Quantized Universal Enveloping Algebras andq-de Rham Cocycles

Abstract

In this work a close connection is established between certain cohomology spaces of quantized universal enveloping algebras and a twisted q-de Ž . Rham Jackson]Aomoto cohomology of configuration spaces. In the Lie algebra case, the idea of such a connection belongs to V. w x w x Ginzburg and V. Schechtman. In 4 and 5 , these authors have constructed canonical morphisms between the de Rham homology of certain local systems over configuration spaces and Ext-spaces between Fock-type modules over Kac]Moody and Virasoro Lie algebras. This construction is, in turn, a generalization of the classical Feigin]Fuchs construction. In their study of representation theory of Virasoro algebras, Feigin and Fuchs have discovered a way of obtaining intertwiners between Fock modules over the Virasoro algebra from the top homology of certain one-dimensional local systems. We investigate this connection between the geometry of configuration spaces and the representation theory in the case of quantum groups. The representations considered here are Verma modules over the quantized enveloping algebras of semisimple Lie algebras. The existence and the w x uniqueness of these modules were established by Lusztig 9 . We consider a family of operators between the Verma modules that satisfy certain difference equations and certain cocycle conditions. These equations are built using a family of q-difference operators that generate a flat connec-