(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 R-bimodules. The functor Π can be chosen such that {Z^n=Twidehat{⊗}_R\cdots widehat{⊗}_R T widehat{⊗}_RX} is the cyclic R-module 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 R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called × 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 anti-Yetter-Drinfel’d {mathcal{B}} -module, the para-cyclic 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 anti-Yetter-Drinfel’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 anti-Yetter-Drinfel’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/s00220-008-0540-3
- arXiv:
- arXiv:0705.3190
- Bibcode:
- 2008CMaPh.282..239B
- Keywords:
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- Mathematics - K-Theory and Homology;
- Mathematics - Quantum Algebra;
- 16W30;
- 16E40
- E-Print:
- 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