A relationship between coseparable corings and separable non-unital rings is established. In particular it is shown that an A-coring C has an associative A-balanced product. A Morita context is constructed for a coseparable coring with a grouplike element. Biseparable corings are defined, and a conjecture relating them to Frobenius corings is proposed.

Characteristic properties of corings with a grouplike element are analysed. Associated differential graded rings are studied. A correspondence between categories of comodules and flat connections is established. A generalisation of the Cuntz-Quillen theorem relating existence of connections in a module to projectivity of this module is proven.

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 descr...

We evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements, and mention results for products of skew-primitive elements. Examples include quantum groups associated to semisimple and abelian Lie algebras, and degenerate affine Hecke algebras of reductive type with trivial parameter.

Abstract We show that if ${\mathcal C}$ is a fusion $2$ -category in which the endomorphism category of unit object or , then indecomposable objects form finite group.

We prove that the delooping, i. e., the classifying space, of a grouplike monoid is an H-space if and only if its multiplication is a homotopy homomorphism, extending and clarifying a result of Sugawara. Furthermore it is shown that the Moore loop space functor and the construction of the classifying space induce an adjunction of the according homotopy categories. 2000 Mathematics Subject Class...

Let H be a cosemisimple Hopf algebra over an algebraically closed field. It is shown that if H has a simple subcoalgebra of dimension 9 and has no simple subcoalgebras of even dimension, then H contains either a grouplike element of order 2 or 3, or a family of simple subcoalgebras whose dimensions are the squares of each positive odd integer. In particular, if H is odd dimensional, then its di...

Let k be a commutative ring, A a k-algebra, C an A-coring that is projective as a left A-module, ∗C the dual ring of C and Λ a right C-comodule that is finitely generated as a left ∗C-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring EndC(Λ). If C contains a grouplike element, we can replace Λ with A.