MIT Infinite Dimensional Algebra Seminar (Spring 2010)

I'll describe some recent results in my program for a proof of Norton's generalized moonshine conjecture. One of the main questions in this program concerns the construction of a special class of generalized Kac-Moody Lie algebras with actions of large finite groups. I will describe a method to construct such algebras using the Frenkel-Szczesny theory of orbifold conformal blocks.

Let A be an associative bialgebra. It has the "deformation complex" D(A); its second cohomology controls the infinitesimal deformations of bialgebra structure on A. It was introduced by Gerstenhaber and Schack and is called the Gerstenhaber-Schack complex.

It is supposed that the Gerstenhaber-Schack complex for any A is a homotopy 3-algebra, which means that there exist a homotopy commutative product on D(A) and a homotopy Lie bracket on D(A)[2], compatible by homotopy Poisson rule. So far, any explicit construction of any of these structures is not known.

I present a construction of (pure) 3-algebra structure on the cohomology of D(A) (I suppose that A is a Hopf algebra). An analogous contruction in the case of Hochschild cohomological complex of an associative algebra B is due to S. Schwede and uses the monoidal structure on the category of B-bimodules. In particular, Schwede gave a conceptual construction of the Gerstenhaber bracket on Hochschild cohomology.

In the case of bialgebras what replaces the category of bimodules is the category of tetramodules, and this category admits two different monoidal structures. These two structures are compatible in a rather non-trivial way such that they form a 2-monoidal category structure.

We prove now the following general theorem: let Q be an n-monoidal abelian category (with some mild assumption), and e be the unit object in Q. Then Ext_Q(e,e) is an (n+1)-algebra. We obtain our theorem when n=2. (The mild assumption is satisfied when the bialgebra A is a Hopf algebra).

The constructions are topological/categorical in their nature, and I am going to explain them.

We will introduce the notion of conformal algebras and Poisson vertex algebras. In particular, we will discuss their role in the theory of Hamiltonian equations and their integrability.

The collection of Feynman graphs of a given quantum field theory can be given the structure of a category somewhat similar to a finitary abelian category. This is a special instance of an "incidence category", which can be constructed from a suitable collection of posets, other examples of which are rooted trees, all finite posets,etc. The Hall algebra of such a category is the enveloping algebra of a Lie algebra which in the case of graphs and rooted trees coincides with the Connes-Kreimer Lie algebras. This perspective makes it natural to apply other notions normally seen in the context of finitary abelian categories, such as stability conditions, correspondences etc. to the setting of combinatorial objects such Feynman graphs. Various parts of this project are joint with Kobi Kremnizer, Dirk Kreimer, and Valerio Toledano-Laredo.

In this talk I would like to discuss Ding-Iohara algebras. In a joint paper with B.Feigin, we construct a natural action of the Ding-Iohara algebra on the sum of the equivariant K-theories of Hilbert schemes. This theoretically gives an action of a Heisenberg algebra in this space. There is also a notion of a Whittaker vector which has a very easy geometric origin. Another very elegant and fruitful approach through spherical DAHA to the same geometric problem is accomplished by O.Schiffmann and E.Vasserot. It also allows to write down the action of a Heisenberg algebra explicitly. Unfortunately, I will not get beyond this on their paper. The second part of the talk is based on the recent paper by B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin, where vector representations, Fock modules and semi-infinite constructions of modules for the Ding-Iohara algebra are constructed. All these representations are parametrized by continuous parameters. In particular, if the parameter is equal to 1, the Fock space coincides with the above mentioned representation in the sum of K-groups. However, under some choice of parameters the action on the Fock space is ill-defined since denominators contain 0. However, in these, so-called resonance cases, there is a subrepresentation, on which the formulas are well defined and which is also of no less interest.

They also checked there is a surjective homomorphism from the Ding-Iohara algebra to the spherical DAHA.

Recently Alday-Gaiotto-Tachikawa (AGT) formulated a conjectural relation between 4-dimensional gauge theory for SU(2) and the so called Liouville theory in 2 dimensions. This conjecture implies (and is more or less equivalent to) the existence of an action of the Virasoro algebra on the cohomology of certain moduli spaces of SU(2)-instantons satisfying certain properties. Further generalizations of this conjecture to other gauge groups suggest that for a simply laced gauge group G there should be an action of the W-algebra of the corresponding affine Lie algebra on the (intersection) cohomology of the appropriate moduli spaces.

In this talk we are going to consider an analog of this construction for finite W-algebras. Namely, let G be a semi-simple group G and let P be a parabolic subgroup G. Let e denote a regular nilpotent element in the corresponding Levi subgroup. To this data one can associate the finite W-algebra W(g,e). We will describe a conjecture saying that there exists an action of W(g,e) on the (intersection) cohomology of certain moduli spaces (related to spaces of maps from a projective line to the partial flag variety G/P), satisfying some natural requirements. We shall explain a proof of this conjecture for g=sl(n) (using the works of Brundan, Kleschev and others, relating W(g,e) to certain Yangians for g=sl(n)). If time permits we shall also explain a relation between these results and a description of the quantum cohomology of G/P.

I will outline an intrinsic proof of Parshin reciprocity laws for two-dimensional tame symbols on an algebraic surface, which generalizes the proof of residue formula on algebraic curves by Tate and the proof of Weil reciprocity laws by Arbarello, De Concini and Kac. The key ingredient is to interpret the 2-dimensional tame symbol as the "commutator" of certain central extension of a group by a Picard groupoid. This is a joint work with D. Osipov.

Let g be a complex, simple Lie algebra, G the corresponding simply connected Lie group and H a maximal torus in G. I will describe a flat connection on H with logarithmic singularities on the root hypertori in H and values in the Yangian of g. Conjecturally, its monodromy is described by the quantum Weyl group operators of the quantum loop algebra U_h(Lg).

Let F be the category of finite dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S₁ &otimes ... &otimes S_N of simple objects of F is simple if and only if for any i < j, S_i&otimes S_j is simple.
We will discuss motivations, applications and the proof of this result.

I'll report on some old ideas of Drinfeld's that have recently received some development:

When one discusses automorphic forms for a non-compact group, one needs to specify growth conditions at infinity; classes of functions corresponding to different growth conditions behave very differently. An analogous phenomenon happens when one considers automorphic sheaves. We'll introduce two natural category of D-modules on Bun_G, one that we'll call D(Bun_G)_! and another that we'll call D(Bun_G)_*. However, following Drinfeld, these categories are related by a functor that we'll denote ~F. The functor ~F acts as identity on the cuspidal part of the category, but performs something non-trivial on Eisenstein series; in fact the latter operation can be seen as a new kind of functional equation.

Moduli spaces of framed torsion free sheaves on C^2 have rich geometry and important applications in supersymmetric gauge theories. The study of quantum cohomology of these spaces lead us to certain solutions of the Yang-Baxter equation and other curious algebraic structures. These fit very well with some perspectives and conjectures proposed in the physics literature, in particular, by Nekrasov--Shatashvili and Alday--Gaiotto--Tachikawa. Joint work with Davesh Maulik.

In this talk we will define and describe completions of symplectic reflection algebras (SRA), generalizing a result of Bezrukavnikov and Etingof. We will use this description to relate arbitrary primitive ideals in the SRA to those of finite codimension. Time permitting, we will also explain another application: an analog of the Kac-Weisfeller conjecture for SRA.

The talk is based on arXiv:1001.0239. All necessary information about SRA will be recalled.