# On the anti-Yetter-Drinfeld module-contramodule correspondence

## TL;DR

This paper explores the relationship between anti-Yetter-Drinfeld modules and contramodules over Hopf algebras, establishing conditions for their equivalence and revealing new structural insights and examples.

## Contribution

It introduces a functor connecting anti-Yetter-Drinfeld modules to contramodules, identifies conditions for equivalence, and provides novel examples and periodicity observations.

## Key findings

- Conditions under which the functor is an equivalence
- Equivalence of the center of the opposite category of comodules to anti-Yetter Drinfeld modules
- Existence of a symmetric 2-contratrace not arising from anti-Yetter Drinfeld modules

## Abstract

We study a functor from anti-Yetter Drinfeld modules to contramodules in the case of a Hopf algebra $H$. Some byproducts of this investigation are the establishment of sufficient conditions for this functor to be an equivalence, verification that the center of the opposite category of $H$-comodules is equivalent to anti-Yetter Drinfeld modules, and the observation of two types of periodicities of the generalized Yetter-Drinfeld modules introduced previously. Finally, we give an example of a symmetric $2$-contratrace on $H$-comodules that does not arise from an anti-Yetter Drinfeld module.