Hopf cyclic cohomology in braided monoidal categories

TL;DR

This paper generalizes Hopf cyclic cohomology to braided monoidal categories, introducing stable anti-Yetter-Drinfeld modules and associating cocyclic objects, thereby extending the theory beyond symmetric braidings.

Contribution

It develops a new framework for Hopf cyclic cohomology in braided categories, including the definition of stable anti-Yetter-Drinfeld modules and cocyclic objects in this setting.

Findings

Defined stable anti-Yetter-Drinfeld modules in braided categories

Constructed para-cocyclic and cocyclic objects for braided Hopf algebras

Extended Hopf cyclic cohomology formalism to non-symmetric braidings

Abstract

We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.