(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads
Abstract
For a (co)monad T _{ l } on a category {mathcal{M}}, an object X in {mathcal{M}} , and a functor {{\varvec {Pi}}:mathcal{M} to mathcal{C}} , there is a (co)simplex {Z^ast:={\varvec {Pi} {T_l}}^{ast +1} X} in {mathcal{C}} . The aim of this paper is to find criteria for para(co)cyclicity of Z ^{*}. Our construction is built on a distributive law of T _{ l } with a second (co)monad T _{ r } on {mathcal{M}} , a natural transformation {i:{\varvec {Pi} {T_l} to {Pi} {T_r}}} , and a morphism {w:{\varvec {T_r}}X to {\varvec {T_l}}X} in {mathcal{M}} . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads {{\varvec {T_l}}=T ⊗_R ()} and {{\varvec {T_r}}=()⊗_R T} on the category of Rbimodules. The functor Π can be chosen such that {Z^n=Twidehat{⊗}_R\cdots widehat{⊗}_R T widehat{⊗}_RX} is the cyclic Rmodule tensor product. A natural transformation {{i}:T widehat{⊗}_R () to () widehat{⊗}_R T} is given by the flip map and a morphism {w: X ⊗_R T to T⊗_R X} is constructed whenever T is a (co)module algebra or coring of an Rbialgebroid. The notion of a stable antiYetterDrinfel’d module over certain bialgebroids, the socalled × _{ R }Hopf algebras, is introduced. In the particular example when T is a module coring of a × _{ R }Hopf algebra {mathcal{B}} and X is a stable antiYetterDrinfel’d {mathcal{B}} module, the paracyclic object Z _{*} is shown to project to a cyclic structure on {T^{⊗_R ast+1} ⊗_{mathcal{B}} X} . For a {mathcal{B}} Galois extension {S subseteq T} , a stable antiYetterDrinfel’d {mathcal{B}} module T _{ S } is constructed, such that the cyclic objects {mathcal{B}^{⊗_R ast+1} ⊗_{mathcal{B}} T_S} and {T^{widehat{⊗}_S ast+1}} are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable antiYetterDrinfel’d module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 August 2008
 DOI:
 10.1007/s0022000805403
 arXiv:
 arXiv:0705.3190
 Bibcode:
 2008CMaPh.282..239B
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Quantum Algebra;
 16W30;
 16E40
 EPrint:
 LaTeX file, 39 pages, 4 eps figures. v2: significantly extended, a new section about cyclic homology of groupoids added. v3: final version, to appear in Commun. Math. Phys