Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra.
We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be defined as a derived functor on the category of representations of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf--Hochschild (co)homology.
For module algebras and module coalgebras over an arbitrary bialgebra, we define two types of bivariant cyclic cohomology groups called bivariant Hopf cyclic cohomology and bivariant equivariant cyclic cohomology. These groups are defined through an extension of Connes' cyclic category $Λ$. We show that, in the case of module coalgebras, bivariant Hopf cyclic cohomology specializes to Hopf cyclic cohomology of Connes and Moscovici and its dual version by fixing either one of the variables as the ground field. We also prove an appropriate version of Morita invariance for both of these theories.
Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M an E-bimodule. We obtain a complex simpler than the canonical one, giving the Hochschild homology of E with coefficients in M. This complex is eqquiped with a natural filtration. We prove that the associated spectral sequence coincides with that obtained by either, the Hochschild-Serre direct method or the Cartan-Leray-Grothendieck method. We also get similar results for the cohomology.
In this paper we show that both variants of the Hopf cyclic homology has excision under some natural homological conditions on the objects and the coefficient module.
We obtain a mixed complex, simpler that the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. Actually, we work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homol…
A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions.
We define a new cyclic module, dual to the Connes-Moscovici cyclic module, for Hopf algebras, and give a characteristric map for the coaction of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.
In this paper we study the cyclic cohomology of certain x-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici x-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of x-Hopf algebras which generalizes the Connes-Moscovici characteristic map to x-Hopf algebras. This enables us to transfer the x-Hopf algebra cyclic cocycles to algebra cyclic cocycles.
In this paper we construct a cylindrical module $A \natural \mathcal{H}$ for an $\mathcal{H}$-comodule algebra $A$, where the antipode of the Hopf algebra $\mathcal{H}$ is bijective. We show that the cyclic module associated to the diagonal of $A \natural \mathcal{H}$ is isomorphic with the cyclic module of the crossed product algebra $A \rtimes \mathcal{H}$. 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 coho…
We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes-Moscovici Hopf algebra $\mathcal{H}_n$. More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of $\mathcal{H}_n$, and we show that the van Est type characteristic homomorphism from the Hopf-cyclic complex of $\mathcal{H}_n$ to the Gelfand-Fuks cohomology of the Lie algebra $W_n$ of formal vector fields on $\mathbb{R}^n$ respects this multiplicative structure. We then illustrate the machinery for $n=1$.
We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.
To any Hopf algebra H we associate two commutative Hopf algebras, which we call the first and second lazy homology Hopf algebras of H. These algebras are related to the lazy cohomology groups based on the so-called lazy cocycles of H by universal coefficient theorems. When H is a group algebra, then its lazy homology can be expressed in terms of the 1- and 2-homology of the group. When H is a cosemisimple Hopf algebra over an algebraically closed field of characteristic zero, then its first lazy homology is the Hopf algebra of the universal abelian grading group of the category of corepresenta…
We review recent progress in the study of cyclic cohomology of Hopf algebras, Hopf algebroids, and invariant cyclic homology starting with the pioneering work of Connes-Moscovici.
We develop an appropriate dihedral extension of the Connes-Moscovici characteristic map for Hopf *-algebras. We then observe that one can use this extension together with the dihedral Chern character to detect non-trivial $L$-theory classes of a *-algebra that carry a Hopf symmetry over a Hopf *-algebra. Using our machinery we detect a previously unknown $L$-class of the standard Podleś sphere.
Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M) extends the usual cyclic homology of the algebra structure of H, and the relative cyclic homology of an H-Galois extension. For a Hopf subalgebra K we compute, under some assumptions, the cyclic homology of an induced modular crossed module. As a direct application of this computation, we describe the relative cyclic homology of strongly graded algebras. In par…
A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of coefficients (AYD modules) over a topological Lie algebra and those over its universal enveloping (Hopf) algebra are isomorphic. For topological Hopf algebras, the category of coefficients is identified with the representation category of a topological algebra called the anti-Drinfeld double. Finally, a topological van Est type isomorphism is detailed, connecting the…
We show that all pairings defined in the literature extending Connes-Moscovici characteristic map in Hopf-cyclic cohomology are isomorphic as natural transformations of derived double functors.
We define a Hopf cyclic (co)homology theory in an arbitrary symmetric strict monoidal category. Thus we unify all different types of Hopf cyclic (co)homologies under one single universal theory. We recover Hopf cyclic (co)homology of module algebras, comodule algebras and module coalgebras along with Hopf-Hochschild (co)homology of module algebras. This universal theory also allows us to define the missing Hopf cyclic theory for comodule coalgebras.
