Batalin-Vilkovisky Algebras and Cyclic Cohomology of Hopf Algebras

We show that the Connes–Moscovici negative cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree -2. More generally, we show that a cyclic operad with multiplication is a cocyclic module whose simplicial cohomology is a Batalin–Vilkovisky algebra and whose negative cyclic cohomology is a graded Lie algebra of degree -2. This generalizes the fact that the Hochschild cohomology algebra of a symmetric algebra is a Batalin–Vilkovisky algebra.     

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

For an algebra with an action of a Hopf algebra we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for . We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg's theorem relating equivariant K-theory to ordinary K-theory of the C^*-algebra crossed product, and characterize equivariant vector bundles on quantum homogeneous spaces.     

New coefficients for Hopf Cyclic Cohomology

In this note, the categories of coefficients for Hopf cyclic cohomology of comodule algebras and comodule coalgebras are extended. We show that these new categories have two proper different subcategories where the smallest one is the known category of stable anti Yetter–Drinfeld modules. We prove that components of Hopf cyclic cohomology such as cup products work well with these new coefficients.     

A NOTE ON CYCLIC DUALITY AND HOPF ALGEBRAS

ABSTRACT

We show that various cyclic and cocyclic modules attached to Hopf algebras and Hopf modules are related to each other via Connes’ duality isomorphism for the cyclic category.     

Crossed Products over Weak Hopf Algebras Related to Cleft Extensions and Cohomology

The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H.For a right H-comodule algebra,they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions AH → A,where AH is the subalgebra of coinvariants,and the equivalence classes of crossed systems for H over AH.Finally,they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group HφZ(AH)2(H,Z(AH)),where Z(AH)is the center of AH.     

Cohomology Theories of Hopf Bimodules and Cup-Product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.     

Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1

We develop intrinsic tools for computing the periodic Hopf cyclic cohomology of Hopf algebras related to transverse symmetry in codimension 1. Besides the Hopf algebra found by Connes and the first author in their work on the local index formula for transversely hypoelliptic operators on foliations, this family includes its ‘Schwarzian’ quotient, on which the Rankin-Cohen universal deformation formula is based, the extended Connes-Kreimer Hopf algebra related to renormalization of divergences in QFT, as well as a series of cyclic coverings of these Hopf algebras, motivated by the treatment of transverse symmetry for non-orientable foliations.The method for calculating their Hopf cyclic cohomology is based on two computational devices, which work in tandem and complement each other: one is a spectral sequence for bicrossed product Hopf algebras and the other a Cartan-type homotopy formula in Hopf cyclic cohomology.     

Equivariant Cyclic Cohomology of H-Algebras

We define an equivariant K ₀-theory for Yetter–Drinfeld algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.     

Bivariant cyclic theory

In this paper we construct a bivariant version of cyclic cohomology and study its fundamental properties. We prove universal coefficient theorems relating the bivariant theory with cyclic homology and cohomology, we construct products in the bivariant theory, and we analyse the notion of an HC-equivalence.Dedicated to Alexander Grothendieck     

一类Hopf曲面上的全纯线丛的上同调

Let X be a non-primary Hopf Surface with Abelian fundamental groupπ_1 (X)(?) Z(?)Z_m, L a line bundle on X, we give a formula for computing the dimension of cohomology H~q(X,Ω~P(L)) and the explicit results for non-primary exceptional Hopf surface.     

Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_p$ for any integer $p>2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_p$, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.     

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

Abstract

In this paper we construct a cylindrical module A ? ? for an ?-comodule algebra A, where the antipode of the Hopf algebra ? is bijective. We show that the cyclic module associated to the diagonal of A ? ? is isomorphic with the cyclic module of the crossed product algebra A ? ?. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic cohomology of crossed product coalgebras.     

On the Cohomology of Relative Hopf Modules

Let H be a Hopf algebra over a field k, and A an H-comodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.     

Excision in Hopf Cyclic Homology

In this paper, we show that, under natural homological conditions, Hopf cyclic homology theory has excision.     

Excision in Bivariant Periodic Cyclic Cohomology: A Categorical Approach

We extend Cuntz and Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.CONICET Researcher and ICTP Associate. Partially supported by grant UBACyTX066.IMCAIMCA-PUCP. Partially supported by CONCYTEC grant CS017-2002-OAJ.     

A New Cyclic Module for Hopf Algebras

We define a new cyclic module, dual to the Connes–Moscovici cocyclic module, for Hopf algebras, and give a characteristic map for coactions of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.     

Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology

We associate to each infinite primitive Lie pseudogroup a Hopf algebra of ‘transverse symmetries,’ by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a ‘quantum group’ counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on the étale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.     

The Cohomology of Line Bundles on Non-primary Hopf Manifolds

The purpose of the present paper is to give an elementary method for the computation of the cohomology groups Hq（X,Ω^p X（L））, （0 ≤q ≤ n） of an n-dimensional non-primary Hopf manifold X with arbitrary fundamental group. We use the method of Zhou to generalize the results for primary Hopf manifolds and non-primary Hopf manifold with an Abelian fundamental group.