MIT Infinite Dimensional Algebra Seminar (Fall 2009)

The unversal Dunkl elements are a family of pairswise commuting elements in a certain noncommutaive quadratic algebra. Classical rational, trigonometric and elliptic (level 0) Dunkl operators happened to be the images of the universal ones in the corresponding representations of the quadratic algebra in question. The former quadratic algebra admits also the so-called Bruhat representation, which gives rise to connections with theclassical and quantum Schubert and Grothendieck Calculi.

In this talk I will consider, mainly, the case of the Dunkl operators at a critical level and their connections with classical and quantum (and elliptic ?) cohomology and K-theory of the type A flag varieties.

We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM-structures naturally leads to affine Dynkin diagrams of A, D, E-type.

Kontsevich's proof of his formality theorem, implying the quantizability of any Poisson manifold, is based on the Feynman graph expansion of a certain topological sigma model on the disk. It is natural to consider this sigma model with D-branes boundary conditions. The data are a manifold and a collection of submanifolds (branes). For one submanifold there is a relative formality theorem that is related to quantization of hamiltonian reductions. The case of two submanifolds is related to the quantization of modules and bimodules and will be the subject of this talk. I will first consider the simplest case of the two subspaces 0, V of a vector space V, following an idea of B. Shoikhet. The result is that the Kontsevich deformation quantization preserves the Koszul duality between the symmetric algebra of V* and the exterior algebra of V. More general cases relevant to representation theory will be discussed. The talk is based on joint work with Damien Calaque, Carlo Rossi and Andrea Ferrario.

In the late 1990s, Borcherds introduced an interpretation of vertex algebras as commutative rings with singularities. The multiplication laws for these objects behave like the multiplication on quantum fields, which is commutative and associative when fields have spacelike separated support, but is not necessarily defined otherwise. The theory of singular commutative rings has an extension to braided tensor categories, and the braided setting is a natural home for orbifold models and generalized lattices. I'll describe the formalism, together with a rudimentary obstruction theory involving double loop spaces that is useful for constructing new examples.

Weyl denominator identities for finite-dimensional Lie superalgebras and their affinizations were formulated by V. Kac and M. Wakimoto and were proven by them for defect one case. I will review a proof for other cases.

The category O of Bernstein-Gelfand-Gelfand is of major importance for the study of universal enveloping algebras of semisimple Lie algebras (over,say, the field of complex numbers). There is an analog of this category for W-algebras introduced by Brundan, Goodwin, and Kleshchev. In my talk I will explain an equivalence between this category and a certain category of generalized Whittaker modules for the universal enveloping algebra. In many cases, this equivalence allows to compute multiplicities in the category O for the W-algebra. Also this category equivalence has some surprising applications to the problem of describing completely prime primitive ideals in the universal enveloping algebra. The talk is based on the arXiv preprints 0812.1584, 0906.0157. All necessary facts on W-algebras will be recalled.

A Poisson trace on a Poisson variety is a functional which is invariant under the flow of Hamiltonian vector fields. Such traces are defined only globally, not locally. In this talk, I will consider the local approach, by studying the D-module which is the quotient of all differential operators by the Hamiltonian vector fields. Using this, one can prove that the space of Poisson traces is finite-dimensional when the variety has finitely many symplectic leaves, and that quantizations in this case have finitely many irreducible representations.

If time permits, I will explain how to use related ideas to prove a conjecture of Alev, that the space of Poisson traces is equal to the space of Hochschild traces of the quantization in the case when the variety is a symmetric power of a surface in three-dimensional space cut out by a quasihomogeneous polynomial, which has an isolated singularity at the origin. This includes the Kleinian singularities, which are the quotients of two-dimensional space by finite subgroups of SL(2).

This is joint work with P. Etingof.

It is long expected that to study the double loop groups, the third cohomology classes in their group cohomolgy should play important roles. In representation theory, such classes appear when groups acts on projectively categories rather than vector spaces. In this talk, I will construct certain abelian category, on which all the double loop groups act such that their certain nontrivial third cohomology classes are realized. This category is the analogue of the fermonic Fock module in the ordinary loop case. At the end of the talk, I will also give a proposal to realize this category geometrically. This is a joint work with Edward Frenkel.

In 1990, Reshetikhin and Turaev introduced knot invariants attached to each highest weight for a simple Lie group using the natural structures of the monoidal category of representations using the natural structure maps of this category corresponding to ribbon tangles. We show how to lift this entire construction up a layer of categorical structure: to each sequence of highest weights, we assign a category whose Grothendieck group is the corresponding tensor product of quantum group representations, equipped with natural functors corresponding to the action of the quantum group, braiding and other structure maps. Better yet, we achieve this entirely by drawing pictures, not using any perverse sheaves or quivers.

Extending this to ribbon tangles we obtain a knot invariant valued in bigraded vector spaces with graded Euler characteristic given by the knot invariants of Reshetikhin and Turaev, which generalizes Khovanov homology, a well-known categorification of the Jones polynomial.

Our aim will be to present a proof of Braverman's conjecture concerning Laumon quasiflag spaces. We consider the generating function of the integrals of the equivariant Chern polynomial of the tangent bundles to the Laumon spaces. We will prove Braverman's conjecture, which states that this generating function coincides with the eigenfunction of the Calogero-Sutherland hamiltonian, up to a simple factor. This conjecture was inspired by the work of Nekrasov in the affine \hat{sl}_n setting, where a similar conjecture is still open. Joint work with Andrei Okounkov.

In 1939 Rademacher derived a conditionally convergent series expression for Klein's j-invariant, and used this expression---the first Rademacher sum---to verify its modular invariance. We will explain how to attach Rademacher sums to an arbitrary group commensurable with the modular group, and we will discuss how the automorphy of the resulting functions reflects the geometry of the group in question.

In the case of a group of genus zero the relationship is particularly striking. On the other hand, of all the properties of the groups of isometries of the hyperbolic plane that arise in moonshine, the genus zero property is perhaps the most elusive. We will show how Rademacher sums can be used to formulate a convenient characterization of the discrete groups of monstrous moonshine.

A physical interpretation of the Rademacher sums comes to light when we consider black holes in the context of three dimensional quantum gravity. This observation amounts to a new connection between moonshine and physics, and promises applications in both directions.

In the first part of my talk I am going to speak about so called large Lie algebras and how they can be used to produce extensive arrays of examples of finitely generated infinite-dimensional Lie algebras satisfying the Engel condition. In the second part I will talk about the distortion of degrees of elements that may arise when one finitely generated Lie algebra is embedded in another. One of the results is that any finitely generated Lie algebra can be embedded without distortion in a finitely generated simple Lie algebra. All results are joint with Alexander Olshanskii.

I will describe a joint work with Finkelberg and Ostrik on understanding character sheaves as objects of the center of the categorified Hecke algebra. (Thus there will be some intersection with the talks I have given in Vogan's seminar, though I will try to concentrate on different aspects.) Time permitting, I will describe an attempt to partly generalize the construction to loop groups, leading to (still mostly conjectural) construction of stable distributions on reductive p-adic groups (joint with Varshavsky).