Gerstenhaber and Batalin-Vilkovisky structures on modules over operads

In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild theory of associative algebras, for Poisson structures, but above all the calculus for general left Hopf algebroids with respect to general coefficients (in which the classical calculus of vector fields and differential forms is contained).Comment: 33 pages; minor revision, to appear in Int. Math. Res. No

Batalin-Vilkovisky algebra structures on (Co)Tor and Poisson bialgebroids

In this article, we extend our preceding studies on higher algebraic structures of (co)homology theories defined by a left bialgebroid (U,A). For a braided commutative Yetter-Drinfel'd algebra N, explicit expressions for the canonical Gerstenhaber algebra structure on Ext_U(A,N) are given. Similarly, if (U,A) is a left Hopf algebroid where A is an anti Yetter-Drinfel'd module over U, it is shown that the cochain complex computing Cotor_U(A,N) defines a cyclic operad with multiplication and hence the groups Cotor_U(A,N) form a Batalin-Vilkovisky algebra. In the second part of this article, Poisson structures and the Poisson bicomplex for bialgebroids are introduced, which simultaneously generalise, for example, classical Poisson as well as cyclic homology. In case the bialgebroid U is commutative, a Poisson structure on U leads to a Batalin-Vilkovisky algebra structure on Tor_U(A,A). As an illustration, we show how this generalises the classical Koszul bracket on differential forms, and conclude by indicating how classical Lie-Rinehart bialgebras (or, geometrically, Lie bialgebroids) arise from left bialgebroids.Comment: 37 pages; minor revision (v3: added table of contents), to appear in J. Pure Appl. Algebr

Cyclic structures in algebraic (co)homology theories

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd modules

Batalin-Vilkovisky Structures on Ext and Tor

This article studies the algebraic structure of homology theories defined by a left Hopf algebroid U over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid (in particular Lie algebra and Poisson), or group and etale groupoid (co)homology. Explicit formulae for the canonical Gerstenhaber algebra structure on Ext_U(A,A) are given. The main technical result constructs a Lie derivative satisfying a generalised Cartan-Rinehart homotopy formula whose essence is that Tor^U(M,A) becomes for suitable right U-modules M a Batalin-Vilkovisky module over Ext_U(A,A), or in the words of Nest, Tamarkin, Tsygan and others, that Ext_U(A,A) and Tor^U(M,A) form a differential calculus. As an illustration, we show how the well-known operators from differential geometry in the classical Cartan homotopy formula can be obtained. Another application consists in generalising Ginzburg's result that the cohomology ring of a Calabi-Yau algebra is a Batalin-Vilkovisky algebra to twisted Calabi-Yau algebras.Comment: 50 pages; minor modifications, to appear in J. Reine Angew. Mat

The cyclic theory of Hopf algebroids

We give a systematic description of the cyclic cohomology theory of Hopf algebroids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic object. We derive general structure theorems for these theories in the special cases of commutative and cocommutative Hopf algebroids. Finally, we compute the cyclic theory in examples associated to Lie-Rinehart algebras and \'etale groupoids.Comment: 44 pages; to appear in Journal of Noncommutative Geometr

Morita theory for Hopf algebroids, principal bibundles, and weak equivalences

We show that two flat commutative Hopf algebroids are Morita equivalent if and only if they are weakly equivalent and if and only if there exists a principal bibundle connecting them. This gives a positive answer to a conjecture due to Hovey and Strickland. We also prove that principal (left) bundles lead to a bicategory together with a 2-functor from flat Hopf algebroids to trivial principal bundles. This turns out to be the universal solution for 2-functors which send weak equivalences to invertible 1-cells. Our approach can be seen as an algebraic counterpart to Lie groupoid Morita theory.Comment: 50 pages; v2: added a section in which we exhibit the categorical group structure of monoidal symmetric autoequivalences. v3: added a section which explains the abstract groupoid case as a guideline and for motivation. To appear in Doc. Mat

Cyclic homology arising from adjunctions

Given a monad and a comonad, one obtains a distributive law between them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfel'd distributive law between the comonad given by a module coalgebra and the monad given by a comodule algebra. It is this self-dual setting that reproduces the cyclic homology of associative and of Hopf algebras in the monadic framework of Boehm and Stefan. In fact, their approach generates two duplicial objects and morphisms between them which are mutual inverses if and only if the duplicial objects are cyclic. A 2-categorical perspective on the process of twisting coefficients is provided and the role of the two notions of bimonad studied in the literature is clarified.Comment: 24 page

Duality features of left Hopf algebroids

We explore special features of the pair (U^*, U_*) formed by the right and left dual over a (left) bialgebroid U in case the bialgebroid is, in particular, a left Hopf algebroid. It turns out that there exists a bialgebroid morphism S^* from one dual to another that extends the construction of the antipode on the dual of a Hopf algebra, and which is an isomorphism if U is both a left and right Hopf algebroid. This structure is derived from Phung's categorical equivalence between left and right comodules over U without the need of a (Hopf algebroid) antipode, a result which we review and extend. In the applications, we illustrate the difference between this construction and those involving antipodes and also deal with dualising modules and their quantisations.Comment: 25 pages; modified version in which the relation to Phung Ho Hai's comodule equivalence is established; to appear in Algebr. Represent. Theor

Centres, trace functors, and cyclic cohomology

We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with the bimodule category centres of the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel'd contramodules and anti Yetter-Drinfel'd modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.Comment: 41 page