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

Abstract

We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules for Hopf algebras.

Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

“…An interesting observation is the appearance of two distinct ways of writing down the formulas; these are identical for Hopf algebras but very different here. This also occurs in the contramodule case [11].…”

“…In this followup paper to [11], we make the modifications necessary to deal with the definitions of anti-Yetter-Drinfeld modules for quasi-Hopf algebras, which generalize Hopf algebras by relaxing the coassociativity condition to coassociativity up to a specified isomorphism. This isomorphism complicated matters sufficiently that a generalization of the formulaic approach used for Hopf algebras, via the usual method of educated guessing, is not possible.…”

“…The main theme of [11] was exploiting the fact that the category of modules is biclosed, i.e., it possesses internal Homs. This allows for a definition of a natural bimodule category over it, the center of which is what we are looking for.…”

“…As mentioned above, historically anti-Yetter-Drinfeld modules appeared before their contramodule versions, but in this paper we rely on the conceptual definition of generalized anti-Yetter-Drinfeld contramodules as in [11] to obtain the module version. In particular the question of stability is explicitly reduced to the contramodule case, though it is immediate that it is equivalent to the stability in [9], though not completely analogous to what is possible to do in the Hopf algebra case; see Remark 2.7.…”

“…In Section 2 we review and augment some generalities from [11] that we use subsequently. Section 3 is a review of the basics of quasi-Hopf algebras and recalls the definition of internal Homs, from [11], for them. Finally in Section 4 we unravel the conceptual definitions of Section 2 into formulas.…”