An A-infinity algebra is given by a codifferential on the tensor coalgebra of a (graded) vector space. An associative algebra is a special case of an A-infinity algebra, determined by a quadratic codifferential. The notions of Hochschild and cyclic cohomology generalize from associative to A-infinity algebras, and classify the infinitesimal deformations of the algebra, and those deformations preserving an invariant inner product, respectively. Similarly, an L-infinity algebra is given by a codifferential on the exterior coalgebra of a vector space, with Lie algebras being special cases given by quadratic codifferentials. There are natural definitions of cohomology and cyclic cohomology, generalizing the usual Lie algebra cohomology and cyclic cohomology, which classify deformations of the algebra and those which preserve an invariant inner product. This article explores the definitions of these infinity algebras, their cohomology and cyclic cohomology, and the relation to their infinitesimal deformations.
In this paper, we introduce the cohomology theory of $\mathcal{O}$-operators on Hom-associative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an $\mathcal{O}$-operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an $\mathcal{O}$-operator is introduced to characterize trivial infinitesimal deformations.
The title refers to the nilcommutative or $NC$-schemes introduced by M. Kapranov in math.AG/9802041. The latter are noncommutative nilpotent thickenings of commutative schemes. We consider also the parallel theory of nil-Poisson or $NP$-schemes, which are nilpotent thickenings of commutative schemes in the category of Poisson schemes. We study several variants of de Rham cohomology for $NC$- and $NP$-schemes. The variants include nilcommutative and nil-Poisson versions of the de Rham complex as well as of the cohomology of the infinitesimal site introduced by Grothendieck. It turns out that ea…
We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example we show that hypercohomology with coefficients in $K…
The purpose of this paper is to study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We introduce a suitable cohomology and discuss Infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide some examples.
In this note we prove that the constant and equivariant cyclic cohomology of algebras coincide. This shows that constant cyclic cohomology is rich and computable.
We show that the Connes-Moscovici 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 cohomology is a Batalin-Vilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree -2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a Batalin-Vilkovisky algebra.
We present results relating the Hopf cyclic cohomology of the Hopf algebra of moving frames with that of the DG Hopf algebra of moving coframes, analogous to the van Est isomorphism between Lie algebra cohomology and continuous group 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.
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 investigate the equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids and derive their Morita invariance. For this, we use the tools developed by McCarthy for $k$-linear categories and subsequently by Kaygun and Khalkhali for a certain class of categories, called as $H$-categories, for a Hopf algebra $H$.
We define cyclic cohomology of corings over not necessarily commutative algebras. We observe that the key fact which allows us to define this complex is that enveloping algebra of an algebra is a para Hopf algebroid. This observation enables us to define Hopf cyclic cohomology of corings on which para Hopf algebroids act.
A generalization of Connes-Thom isomorphism is given for stable, homotopy invariant, and split exact functors on separable $C^*$-algebras. As examples of these functors, we concentrate on asymptotic and local cyclic cohomology and the result is applied to improve some formulas in asymptotic and local cyclic cohomology of $C^*$-algebras. As an other application, it is shown that these cyclic theories are rigid after Rieffel's deformation quantizations.
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.
We give a construction of cyclic cocycles representing the equivariant characteristic classes of equivariant bundles. Our formulas generalize Connes' Godbillon-Vey cyclic cocycle. An essential tool of our construction is Connes-Moscovici's theory of cyclic cohomology of Hopf algebras.
In this paper, we compute the entire cyclic cohomology of noncommutative 3-spheres. First of all, we verify the Mayer-Vietoris exact sequence of entire cyclic cohomology in the framework of Fréchet $^*$-algebras. Applying it to their noncommutative Heegaard decomposition, we deduce that their entire cyclic cohomology is isomorphic to the d'Rham homology of the ordinary 3-sphere with the complex coefficients.
Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel of a smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finite dimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) is finitely generated and that H*(DG_r,M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θ_r:H*(G_r,k) \otimes S(g) \to H*(DG_r,k) which offers an approach to support varieties for DG_r-modules. For many examples of interest, θ_r is injective and induces an isomorphism of associated reduced schemes. Additionally, for M an …
Our aim in this paper is to compute the entire cyclic cohomology of noncommutative 2-tori. First of all, we clarify their algebraic structure of noncommutative 2-tori as a $F^*$-algebra, according to the idea of Elliott-Evans. Actually, they are the inductive limit of subhomogeneous $F^*$-algebras. Using such a result, we compute their entire cyclic cohomology, which is isomorphic to their periodic one as a complex vector space.
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 present a geometric approach, in the spirit of the Chern-Weil theory, for constructing cocycles representing the classes of the Hopf cyclic cohomology of the Hopf algebra H(n) relative to GL(n, R). This provides an explicit description of the universal Hopf cyclic Chern classes, which complements our earlier geometric realization of the Hopf cyclic characteristic classes of foliations.
Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podleś spheres and from $SU_q(2)$.
Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows to expand the range of applications of cyclic cohomology. It is the goal of the present paper to illustrate these recent developments, with special emphasis on the application to transverse index theory, and point towards future directions. In particular, we highlight the remarkable accord between our framework for cyclic cohomology of Hopf algebras on one hand and both the algebraic as we…
By properly specializing the parameters irreducible modules of maximal dimension for the De Concini-Kac version of the Drinfeld-Jimbo quantum algebra in type $A$ may be transformed into modules over Lusztig's infinitesimal quantum algeba. Thus obtained modules have a simple socle and a simple head, and share the same dimension as the infinitesimal Verma modules. Despite these common features we find that they are never isomorphic to infinitesimal Verma modules unless they are irreducible. The same carry over to the modular setup for the special linear groups in positive characteristic.
We use stable anti Yetter-Drinfeld contramodules to improve the cup products in Hopf cyclic cohomology. The improvement fixes the lack of functoriality of the cup products previously defined and show that the cup products are sensitive to the coefficients.
A new quantization of groupoids under the name of \times-Hopf coalgebras is introduced. We develop a Hopf cyclic theory with coefficients in stable-anti-Yetter-Drinfeld modules for \times-Hopf coalgebras. We use \times-Hopf coalgebras to study coextensions of coalgebras. Finally, equivariant \times-Hopf coalgebra Galois coextensions are defined and applied as functors between categories of stable anti-Yetter-Drinfeld modules over \times-Hopf coalgebras involved in the coextension.
In this note we discuss the possibility of constructing the cosimplicial complex for the multiplier Hopf algebras and extending the cyclicity operator to obtain the Hopf-cyclic cohomology for them. We show that the definition of modular pairs in involution for multiplier Hopf algebras and provide the definition of Hopf-cyclic cohomology for algebras of functions over discrete groups.
This paper is concerned with the theory of cup-products in Hopf-type cyclic cohomology of algebras and coalgebras. Here we give detailed proofs of the statements, announced in our previous paper. We show that the cyclic cohomology of a coalgebra can be obtained from a construction involving noncommutative Weil algebra. Then we use a generalization of Quillen and Crainic's construction to define the cup-product. We discuss the relation of the introduced cup-product and $S$-operations on cyclic cohomology. After this we describe the relation of this type of product and bivariant cyclic cohomolog…
This paper introduces a chain model for the Deligne-Mumford operad formed by homotopically trivializing the circle in a chain model for the framed little disks. We then show that under degeneration of the Hochschild to cyclic cohomology spectral sequence, a known action of the framed little disks on Hochschild cochains lifts to an action of this new chain model. We thus establish homotopy hyper-commutative algebra structures on both Hochschild and cyclic cochain complexes, and we interpret the gravity brackets on cyclic cohomology as obstructions to degeneration of this spectral sequence. Our …
In this paper we calculate both the periodic and non-periodic Hopf-cyclic cohomology of Drinfeld-Jimbo quantum enveloping algebra $U_q(\mathfrak{g})$ for an arbitrary semi-simple Lie algebra $\mathfrak{g}$ with coefficients in a modular pair in involution. We show that its Hochschild cohomology is concentrated in a single degree determined by the rank of the Lie algebra $\mathfrak{g}$.
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…
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…
In this paper we propose still another approach to the Hopf-type cyclic homology of Hopf algebras, introduced by A. Connes and H. Moscovici. Our construction is based on the notion of "universal differential calculus" on an algebra. Few remarks, concerning the possible generalizations and applications of this approach are made. (This is only a preliminary version of the paper. Only outlines of some ideas are given.)
We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^δ$. We give an explicit way to construct Hopf cyclic cohomology classes of our Hopf algebra $\cH$ and then realize these classes in terms of explicit representative cocycles in the cyclic cohomology of the convolution algebra $\cA$.
In this paper, we further explore the conceptual approach to cyclic cohomology with coefficients. In particular we give a derived version of the definition with better invariance properties. We show that the new definition agrees with the old under certain conditions and we prove, for the new definition, both its Morita invariance and its $2$-Morita invariance, under suitable interpretations of these terms. More generally, we prove a version of Shapiro's lemma for cyclic cohomology with coefficients.
A category of coefficients for Hopf cyclic cohomology is defined. It is shown that this category has two proper subcategories of which the smallest one is the known category of stable anti Yetter-Drinfeld modules. The middle subcategory is comprised of those coefficients which satisfy a generalized SAYD condition depending on both the Hopf algebra and the (co)algebra in question. Some examples are introduced to show that these three categories are different. It is shown that all components of Hopf cyclic cohomology work well with the new coefficients we have defined.
We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.
