Tensor categories, dynamical R-matrices and double Hecke algebras

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The current project proposes research on three subjects: 1) tensor categories and Hopf algebras; 2) dynamical R-matrices and special functions, and 3) Cherednik algebras, Calogero-Moser systems. In the first area, the PI plans to construct new examples of finite tensor categories, both semisimple and nonsemisimple, and classify module categories over given tensor categories. In the second area, the PI plans to study elliptic hypergeometric functions of Felder and Varchenko using representation theory of quantum affine algebras and dynamical R-matrices, and to prove the De Concini conjecture about quantum Weyl groups and the Felder-Varchenko conjectures on the properties of elliptic hypergeometric functions. In the third area, the PI plans to study finite dimensional representations of symplectic reflection algebras, their large $N$ limits (double Yangians), hyperbolic analogs (Gan-Ginzburg algebras), and ``topological analogs'' (Hecke algebras of various braid groups). He plans to prove algebraic PBW theorems for deformations of group algebras which are similar to double affine Hecke algebras. He also plans to study the connections of this subject with representations of quivers. The current project is at the crossroads of several mathematical areas -- representation theory, mathematical physics, theory of integrable systems, special functions. The theory of Cherednik algebras and Calogero-Moser spaces originates from a remarkable family of orthogonal polynomials of several variables called Macdonald polynomials. These polynomials are a very broad generalization of the classical Legendre polynomials arising in the study of the hydrogen atom, and they have an incredibly rich structure. Similarly to this classical case, Macdonald polynomials are eigenstates of a quantum Hamiltonian (Macdonald operator), which defines an integrable quantum-mechanical system, in the sense that one can exactly solve it. Representation theory allows us to use symmetry to understand the structure of Macdonald polynomials. Similarly to how the rotational symmetry of the hydrogen atom is instrumental in the study of its quantum-mechanical properties, the structure of Macdonald polynomials is uncovered using the quantum group and Cherednik algebra symmetry. The theory of tensor categories, which is another subject of this proposal, is also instrumental in understanding quantum systems, since the representation-theoretical structure expressing the symmetry of many such systems takes the shape of a tensor category.