(Co)cyclic (Co)homology of bialgebroids: An approach via (Co)monads

Gabriella Böhm, Dragoş Ştefan

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21 Citations (Scopus)

Abstract

For a (co)monad T l on a category M, an object X in M, and a functor Π : M → C, there is a (co)simplex Zz.ast;:= Π Tl *+1 X in 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 M, a natural transformation i:Π Tl} to Π} Tr, and a morphism {w:{Tr}}X to Tl}X in 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 T l}=T ⊗R (-) and Tr}}=(-) ⊗R T} on the category of R-bimodules. The functor Π can be chosen such that {Zn=T ⊗ RR T ⊗RX} is the cyclic R-module tensor product. A natural transformation {i}:T ⊗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 B and X is a stable anti-Yetter-Drinfel'd B-module, the para-cyclic object Z * is shown to project to a cyclic structure on T R, *+1 ⊗BX. For a B-Galois extension ⊂ T}, a stable anti-Yetter-Drinfel'd B-module T S is constructed, such that the cyclic objects B R., *+1 ⊗B TS and {T S *+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.

Original languageEnglish
Pages (from-to)239-286
Number of pages48
JournalCommunications in Mathematical Physics
Volume282
Issue number1
DOIs
Publication statusPublished - Aug 2008

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ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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