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Using a variety of methods developed in the literature (in particular , the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion category is always positive, and that the S-matrix of any (not necessarily hermitian) modular category is unitary. We(More)
This paper arose from a minicourse given by the first author at MIT in the Spring of 1999, when the second author extended and improved his lecture notes of this minicourse. It contains a systematic and elementary introduction to a new area of the theory of quantum groups – the theory of the classical and quantum dynamical Yang-Baxter equations. The quantum(More)
In the paper [Dr], V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider " set-theoretical " solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set X × X, where X is a fixed set. In this paper we study such solutions, which in addition satisfy the unitarity and(More)
Introduction By spherical functions one usually means functions on the double coset space K\G/K, where G is a group and K is a subgroup of G. This is equivalent to considering functions on the homogeneous space G/K left invariant with respect to K. More generally, if V is a fixed irreducible representation of K, for example, finite-dimensional, one can look(More)
We introduce a class of multidimensional Schrödinger operators with elliptic potential which generalize the classical Lamé operator to higher dimensions. One natural example is the Calogero–Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper(More)
This paper is a continuation of [EK1-4]. The goal of this paper is to define and study the notion of a quantum vertex operator algebra (VOA) in the setting of the formal deformation theory and give interesting examples of such algebras. Our definition of a quantum VOA is based on the ideas of the paper [FrR]. The first chapter of our paper is devoted to the(More)
For any simple Lie algebra g and any complex number q which is not zero or a nontrivial root of unity, we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group U q (g). This dynamical quantum group is obtained from the fusion and exchange relations between(More)
Given a dynamical twist for a finite-dimensional Hopf algebra, we construct two weak Hopf algebras, using methods of P. Xu and of P. Etingof and A. Varchenko, and we show that they are dual to each other. We generalize the theory of dynamical quantum groups to the case when the quantum parameter q is a root of unity. These objects turn out to be(More)