A Note on Cyclic Duality and Hopf Algebras

Abstract

We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes' duality isomorphism for the cyclic category.

Bialgebra Cyclic Homology with Coefficients

“…Our theory is an extension of the theory developed in [2] by lifting two restrictions: (i) our theory uses bialgebras as opposed to Hopf algebras (ii) the coefficient module/comodules are just stable as opposed to stable antiYetter-Drinfeld. In the second part of this paper, we recover the main result of [4]. Namely, these two cyclic theories are dual in the sense of (co)cyclic objects, whenever the input pair (H, X) has the property that H is a Hopf algebra and X is a stable anti-Yetter-Drinfeld module.…”

A categorical approach to cyclic duality

“…In its original form, it is an isomorphism between the category of cyclic objects and the category of cocyclic objects in a given category. It was extended in [18] to an isomorphism between certain full subcategories of the categories of para-cyclic, and of para-cocyclic objects. The objects of these full subcategories are those para-(co)cyclic objects whose para-(co)cyclic morphisms are isomorphisms at all degrees.…”

“…Connes's cyclic duality functor (in the extended form in [18]) and also its dual version (from a subcategory of the category of para-cyclic objects to a subcategory of the category of para-cocyclic objects) both will be denoted by ' (−). Denote by A × the full subcategory of A in Definition 2.1, whose objects (T l , T r , Φ, ⊓, i, ⊔, w) obey the property that Φ, i and w are natural isomorphisms.…”

“…These (full) subcategories have those objects whose para-(co)cyclic morphisms are isomorphisms at each grade, cf. Khalkhali and Rangipour's work [18].…”

The universal Hopf-cyclic theory

“…Khalkhali and Rangipour [16] showed that the cyclic dual of the canonical cocyclic object of the (MC)-type symmetry evaluated at H is functorially isomorphic to the canonical cyclic object of the (CA)-type symmetry evaluated at the same Hopf algebra H . Then the author and Khalkhali [14] successfully unified the cyclic theories for the (MA) and (MC)-type symmetries and their cyclic duals under the banner of bivariant Hopf-cyclic cohomology.…”