Invariance properties of cyclic cohomology with coefficients

A Categorical Approach to Cyclic Cohomology of Quasi-Hopf Algebras and Hopf Algebroids

“…, where Hom l (V, M ) and Hom r (V, M ) are left and right internal homomorphisms respectively. As in [22], we can introduce the contragradient M-bimodule category M op . Specifically, for M ∈ M op and V ∈ M, the actions are given by:…”

“…The paper is organized as follows. In Section 2 we discuss some generalities as in [12,22] concerning a conceptual definition of contramodule coefficients for a suitable monoidal category. A new phenomenon that arises in this paper is the difficulty of proving that a certain natural map is an isomorphism.…”

Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

“…As in [16], we can introduce the contragradient M-bimodule category M op . Specifically, for M ∈ M op and V ∈ M, the actions are given by…”

On the anti-Yetter–Drinfeld module-contramodule correspondence

“…The introduction of anti-Yetter-Drinfeld contramodule coefficients to the Hopf-cyclic cohomology theory in [2] that followed the definition of anti-Yetter-Drinfeld module coefficients in [3] can in retrospect be conceptually understood as being completely natural since they are seen to be exactly corresponding to the representable symmetric 2-contratraces, see [7] and [5]. The latter form a well behaved class of Hopf-cyclic coefficients explored in [4] and [7], that lead directly to Hopf-cyclic type cohomology theories.…”

“…

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 [7]. 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 [5] where the question of H-modules was addressed, and the observation of two types of periodicities of the generalized Yetter-Drinfeld modules introduced in [4].

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