We introduce the concept of {\it para-Hopf algebroid} and define their cyclic cohomology in the spirit of Connes-Moscovici cyclic cohomology for Hopf algebras. Para-Hopf algebroids are closely related to, but different from, Hopf algebroids. Their definition is motivated by attempting to define a cyclic cohomology theory for Hopf algebroids in general. We show that many of Hopf algebraic structures, including the Connes-Moscovici algebra $\mathcal{H}_{FM}$, are para-Hopf algebroids.
We define cup coproducts for Hopf cyclic cohomology of Hopf algebras and for its dual theory. We show that for universal enveloping algebras and group algebras our coproduct recovers the standard coproducts on Lie algebra homology and group homology, respectively.
This is the first one in a series of two papers on the continuation of our study in cup products in Hopf cyclic cohomology. In this note we construct cyclic cocycles of algebras out of Hopf cyclic cocycles of algebras and coalgebras. In the next paper we consider producing Hopf cyclic cocycle from "equivariant" Hopf cyclic cocycles. Our approach in both situations is based on (co)cyclic modules and bi(co)cyclic modules together with Eilenberg-Zilber theorem which is different from the old definition of cup products defined via traces and cotraces on DG algebras and coalgebras.
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.
We introduce the cylindrical module $A \natural \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between $\mathsf{C}_{\bullet}(A^{op} \rtimes \mathcal{H}^{cop})$ the cyclic module of the crossed product algebra $A^{op} \rtimes \mathcal{H}^{cop} $, and $Δ(A \natural \mathcal{H}) $, the cyclic module related to the diagonal of $A \natural \mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it follows that $\mathsf{C}_{\bullet}(A \rtimes \mathcal{H}) \simeq Δ(A^{op} \natural \mathcal{H}…
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.
Let k be a field, A a unitary associative k-algebra and V a k-vector space endowed with a distinguished element 1_V. We obtain a mixed complex, simpler that the canonical one, that gives the Hochschild, cyclic, negative and periodic homology of a crossed product E:=A#_f V, in the sense of Brzezinski. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A that satisfies suitable hypothesis and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. Then, when E is a cleft braide…
The aim of this note is to show that the generalized supertrace, constructed in another paper of the author, inducing an isomorphism between the Hochschild homology of a superalgebra and that of the superalgebra of square supermatrices of a given type over $A$, induces, also, an isomorphism between the cyclic homologies of the two superalgebras.
We consider Hopf crossed products of the the type $A#_σ\mathcal{H}$, where $\mathcal{H}$ is a cocommutative Hopf algebra, $A$ is an $\mathcal{H}$-module algebra and $σ$ is a "numerical" convolution invertible 2-cocycle on $\mathcal{H}$. we give an spectral sequence that converges to the cyclic homology of $A#_σ\mathcal{H}$ and identify the $E^1$ and $E^2$ terms of the spectral sequence.
We calculate the bivariant local cyclic cohomology of the Banach convolution algebra of summable functions on a word-hyperbolic group. Our result implies that the Banach algebraic assembly map in local cyclic homology is an isomorphism for such a group.
We study the curved Koszul duality theory for associative algebras presented by quadratic-linear-constant (QLC) relations. As an application, we investigate the cyclic (co)homology of a QLC algebra and its Koszul dual curved DG algebra, and extend a result due to Feigin and Tsygan. We also study a commutative/Lie analog of this curved Koszul duality theory.
We define and study equivariant analytic and local cyclic homology for smooth actions of totally disconnected groups on bornological algebras. Our approach contains equivariant entire cyclic cohomology in the sense of Klimek, Kondracki and Lesniewski as a special case and provides an equivariant extension of the local cyclic theory developped by Puschnigg. As a main result we construct a multiplicative Chern-Connes character for equivariant KK-theory with values in equivariant local cyclic homology.
We prove that continuous Hochschild and cyclic homology satisfy excision for extensions of nuclear H-unital Frechet algebras and use this to compute them for the algebra of Whitney functions on an arbitrary closed subset of a smooth manifold. Using a similar excision result for periodic cyclic homology, we also compute the periodic cyclic homology of algebras of smooth functions and Whitney functions on closed subsets of smooth manifolds.
We construct cup products of two different kinds for Hopf-cyclic cohomology. When the Hopf algebra reduces to the ground field our first cup product reduces to Connes' cup product in ordinary cyclic cohomology. The second cup product generalizes Connes-Moscovici's characteristic map for actions of Hopf algebras on algebras.
