveloping algebras. The second talk will be devoted to finite dimensional modules and bimodules over W-algebras.

(continued from Oct 24)

The BBD decomposition theorem is a powerful tool in the study of l-adic cohomology. In the case of the Hitchin fibration, we have a rather precise description of all the pieces in the decomposition. This description turns out to be crucial in the proof of the fundamental lemma.

We will explain the theory of topological quantum computation and how it relates to modular categories. We first explain how theories of anyons in physics are related to modular categories, and give Kitaev's example of an explicit graph Hamiltonian that realizes the Drinfel'd double of a finite group. We then give an introduction to quantum computation, and show how anyons (i.e., modular categories) can be used to realize universal quantum computation.

I will report on the work in progress joint with A. Okounkov. The goal is to prove a conjecture relating representations of rational Cherednik algebra in positive characteristic to quantum cohomology of the Hilbert scheme of points on the plane; the conjecture is partly inspired by T. Bridgeland's ideas on stability coniditions in triangulted categories. The representation theoretic side is in many respects similar to representations of semi-simple Lie algebras over fields of positive characteristic studied (in response to conjectures by Lusztig) earlier in a joint work with Mirkovic and Rumynin.

The q-Whittaker functions, which are eigenfunctions of the q-Toda difference operators, are the limits of the q,t-spherical functions, generalizing the Macdonald polynomials, as t->0 under the Inozemtsev-Etingof procedure. In contrast to the spherical functions, they are not symmetric. However their coefficients have important integrality-positivity properties (which are, generally, missing in the Macdonald theory) and are closely connected with the Demazure characters. In known examples, these functions serve the quantum cohomology of the flag varieties; they are expected to be related to the quantum Langlands program. We will discuss them in the 1D case almost from scratch, beginning with the p-adic Shintani formula.