Quantized universal enveloping algebras and \(q\)-de Rham cocycles (Q1279930)

The author describes a connection between certain cohomology spaces of quantized universal enveloping algebras and a twisted \(q\)-de Rham cohomology of configuration spaces. The connection between the geometry of configuration spaces and the representation theory (Verma modules) is investigated. A family of operators between the Verma modules that satisfy certain difference equations and certain cocycle conditions is considered. These equations are built using a family of \(q\)-difference operators that are the differentials of a \(q\)-de Rham complex of the space of formal algebraic \(q\)-differential forms over the \(n\)-torus. Canonical \(q\)-de Rham cocycles are constructed and as a consequence the author obtains canonical maps between the homology of the local systems and the Ext-spaces between the Verma modules.