In our previous work, Hopf cyclic cohomology in braided monoidal categories, we extended the formalism of Hopf cyclic cohomology due to Connes and Moscovici and the more general case of Hopf cyclic cohomology with coefficients to the context of abelian braided monoidal categories. In this paper we go one step further in reducing the restriction of the ambient category being symmetric. We let the ambient category to be non-symmetric but assume only the restriction on the braid map for the Hopf algebra object (in that category) which is the main player in the theory. In the case of Hopf cyclic c…
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 …
We show by a direct computation that, for any Hopf algebra with a modulus-like character, the formulas first introduced in [CM] in the context of characteristic classes for actions of Hopf algebras, do define a cyclic module. This provides a natural generalization of Lie algebra cohomology to the general framework of Noncommutative Geometry, which covers the case of the Hopf algebra associated to n-dimensional transverse geometry [CM] as well as the function algebras of the classical quantum groups.
We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair, consisting of a group-like element and a character, in involution. This provides the key construct allowing to extend cyclic cohomology to Hopf algebras in the non-unimodular case and further to develop a theory of characteristic classes for actions of Hopf algebras compatible not only with traces but also with the modular theory of weights. It applies to ribbon and to coribbon algebras, as well as to quantum groups and their duals.
We construct a duality isomorphism in equivariant periodic cyclic homology analogous to Baaj-Skandalis duality in equivariant Kasparov theory. As a consequence we obtain general versions of the Green-Julg theorem and the dual Green-Julg theorem in periodic cyclic theory. Throughout we work within the framework of bornological quantum groups, thus in particular incorporating at the same time actions of arbitrary classical Lie groups as well as actions of compact or discrete quantum groups. An important ingredient in the construction of our duality isomorphism is the notion of a modular pair for…
We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Ger…
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.
Global constructions of quantization deformation and obstructions are discussed for an arbitrary complex analytic space in terms of adapted (analytic) Hochschild cohomology. For K3-surfaces an explicit global construction of a Poisson bracket is given. It is shown that the analytic Hochschild (co)homology on a complex space has structure of coherent analytic sheaf in each degree.
We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss-Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.
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…
REVISED VERSION: We have re-organized the paper, and included some new results. Most important, we prove that the (truncated) Weil complexes compute the cyclic cohomology of the Hopf algebra (see the new Theorem 7.3). We also include a short discussion on the uni-modulare case, and the computation for $H= U_q(sl_2)$. THE OLD ABSTRACT: We give a construction of Connes-Moscovici's cyclic cohomology for any Hopf algebra equipped with a twisted antipode. Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the …
We prove a cyclic cohomological analogue of Haefliger's van Est-type theorem for the groupoid of germs of diffeomorphisms of a manifold. The differentiable version of cyclic cohomology is associated to the algebra of transverse differential operators on that groupoid, which is shown to carry an intrinsic Hopf algebraic structure. We establish a canonical isomorphism between the periodic Hopf cyclic cohomology of this extended Hopf algebra and the Gelfand-Fuchs cohomology of the Lie algebra of formal vector fields. We then show that this isomorphism can be explicitly implemented at the cochain …
We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of `modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class w…
Let $H$ be a Hopf algebra and let $\mathcal D_H$ be a Hopf-module category. We describe the cocycles and coboundaries for the Hopf cyclic cohomology of $\mathcal D_H$, which correspond respectively to categorified cycles and vanishing cycles over $\mathcal D_H$. An important role in our work is played by semicategories, which are categories that may not contain identity maps. In particular, a cycle over $\mathcal D_H$ consists of a differential graded $H$-module semicategory equipped with a trace on endomorphism groups satisfying some conditions. Using a pairing on cycles, we obtain a pairing …
We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the literature, and while a definition exists for the latter, we feel that our approach demystifies the seemingly arbitrary formulas present there. This paper emphasizes the importance of working with a biclosed monoidal category in order to obtain natural coefficients for a cyclic theory that are analogous to the stable anti-Yetter-Drinfeld contramodules for Hopf…
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 …
The purpose of this article is to embed the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in wh…
We examine Hopf cyclic cohomology in the same context as the analysis of the geometry of loop spaces $LX$ in derived algebraic geometry and the resulting close relationship between $S^1$-equivariant quasi-coherent sheaves on $LX$ and $D_X$-modules. Furthermore, the Hopf setting serves as a toy case for the categorification of Chern character theory. More precisely, this examination naturally leads to a definition of mixed anti-Yetter-Drinfeld contramodules which reduces to that of the usual mixed complexes for the trivial Hopf algebra and generalizes the notion of stable anti-Yetter-Drinfeld c…
In this dissertation we study the coefficients spaces (SAYD modules) of Hopf-cyclic cohomology theory over a certain family of bicrossed product Hopf algebras, and we compute the Hopf-cyclic cohomology of such Hopf algebras with coefficients. We associate a Hopf algebra, what we call a Lie-Hopf algebra, to any matched pair of Lie groups, Lie algebras and affine algebraic groups via the semi-dualization procedure of Majid. We then identify the SAYD modules over Lie-Hopf algebras with the representations and corepresentations of the total Lie group, Lie algebra or the affine algebraic group of t…
