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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 consider correlation functions for the Wess-Zumino-Witten model on the torus with the insertion of a Cartan element; mathematically this means that we consider the function of the form F = Tr(Φ 1 (z 1). .. Φ n (z n)q −∂ e h) where Φ i are intertwiners between Verma modules and evaluation modules over an affine Lie algebrâ g, ∂ is the grading operator in(More)
This is the second part of the paper (the rst part is published in Journal of AMS 9 1135. In the rst part, we deened for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space Hom(L X i X i ; U) is modular invariant. Also, we have shown that in the case of the MTC arising from the(More)
Dedicated to I.M.Gel'fand on the occasion of his 80th birthday A representation-theoretic approach to special functions was developed in the 40-s and 50-s in the works of I. and their collaborators (see [V],[VK]). The essence of this approach is the fact that most classical special functions can be obtained as suitable specializations of matrix elements or(More)
Classical and quantum family algebras, previously introduced by the author and playing an important role in the theory of semi-smiple Lie algebras and their representations are studied. Basic properties, structure theorems and explicit fomulas are obtained for both types of family algebras in many significant cases. Exact formulas (based on experimental(More)
In this paper we study some properties of tensor categories that arise in 2-dimensional conformal and 3-dimensional topological quantum field theory – so called modular tensor categories. By definition, these categories are braided tensor categories with duality which are semisimple, have finite number of simple objects and satisfy some non-degeneracy(More)