A categorical approach to cyclic cohomology of quasi-Hopf algebras and Hopf algebroids

When Ext is a Batalin–Vilkovisky algebra

“…2.7] it was shown that at least for a Hopf algebra H the category of aYD modules can be identified with the centre of the bimodule category given by the opposite category of H-comodules. We presume that an analogous statement holds for bialgebroids when using the weak centre of a bimodule category [KobSh,Def. 2.1], see also Remark 4.17 at the end of section §4.3.…”

“…Definition 4.14. A trace functor consists of a functor T : C Ñ E between a (unital, associative) monoidal category pC, b, 1q and a category E, together with isomorphisms τ X,Y : T pX b Y q » T pY b Xq for all X, Y P C that are unital (that is, τ 1,Y " id), functorial in X and Y , as well as fulfil the property τ Z,XbY ˝τY,ZbX ˝τX,Y bZ " id for all X, Y, Z P C. We illustrate the above by looking at aYD contramodules, inspired by but slightly generalising the approach given in [KobSh,§7]. Let C op denote the opposite category to a given category C.…”

“…Remark 4.17. In [KobSh,§6], the biclosedness of U -Mod is proven if U is assumed to be a full Hopf algebroid with invertible antipode; Lemma 4.16 is slightly more general inasmuch as there are left Hopf algebroids which are not full Hopf algebroids (in particular, do not carry neither a right bialgebroid structure nor an antipode): an example is given by the universal enveloping algebra of a Lie-Rinehart algebra [KoP]. Still in [KobSh], it is shown that the category of aYD contramodules over a full Hopf algebroid is equivalent to the (weak) center (in the sense of [EtNiOs,§2.8]) of pU -Modq op as a bimodule category over U -Mod.…”

Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras

“…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.…”