Noncommutative Algebra Seminar

The talk will focus on finite-dimensional representations of hyper loop algebras over arbitrary fields. Hyperalgebras are certain Hopf algebras related to algebraic groups. When the field is of characteristic zero, a given hyper loop algebra coincide with the universal enveloping algebra of a certain "classical" loop algebra. The main results we will discuss are: the classification of the irreducible representations, construction of the Weyl modules, a study of base change (forms), and tensor products of irreducible modules. Some of these results are more interesting when the field is not algebraically closed and are beautifully related to the study of irreducible representations of polynomial algebras and field theory. If time allows, we shall also discuss the relation with finite-dimensional representations of quantum affine algebras.

In this talk, we will give an introduction to the problem of Fedosov quantization, as well as an introduction to the machinery of algebraic geometry in positive characteristic. In particular, we will discuss Frobenius morphisms, Cartier operations, symplectic manifolds, and the notion of flat descent.

In this part of the talk, we will discuss the main reaults on Fedesov quantization in positive characteristic. In particular, we will describe the cohomological obstructions to quantizing a symplectic manifold in characterisitic p, and we will discuss the technique of formal geometry and how it is related to quantization.

For each partition we construct a natural representation of the Lie algebra of two-variable matrix-valued polynomials. We discuss universality properties of these repreresntations as well as combinatorics of their characters. We present an explicit answer for the case of partition $(n)$, when the dimension is the higher Catalan number, as well as a simple recurrance relation in general case. At last we discuss possible generalizations for a bigger number of variables.

In this talk, we begin with the definitions of the degenerated affine Hecke algebra (dAHA) and the degenerated double affine Hecke algebra (dDAHA). We will describe a Lie-theoretic construction of representations of the dAHA of type $A_{n-1}$ (given by T. Arakawa and T. Suzuki), the dDAHA of type $A_{n-1}$ (given by D. Calaque, B. Enriquez and P. Etingof) and the dAHA and dDAHA of type $BC_{n}$ (given by P. Etingof, R. Freund and X.Ma). Then we will talk about what kinds of dAHA (dDAHA) modules we get from above Lie-theoretic constructions.

The semisimple representations of (double) affine Hecke algebras are defined w.r.t. the Bernstein-Zelevinsky commutative subalgebra. In the affine GL-case, the irreducible ones are described by blocks of skew Young diagrams (I.Ch., M.Nazarov) and have important relations to the classical representation theory and quantum groups. They are all pseudo-unitary, however the corresponding anti-involution of the affine Hecke algebra has little to do with the one that naturally comes from the p-adic theory. It may explain why the geometric interpretation of the irreducible semisimple representations via the general classification due to Kazhdan-Lusztig and Ginzburg remains an open question for arbitrary root systems.

In the daha theory, the pseudo-unitary representations are of much greater importance than in the affine case. The examples of the pseudo-unitary structures are a) the Macdonald inner product, b) the Verlinde pairing for conformal blocks, c) the Harish-Chandra pairing in the theory of spherical functions. The GL-classification is given in terms of the blocks of infinite periodic skew Young diagrams (I.Ch, continued by T.Suzuki, M.Vazirani). Such class of diagrams is a natural challenge for the specialists in combinatorics.

This classification will be explained in detail (and from scratch) in the first lecture. The justification and the problems with extending this construction to arbitrary root systems will be discussed in the second lecture. Combinatorially, the main problem is that the theory of reduced decompositions (words) becomes significantly more involved for the root systems apart from type A and the rank two systems; it happens at level of "relations of relations". Presumably, these combinatorial obstacles are the main reason why we do not have the notion of dominant weight in the theory of affine and double affine Hecke algebras. It will be discussed in the lecture, as well as the related elements of the technique of intertwiners in the non-semisimple variant from my latest paper "Non-semisimple Macdonald polynomials".

In this talk we present a new construction for building representations of the elliptic braid group from the data of a quantum D-module over a ribbon Hopf algebra U. The construction is modelled on - and generalizes - geometric constructions by Calaque, Enriquez and Etingof in the classical setting, and is also related to constructions of Lyubashenko and Majid in the braided setting. The latter construction is a special case where the D-module is the basic representation, while the former can be recovered as a quasi-classical limit of $U=U_t(sl_N)$, as $t\to 1$. In the first half of the talk, we'll recall the graphical calculus of braided tensor categories, and some of the fundamental constructions we'll need. In the second half, we'll construct the representations mentioned in the title.

In this talk, I will give an introduction to localization theory for lie algebras. In particular, I will discuss differential operators on the flag variety , hamiltonian reduction, and the geometry of the adjoint quotient and the springer resolution, with an eye towards the general notion of a "non-commutative resolution of singularities."

In this part of the talk, we will define the finite W-algebras using quantum hamiltonian reduction. We will describe these algebras as deformations of certain affine varieties, and describe the resolution of singularities of these varieties. We will then explain a localization theorem for these algebras, in parallel with the theory from last week.