Flat Connections and Comodules

This is the text of a talk given at the Isaac Newton Institute in Cambridge on 4th August, 2006. This talk is motivated by a recent paper [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential geometry, Lett. Math. Phys. 76 (2006), 77–91] in which Hopf modules appearing as coefficients in Hopf-cyclic cohomology are interpreted as modules with flat connections. We start by describing how all the algebraic structure involved in a universal differential calculus fits in a natural way into the notion of a coring (or a coalgebra in the category of bimodules). We recall the theorem of Roiter [A.V. Roiter, Matrix problems and representations of BOCS’s. [in:] Lecture Notes in Mathematics, vol. 831, Springer-Verlag, Berlin and New York, 1980, pp. 288-324] in which a bijective correspondence is established between semi-free differential graded algebras and corings with a grouplike element. A brief introduction to the theory of comodules is given and the theorem establishing a bijective correspondence between comodules of a coring with a grouplike element and flat connections (with respect to the associated differential graded algebra) is given [T Brzeziński, Corings with a grouplike element, Banach Center Publ., 61 (2003), 21-35]. Finally we specialise to corings which are built on a tensor product of algebra and a coalgebra. Such corings are in one-to-one correspondence with so-called entwining structures, and their comodules are entwined modules. The latter include all known examples of Hopf-type modules such as Hopf modules, relative Hopf modules, Long dimodules, Doi-Koppinen and alternative Doi-Koppinen modules. In particular they include Yetter-Drinfeld and anti-Yetter-Drinfeld modules and their generalisations, hence all the modules of interest to Hopf-cyclic cohomology. In this way the interpretation of the latter as modules with flat connections is obtained as a corollary of a more general theory.