The universal Hopf-cyclic theory

2008

Commun. Math. Phys.

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 Z * := ΠT l * +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 Tr on M, a natural transformation i : ΠT l → ΠTr , and a morphism w : Tr X → T l 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 Z n = T b ⊗ R . . . b ⊗ R T b ⊗ R X is the cyclic R-module tensor product. A natural transformation i : T b ⊗ R (−) → (−) b ⊗ R T is given by the flip map and a morphism w : X ⊗ R T → 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, 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 ⊗ B X. For a B-Galois extension S ⊆ T , a stable anti Yetter-Drinfel'd B-module T S is constructed, such that the cyclic objects B ⊗ R * +1 ⊗ B T S and T b ⊗ 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. Latter extends results of Burghelea on cyclic homology of groups. Contents Introduction = H⊗n , associated to the coalgebra underlying a Hopf algebra H over a field K. The cocyclic operator was given in terms of a so called modular pair in involution.In the subsequent years the Connes-Moscovici cocyclic module was placed in a broader and broader context. In [KR03] (see also [HKRS2]) to any (co)module algebra T of a Hopf algebra H, and any H-(co)module X, there was associated a para-cyclic module with components T ⊗ * +1 ⊗ X. Dually, for any (co)module coalgebra T of a Hopf algebra H, and any H-(co)module X, there is a para-cocyclic module with components T ⊗ * +1 ⊗ X. The Connes-Moscovici cosimplex Z CM * turns out to be isomorphic to a quotient of the para-cocyclic module associated to the regular module coalgebra T := H and an H-comodule defined on K. , a modular pair in involution was proven to be equivalent to a stable anti Yetter-Drinfel'd module structure on the ground field K. In [HKRS2], the para-cocyclic module T ⊗ * +1 ⊗ X, associated to an H-module coalgebra T and a stable anti Yetter-Drinfel'd H-module X, was shown to project to a cocyclic obj...

“…In contrast, the category of R-bimodules is not even braided in general. Hence Kaygun's elegant theory [Kay06], formulated in a symmetric monoidal category S, is not applicable.…”

Section: = Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads

2008

Commun. Math. Phys.

“…As a most important achievement, also criteria (on the coefficients) for the existence of truly cocyclic subobjects and quotients were found. These constructions were extended to bialgebras (extending Hopf algebras) by Kaygun in [16] and [17]. A new type of coefficients, so-called contramodules, was proposed by Brzeziński in [5].…”

Section: Introductionmentioning

confidence: 99%

“…An important antecedent work of similar aims is Kaygun's paper [17], where a universal construction of para-(co)cyclic objects, including examples from bialgebras, was presented. The construction in this work is built on monoids and comonoids in symmetric monoidal categories.…”

Section: Introductionmentioning

confidence: 99%

A categorical approach to cyclic duality

2012

J. Noncommut. Geom.

Examples of Para-cocyclic Objects Induced by BD-Laws

2009

Algebr Represent Theor

Abstract. In a recent paper [BŞ], we gave a general construction of a para-cocyclic structure on a cosimplex, associated to a so called admissible septuple -consisting of two categories, three functors and two natural transformations, subject to compatibility relations. The main examples of such admissible septuples were induced by algebra homomorphisms. In this note we provide more general examples coming from appropriate ('locally braided') morphisms of monads.