Cyclic Cohomology Research Articles

Smooth Crossed Products of Rieffel’s Deformations

Assume A is a Frechet algebra equipped with a smooth isometric action of a vector group V, and consider Rieffel's deformation A_J of A. We construct an explicit isomorphism between the smooth crossed products V\ltimes\A_J and V\ltimes\A. When combined with the Elliott-Natsume-Nest isomorphism, this immediately implies that the periodic cyclic cohomology is invariant under deformation. Specializing to the case of smooth subalgebras of C*-algebras, we also get a simple proof of equivalence of Rieffel's and Kasprzak's approaches to deformation.

Hopf cyclic cohomology and Hodge theory for proper actions

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of invariant differential forms. We prove that every cyclic cohomology class of the Hopf algebroid is represented by a generalized harmonic form. This implies that the space of cyclic cohomology of the Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we discuss properties of the Euler characteristic for a proper cocompact action.

Pseudodifferential extension and Todd class

Let M be a closed manifold. Wodzicki shows that, in the stable range, the cyclic cohomology of the associative algebra of pseudodifferential symbols of order ≤0 is isomorphic to the homology of the cosphere bundle of M. In this article we develop a formalism which allows to calculate that, under this isomorphism, the Radul cocycle corresponds to the Poincaré dual of the Todd class. As an immediate corollary we obtain a purely algebraic proof of the Atiyah–Singer index theorem for elliptic pseudodifferential operators on closed manifolds.

Twisted Cyclic Cohomology and Modular Fredholm Modules

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 Podle\'s spheres and from $SU_q(2)$.

Projective spectrum and cyclic cohomology

For a tuple A=(A1,A2,…,An) of elements in a unital algebra B over C, its projective spectrumP(A) or p(A) is the collection of z∈Cn, or respectively z∈Pn−1 such that the multi-parameter pencil A(z)=z1A1+z2A2+⋯+znAn is not invertible in B. B-valued 1-form A−1(z)dA(z) contains much topological information about Pc(A):=Cn∖P(A). In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of B does a similar job. In fact, a Chen–Weil type map κ from the cyclic cohomology of B to the de Rham cohomology Hd⁎(Pc(A),C) is established. As an example, we prove a closed high-order form of the classical Jacobiʼs formula.

ON THE JLO COCYCLE AND ITS TRANSGRESSION IN ENTIRE CYCLIC COHOMOLOGY

The JLO character formula due to Jaffe–Lesniewski–Osterwalder [Quantum K-theory: the Chern character, Commun. Math. Phys.112 (1988) 75–88] assigns to each Fredholm module a cocycle in entire cyclic cohomology. It descends to define a cohomological Chern character on K-homology. This paper extends the definition of the JLO character formula for Breuer–Fredholm modules, the modules that represent type II noncommutative geometry; and shows that the JLO character formula coincides with the Connes character formula [see M. Benameur and T. Fack, Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math.199 (2006) 29–87] at the level of entire cyclic cohomology.

Cyclic cohomology after the excision theorem of Cuntz and Quillen

AbstractThe excision theorem of Cuntz and Quillen established the existence of a six term exact sequence in the bivariant periodic cyclic cohomology HP*(–,–) associated with an arbitrary algebra extension 0 →S→P→Q→ 0. This remarkable result enabled far reaching developments in the purely algebraic periodic cyclic cohomology. It also provided a new formalism that led to the creation of new versions of this theory for topological and bornological algebras. In this article we outline some of the developments that resulted from this breakthrough.

K-theory, cyclic cohomology and pairings for quantum Heisenberg manifolds

The C*-algebras called quantum Heisenberg manifolds (QHMs) were introduced by Rieffel in 1989 as strict deformation quantizations of Heisenberg manifolds. It was later shown that they are also examples of generalized crossed products. In this article, we compute the pairings of K-theory and cyclic cohomology on the QHM. Combining these calculations with other results proved elsewhere, we also determine the periodic cyclic homology and cohomology of these algebras, and obtain explicit bases of the periodic cyclic cohomology of the QHM. We further isolate bases of periodic cyclic homology, expressed as Chern characters of the K-theory.

Six-term exact sequences for smooth generalized crossed products

We define smooth generalized crossed products and prove six-term exact sequences of Pimsner–Voiculescu type. This sequence may, in particular, be applied to smooth subalgebras of the quantum Heisenberg manifolds in order to compute the generators of their cyclic cohomology. Further, our results include the known results for smooth crossed products. Our proof is based on a combination of arguments from the setting of (Cuntz–)Pimsner algebras and the Toeplitz proof of Bott periodicity.

Quillen's work on the foundations of cyclic cohomology

AbstractWe survey Quillen's contributions to the area of cyclic homology, apart from his very first result in [19].

