We use Kazhdan-Lusztig tensoring to, first, describe annihilating ideals of highest weight modules over an affine Lie algebra in terms of the corresponding VOA and, second, to classify tilting functors, an affine analogue of projective functors known
in the case of a simple Lie algebra. For the sake of completeness, the classification of annihilating ideals is borrowed from our previous work, q-alg/9711011; the part on tilting functors is new.
We define and calculate the fusion algebra of WZW model at a rational level by cohomological methods. As a byproduct we obtain a cohomological characterization of admissible representations of $widehat{gtsl}_{2}$.
We explicitly write dowm integral formulas for solutions to Knizhnik-Zamolodchikov equations with coefficients in non-bounded -- neither highest nor lowest weight -- $gtsl_{n+1}$-modules. The formulas are closely related to WZNW model at a rational level.
We introduce and study action of quantum groups on skew polynomial rings and related rings of quotients. This leads to a ``q-deformation of the Gelfand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certai
n non-commutaive rings of quotients coming from complex powers of quantum group generators; this is applied to explicit calculation of singular vectors in Verma modules over $U_{q}(gtsl_{n+1})$. We finally give a definition of a $q-$connection with coefficients in a ring of skew polynomials and study the structure of quantum group modules twisted by a $q-$connection.
We study a family of modules over Kac-Moody algebras realized in multi-valued functions on a flag manifold and find integral representations for intertwining operators acting on these modules. These intertwiners are related to some expressions involv
ing complex powers of Lie algebra generators. When applied to affine Lie algebras, these expressions give integral formulas for correlation functions with values in not necessarily highest weight modules. We write related formulas out in an explicit form in the case of $hat{gtsl_{2}}$. The latter formulas admit q-deformation producing an integral representation of q-correlation functions. We also discuss a relation of complex powers of Lie algebra (quantum group) generators and Casimir operators to ($q-$)special functions.
