RINGS OF SKEW POLYNOMIALS AND GELFAND-KIRILLOV CONJECTURE FOR QUANTUM GROUPS

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 Gel'fand-Kirillov conjecture which we partially prove. We propose a construction of automorphisms of certain non-commutative 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)(sI(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.