New Coefficients For Hopf Cyclic Cohomology

TL;DR

This paper extends the categories of coefficients used in Hopf cyclic cohomology for comodule algebras and coalgebras, introducing new subcategories that preserve key cohomological operations.

Contribution

It introduces new coefficient categories for Hopf cyclic cohomology, expanding beyond stable anti Yetter-Drinfeld modules, and demonstrates their compatibility with cohomological structures.

Findings

New coefficient categories are compatible with cup products.

The smallest new category is the known stable anti Yetter-Drinfeld modules.

Extended categories have two proper subcategories.

Abstract

In this note the categories of coefficients for Hopf cyclic cohomology of comodule algebras and comodule coalgebras are extended. We show that these new categories have two proper different subcategories where the smallest one is the known category of stable anti Yetter-Drinfeld modules. We prove that components of Hopf cyclic cohomology such as cup products work well with these new coefficients.