Let V be a complete discrete valuation ring with uniformiser p. We introduce an invariant of Banach V-algebras called local cyclic homology. This invariant is related to analytic cyclic homology for complete, bornologically torsionfree V-algebras. It is shown that local cyclic homology only depends on the reduction mod p of a Banach V-algebra and that it is homotopy invariant, matricially stable, and excisive.
We analyze the relationship between Bott periodicity in topological $K$-theory and the natural periodicity of cyclic homology. This is a basis for understanding the multiplicativity, in odd dimensions, of a bivariant Chern-Connes character from $K$-theory to cyclic theory.
In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that…
We present an approach to cyclic homology of A_{\infty} algebras. Our main technical tool is the concept of X-complex due to Cuntz and Quillen. This, in particular, enables us to compute the periodic cyclic homology of an A_{\infty} algebra in terms of the periodic cyclic homology of its homology.
We define the notion of equivariant Hopf Galois extension and apply it as a functor between category of SAYD modules of the Hopf algebras involving in the extension. This generalizes the result of Jara-Stefan and Böhm-Stefan on associating a SAYD modules to any ordinary Hopf Galois extension.
Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by…
We prove that for an inclusion of unital associative but not necessarily commutative algebras $B\subseteq A$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient $B$-bimodule $A/B$ is flat. We also prove that for an arbitrary r-flat morphism $f:B\to A$ with an H-unital kernel, one can express the Wodzicki excision sequence and the corresponding Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.
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 étale groupoids.
We define the Hochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.
We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant cyclic (co)homology, and the Yoneda interpretation of Ext-groups. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category o…
We study graded connected algebras over a field of characteristic zero and give an explicit formula for the cyclic homology of a tensor algebra. By means of a slightly new definition of David Anick's notion "strongly free" we are able to prove that cyclic homology of an algebra of global dimension two is zero in homological degree greater than one and is zero also in homological degree equal to one in case the relations are monomials. We give also explicit formulas for the cyclic homology of a tensor algebra modulo one symmetric quadratic form.
The multiplicative group of a global field acts on its adele ring by multiplication. We consider the crossed product algebra of the resulting action on the space of Schwartz functions on the adele ring and compute its Hochschild, cyclic and periodic cyclic homology. We also compute the topological K-theory of the C*-algebra crossed product.
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand-Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes v…
Assuming local one-sided units exist, I give an elementary proof of Wodzicki excision for cyclic homology. The proof is also constructive and provides an explicit inverse excision map. As far as I know, the latter is new.
We define and study equivariant periodic cyclic homology for locally compact groups. This can be viewed as a noncommutative generalization of equivariant de Rham cohomology. Although the construction resembles the Cuntz-Quillen approach to ordinary cyclic homology, a completely new feature in the equivariant setting is the fact that the basic ingredient in the theory is not a complex in the usual sense. As a consequence, in the equivariant context only the periodic cyclic theory can be defined in complete generality. Our definition recovers particular cases studied previously by various author…
Certain cocycles constructed by Connes are characters of $p$-summable Fredholm modules. In this article, we establish some consequences of the universal properties which these characters enjoy. Our main technical result is that the entire cyclic cohomology of the p-th Schatten ideal L^p (respectively, homology) is independent of p and isomorphic to the entire cyclic cohomology (respectively, homology) of the ideal of trace class operators L^1.
This paper is an introduction to Hopf cyclic cohomology with an emphasis on its most recent developments. We cover three major areas: the original definition of Hopf cyclic cohomology by Connes and Moscovici as an outgrowth of their study of transverse index theory on foliated manifolds, the introduction of Hopf cyclic cohomology with coefficients by Hajac-Khalkhali-Rangipour-Sommerhauser, and finally the latest episode on unifying the coefficients as well as extending the notion to more general settings beyond Hopf algebras. In particular, the last section discusses the relative Hopf cyclic t…
We discuss the cyclic homology of crossed product algebras from the Cuntz-Quillen point of view. The periodic cyclic homology of a crossed product algebra $A\rtimes G$ is described in terms of the $G$-action on periodic cyclic bicomplexes of crossed products of $A$ by the cyclic subgroups of $G$ .
We define Getzler's Gauss-Manin connection in cyclic homology at the level of chains and outline some relations of this construction to noncommutative calculus.
A classical result of Loday-Quillen and Tsygan states that the Lie algebra homology of the algebra of stable matrices over an associative algebra is isomorphic, as a Hopf algebra, to the exterior algebra of the cyclic homology of the algebra. In this paper we develop the necessary tools needed to extend extend this result to the category of L_{\infty} algebras.
