Generalized Coefficients for Hopf Cyclic Cohomology

TL;DR

This paper introduces a new category of coefficients for Hopf cyclic cohomology, expanding the known stable anti Yetter-Drinfeld modules and demonstrating their compatibility with existing cohomology components.

Contribution

It defines a generalized SAYD condition for coefficients, creating a new subcategory and showing its distinction from known categories, enhancing the framework of Hopf cyclic cohomology.

Findings

New coefficients satisfy the generalized SAYD condition

All components of Hopf cyclic cohomology are compatible with the new coefficients

The three categories of coefficients are distinct and properly nested

Abstract

A category of coefficients for Hopf cyclic cohomology is defined. It is shown that this category has two proper subcategories of which the smallest one is the known category of stable anti Yetter-Drinfeld modules. The middle subcategory is comprised of those coefficients which satisfy a generalized SAYD condition depending on both the Hopf algebra and the (co)algebra in question. Some examples are introduced to show that these three categories are different. It is shown that all components of Hopf cyclic cohomology work well with the new coefficients we have defined.