On the anti-Yetter–Drinfeld module-contramodule correspondence

Abstract

We study a functor from anti-Yetter–Drinfeld modules to contramodules in the case of a Hopf algebra $H$. This functor is unpacked from the general machinery of [11]. 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 in contrast to [8] where the question of $H$-modules was addressed, and the observation of two types of periodicities of the generalized Yetter–Drinfeld modules introduced in [7]. Finally, we give an example of a symmetric 2-contratrace on $H$-comodules that does not arise from an anti-Yetter–Drinfeld module.