Involutive $$A_\infty $$ -algebras and dihedral cohomology

We define and study the cohomology theories associated to $$A_\infty $$ -algebras and cyclic $$A_\infty $$ -algebras equipped with an involution, generalising dihedral cohomology to the $$A_\infty $$ context. Such algebras arise, for example, as unoriented versions of topological conformal field theories. It is well known that Hochschild cohomology and cyclic cohomology govern, in a precise sense, the deformation theory of $$A_\infty $$ -algebras and cyclic $$A_\infty $$ -algebras and we give analogous results for the deformation theory in the presence of an involution. We also briefly discuss generalisations of these constructions and results to homotopy algebras over Koszul operads, such as $$L_\infty $$ -algebras or $$C_\infty $$ -algebras equipped with an involution.

Connes' Pairings for a New K-Theory over Weak Hopf Algebras

In this article we first study an equivariant cyclic cohomology for weak H-module agebras over a weak Hopf algebra H with a bijective antipode. Then we define an equivariant K-theory for weak quantum Yetter–Drinfeld algebras over H and establish a generalized Connes' pairing between the equivariant cyclic cohomology and the equivariant K-theory. As an application we consider our theory for groupoids.

On the counting of holomorphic discs in toric Fano manifolds

Abstract Open Gromov-Witten invariants in general are not well-defined. We discuss in detail the enumerative numbers of the Clifford torus T2 in ℂP2. For cyclic A∞-algebras, we show that a certain generalized way of counting may be defined up to Hochschild or cyclic boundary elements. In particular we obtain a well-defined function on Hochschild or cyclic homology of a cyclic A∞-algebra, which is invariant under cyclic A∞ homomorphisms. We discuss the example of the Clifford torus T2 and compute the invariant for a specific cyclic cohomology class.

Equivariant Hopf Galois extensions and Hopf cyclic cohomology

We define the notion of equivariant \times -Hopf Galois extension and apply it as a functor between the categories of stable anti-Yetter–Drinfeld (SAYD) modules of the \times -Hopf algebras involved in the extension. This generalizes the result of Jara–Ştefan and Böhm–Ştefan on associating a SAYD modules to any ordinary Hopf Galois extension.

Higher spectral flow and an entire bivariant JLO cocycle

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

Cyclic Cocycles on Twisted Convolution Algebras

We give a construction of cyclic cocycles on convolution algebras twisted by gerbes over discrete translation groupoids. For proper \'etale groupoids, Tu and Xu provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to a construction of Mathai and Stevenson. When the groupoid is not proper, we cannot construct an invariant connection on the gerbe; therefore to study this algebra, we instead develop simplicial techniques to construct a simplicial curvature 3-form representing the class of the gerbe. Then by using a JLO formula we define a morphism from a simplicial complex twisted by this simplicial curvature 3-form to the mixed bicomplex computing the periodic cyclic cohomology of the twisted convolution algebras. The results in this article were originally published in the author's Ph.D. thesis.

SAYD Modules over Lie-Hopf Algebras

In this paper a general van Est type isomorphism is proved. The isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules over the total Lie algebra and those modules over the associated Hopf algebra. In contrast to the non-general case done in our previous work, here the van Est isomorphism is proved at the first level of a natural spectral sequence, rather than at the level of complexes. It is proved that the Connes-Moscovici Hopf algebras do not admit any finite dimensional SAYD modules except the unique one-dimensional one found by Connes-Moscovici in 1998. This is done by extending our techniques to work with the infinite dimensional Lie algebra of formal vector fields. At the end, the one to one correspondence is applied to construct a highly nontrivial four dimensional SAYD module over the Schwarzian Hopf algebra. We then illustrate the whole theory on this example. Finally explicit representative cocycles of the cohomology classes for this example are calculated.

Slowly increasing cohomology for discrete metric spaces with polynomial growth

The authors introduce a kind of slowly increasing cohomology HS* (X) for a discrete metric space X with polynomial growth, and construct a character map from the slowly increasing cohomology HS* (X) into HC*cont(S(X)), the continuous cyclic cohomology of the smooth subalgebra S(X) of the uniform Roe algebra B*(X). As an application, it is shown that the fundamental cocycle, associated with a uniformly contractible complete Riemannian manifold M with polynomial volume growth and polynomial contractibility radius growth, is slowly increasing.

Simplicial cohomology of band semigroup algebras

We establish the simplicial triviality of the convolution algebra l1 (S), where S is a band semigroup. This generalizes some results of Choi (Glasgow Math. J. 48 (2006), 231–245; Houston J. Math. 36 (2010), 237–260). To do so, we show that the cyclic cohomology of this algebra vanishes in all odd degrees, and is isomorphic in even degrees to the space of continuous traces on l(S). Crucial to our approach is the use of the structure semilattice of S, and the associated grading of S, together with an inductive normalization procedure in cyclic cohomology. The latter technique appears to be new, and its underlying strategy may be applicable to other convolution algebras of interest.

A van Est Isomorphism for Bicrossed Product Hopf Algebras

To any locally finite representation of a given double crossed sum (product) Lie algebra (group), we associate a stable anti Yetter-Drinfeld (SAYD) module over the bicrossed product Hopf algebra which arises from the semidualization procedure. We prove a van Est isomorphism between the relative Lie algebra cohomology of the total Lie algebra and the Hopf cyclic cohomology of the corresponding Hopf algebra with coefficients in the associated SAYD module.