MIT Infinite Dimensional Algebra Seminar (Fall 2010)

Universal weight function of a quantum affine algebra is a family of functions with values in its Borel subalgebra satisfying certain co-algebraic properties. The universal weight function happens to be the main tool to establish the algebraic Bethe ansatz which solves the eigenvalue problem for the set of commuting quantum integrals of motion. The general construction of the universal weight function involves different realizations of the quantum affine algebra, namely: the "standard" Chevalley realization, the Drinfeld realization, and the so-called RLL-realization based on Faddeev-Reshetikhin-Takhtajan-Semenov-Tian-Shansky approach.

I am going to tell the whole story for the most simple U_q(\widehat sl_2)-case. Depending on time allowance I will also state the results of joint work with S. Khoroshkin for the U_q(A_2^(2))-case as for the first known case of quantum twisted affine algebra with the explicit formula for its universal weight function.

There are two main "sets of tools" in the field of Deformation Quantization. One uses physical ideas, and one uses methods from homological algebra. We show how to link these approaches at various points. Concretely, we will see how to extend M. Kontsevich's formality morphism to a homotopy Gerstenhaber morphism, using Feynman rules. It can be shown to be equivalent to a formality morphism constructed by D. Tamarkin. On the latter construction acts the Grothendieck-Teichmüller group. Its Lie algebra can be shown to be equal to the zeroth cohomology of M. Kontsevich's graph complex.

The goal of the talk is to give a survey of results and methods shared by random matrix theory and representation theory of inductive limits of finite or compact groups.

Lozenge tilings of planar domains provide a simple, yet sophisticated model of random surfaces. Asymptotic behavior of such models has been extensively studied in recent years. We will start from recent results about q-distributions on tilings of a hexagon or, equivalently, on boxed plane partitions. (This part is based on the joint work with A.Borodin and E.Rains). In the second part of the talk we will explain how representation theory of the infinite-dimensional unitary group is related to random lozenge tilings with a certain Gibbs property. We will discuss applications of this correspondence and results on the classification of Gibbs measures on tilings of the half-plane.

The dimer model on a bipartite graph G on a torus gives rise to a quantum integrable system of a special type, which we call a cluster integrable system.

An open part of the phase space parametrises 1-dimensional bundles with connection on the graph G. The classical Hamiltoniansare (essentially) the coefficients of the partition function of the dimer model. The phase space has a cluster Poisson variety structure, and thus a canonical q-deformation and quantization. The quantum Hamiltonians act naturally in a Hilbert space, defined using the quantum dilogarithm.

This is a joint work with R. Kenyon.

Recently, Kac and Wakimoto established specialized character formulas for irreducible highest weight sl(m,1)^ modules, and later works of the author and Bringmann-Ono show that these characters may be realized as parts of certain non-holomorphic modular functions. We will describe this, and show how the "modularity" of these characters can be exploited to obtain detailed asymptotics.

For a semisimple Lie algebra g, the quantum loop algebra U_h(Lg) and the Yangian Y_h(g) are certain deformations of the loop algebra g[z,z^(-1)] and the current algebra g[u] respectively. These two algebras are very closely related, and are believed to have the same representation theory. To mention a few known results relating the quantum loop algebras and the Yangians, we have the following: (a) the finite-dimensional irreducible representations of both these algebras are parametrized by certain rank(g)-tuple of polynomials, called Drinfeld polynomials (b) both these algebras have geometric realizations on the same Steinberg-type variety, and (c) the Yangian can be obtained by a certain degeneration of the quantum loop algebra. Despite these results, no natural relationship between the two algebras is known. In this talk, I will explain how to construct a functor between the finite-dimensional representation categories of these two algebras. This talk is based on a joint work with Valerio Toledano Laredo.

In 1972 I. G. Macdonald generalized a classical formula of H. Weyl, obtaining, in particular, a formula for certain powers of eta-function which include some classical identities of Jacobi. In 1994 V. Kac and M. Wakimoto stated a super-analogue of Macdonald identities and proved it for some special cases. In contrast to Lie algebra case, the trivial representation is critical for some affine Lie superalgebras and this leads to non-trivial extra-factors in the formulas.

Specializations of these identities give, in particular, Jacobi and Legendre formulas for representing an integer as a sum of squares or a sum of triangular numbers, respectively. In this talk I will outline a proof of these identities.

The q-characters of an untwisted quantum affine algebra are combinatorial objects that describe the structure of irreducible finite-dimensional representations of this algebra. The evaluation map allows to use the q-characters of quantum affine algebra of type A_n as a combinatorial tool for description of representations of underlying Lie algebra sl_n or gl_n. In the talk we will illustrate this application on corresponding examples and compute q-characters of certain class of modules of quantum affine algebra of gl_\infty.

One of the most exciting things to come out of attempts to classify subfactors is the appearance of "exotic subfactors" -- subfactors which don't come from previously understood algebraic structures such as groups, quantum groups or conformal field theories. A subfactor, by the way, is a structure arrising in von Neumann algebras which is closely related to monoidal tensor categories. I'll talk about the invariants of subfactors and how one can begin to classify them, as well as how to construct exotic subfactors using diagramatic techniques (planar algebras).