Centres, trace functors, and cyclic cohomology

TL;DR

This paper explores the categorical structures of modules and comodules over Hopf algebroids, establishing connections to trace functors and cyclic cohomology through the study of centers and anti Yetter-Drinfel'd modules.

Contribution

It introduces a categorical framework linking biclosed monoidal categories, centers, and anti Yetter-Drinfel'd modules to cyclic cohomology for Hopf algebroids.

Findings

Biclosedness of module and comodule categories over Hopf algebroids is established.

Categorical equivalences to anti Yetter-Drinfel'd modules and contramodules are demonstrated.

Trace functors are shown to enable the construction of cyclic operators for cyclic cohomology.

Abstract

We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.