InfiniteDimensional Algebra Seminar
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Math Department at MIT 
Contacts: Pavel Etingof, Victor Kac
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Spring 2010  Fridays 3:00  5:00pm at 2135
February 12
 Scott Carnahan (MIT)
 Recent progress in generalized moonshine
 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 KacMoody Lie algebras with actions of large finite groups. I will describe a method to construct such algebras using the FrenkelSzczesny theory of orbifold conformal blocks.
February 19
 Boris Shoikhet (University of Luxembourg)
 Bialgebras, tetramodules, and nmonoidal categories
 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 GerstenhaberSchack complex.
It is supposed that the GerstenhaberSchack complex for any A is a homotopy 3algebra, 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) 3algebra 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 Bbimodules. 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 nontrivial way such that they form a 2monoidal category structure.
We prove now the following general theorem: let Q be an nmonoidal 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.
February 26
 Alberto De Sole (University of Rome La Sapienza)
 Integrable systems in the theory of Poisson vertex algebras
 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.
March 5
 Matt Szczesny (Boston University)
 Feynman graphs, Hall algebras, and incidence categories
 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 ConnesKreimer 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 ToledanoLaredo.
March 12
 Alexander Tsymbaliuk (MIT)
 DingIohara algebras and their action on the Ktheory of the Hilbert
scheme

In this talk I would like to discuss DingIohara algebras. In a
joint paper with B.Feigin, we construct a natural action of the DingIohara
algebra on the sum of the equivariant Ktheories 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
semiinfinite constructions of modules for the DingIohara 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 Kgroups.
However, under some choice of parameters the action on the Fock space is
illdefined since denominators contain 0. However, in these, socalled
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 DingIohara
algebra to the spherical DAHA.
March 19
 Alexander Braverman (Brown University)
 Cohomology of quasimaps spaces and finite Walgebras: a finite analog of the AGT conjecture
 Recently AldayGaiottoTachikawa (AGT) formulated a conjectural relation between 4dimensional 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 Walgebra 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 Walgebras. Namely, let G be a semisimple 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 Walgebra 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.
April 2, 3:305:00
 Xinwen Zhu (Harvard)
 A categorical approach to Parshin reciprocity laws on algebraic
surfaces
 I will outline an intrinsic proof of Parshin reciprocity laws for twodimensional 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 2dimensional tame symbol as the "commutator" of certain central extension of a group by a Picard groupoid. This is a joint work with D. Osipov.
April 9
 Valerio Taledano Laredo (Northeastern University)
 The trigonometric Casimir connection of a simple Lie algebra
 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).
April 16
 David Hernandez (ENS, Paris)
 Simple tensor products and quantum affine algebras
 Let F be the category of finite dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S_{1} &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.
April 23
 Dennis Gaitsgory (Harvard)
 A strange functional equation for geometric Eisenstein series
 I'll report on some old ideas of Drinfeld's that have recently received some development:
When one discusses automorphic forms for a noncompact 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 Dmodules 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 nontrivial on Eisenstein series; in fact the latter operation can be seen as a new kind of functional equation.
April 30
 Andrei Okounkov (Princeton)
 Quantum cohomology of framed sheaves, Rmatrices, Walgebras, and other capital letters
 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 YangBaxter equation and other curious algebraic structures. These fit very well with some perspectives and conjectures proposed in the physics literature, in particular, by NekrasovShatashvili and AldayGaiottoTachikawa. Joint work with Davesh Maulik.
May 7
 Ivan Losev (MIT)
 Completions of symplectic reflection algebras
 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 KacWeisfeller conjecture for SRA.
The talk is based on arXiv:1001.0239. All necessary information about SRA will be recalled.