Algebraic connections,universal bimodules and entire cyclic cohomology

A new method of constructing homotopies suitable for entire cyclic cohomology is presented. As a result, the periodic and entire cyclic cohomology of Banach algebras of finite cohomological dimension are shown to be isomorphic. The same method can be used to calculate the algebraic entire cyclic cohomology of (non-commutative) tori.     

On the Algebraic Index for Riemannian Étale Groupoids

In this paper, we construct an explicit quasi-isomorphism to study the cyclic cohomology of a deformation quantization over a Riemannian étale groupoid. Such a quasi-isomorphism allows us to propose a general algebraic index problem for Riemannian étale groupoids. We discuss solutions to that index problem when the groupoid is proper or defined by a constant Dirac structure on a 3-dimensional torus.     

Para-Hopf Algebroids and their Cyclic Cohomology

We introduce the concept of 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 , are para-Hopf algebroids     

Cyclic Cohomology and Hopf Algebra Symmetry

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 the expansion of the range of applications of cyclic cohomology. It is the goal of this Letter 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 the one hand and both the algebraic as well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square.     

Closed star products and cyclic cohomology

We define the notion of a closed star product. A (generalized) star product (deformation of the associative product of functions on a symplectic manifold W) is closed iff integration over W is a trace on the deformed algebra. We show that for these products the cyclic cohomology replaces the Hochschild cohomology in usual star products. We then define the character of a closed star product as the cohomology class (in the cyclic bicomplex) of a well-defined cocycle, and show that, in the case of pseudodifferential operators (standard ordering on the cotangent bundle to a compact Riemannian manifold), the character is defined and given by the Todd class, while in general it fails to satisfy the integrality condition.     

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 étale groupoids, Tu and Xu (Adv Math 207(2):455–483, 2006) provide a map between the periodic cyclic cohomology of a gerbe-twisted convolution algebra and twisted cohomology groups which is similar to the construction of Mathai and Stevenson (Adv Math 200(2):303–335, 2006). 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.     

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.     

A Pairing Between Super Lie-Rinehart and Periodic Cyclic Homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K⁰-group of a smooth noncommutative torus. It also produces new homological invariants of proper maps contracting each orbit contained in a closed invariant subset in a manifold acted on smoothly by a connected Lie group. Finally we compare it with the characteristic map for the Hopf-cyclic cohomology. The author was partially supported by the KBN grant 1P03A 036 26.     

Spectral flow invariants and twisted cyclic theory for the Haar state on

In [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], we presented a K-theoretic approach to finding invariants of algebras with no non-trivial traces. This paper presents a new example that is more typical of the generic situation. This is the case of an algebra that admits only non-faithful traces, namely SU_q(2) and also KMS states. Our main results are index theorems (which calculate spectral flow), one using ordinary cyclic cohomology and the other using twisted cyclic cohomology, where the twisting comes from the generator of the modular group of the Haar state. In contrast to the Cuntz algebras studied in [A.L. Carey, J. Phillips, A. Rennie, Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras. arXiv:0801.4605], the computations are considerably more complex and interesting, because there are non-trivial ‘eta’ contributions to this index.     

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.     

Superspace formulation of the Chern character of a theta-summable Fredholm module

We apply the concepts of superanalysis to present an intrinsically supersymmetric formulation of the Chern character in entire cyclic cohomology. We show that the cocycle condition is closely related to the invariance under supertranslations. Using the formalism of superfields, we find a path integral representation of the index of the generalized Dirac operator.Supported in part by the Department of Energy under grant DE-FG02-88ER25065     

On the Chern character of θ summable Fredholm modules

We show that the entire cyclic cohomology class given by the Jaffe-Lesniewski-Osterwalder formula is the same as the class we had constructed earlier as the Chern character of -summable Fredholm modules.     

Cyclic Cohomology and Hopf Algebras

We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair in involution, consisting of a group-like element and a character. This provides the key construction for allowing the extension of cyclic cohomology to Hopf algebras in the nonunimodular case and, further, to developing a theory of characteristic classes for actions of Hopf algebras compatible not only with traces but also with the modular theory of weights. This applies to both ribbon and coribbon algebras as well as to quantum groups and their duals.     

Boundary Maps for <Emphasis Type="Italic">C</Emphasis>*-Crossed Products with with an Application to the Quantum Hall Effect

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t. Connes pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators.     

Quasi homomorphismes,cohomologie cyclique et positivité

We show that cyclic cohomology of an algebraA is obtained from traces with suitable domains on the algebraqA of the second author. WhenA is aC* algebra so isqA and the notion of positive trace makes sense. We hence get a notion of positivity for cyclic cocycles. We prove that a positive trace onqA defines a type I or II Fredholm module onA.     

Geometric invariants of the quantum Hall effect

We study the two-dimensional Hall effect with a random potential. The Hall conductivity is identified as a geometric invariant associated with an algebra of observables. Using the pairing betweenK-theory and cyclic cohomology theory, we identify this geometric invariant with a topological index, thereby giving the Hall conductivity a new interpretation.Supported in part by the National Science Foundation under Grant No. DMS-8717185     

Currents on Grassmann algebras

We define currents on a Grassmann algebra Gr(N) with N generators as distributions on its exterior algebra (using the symmetric wedge product). We interpret the currents in terms of ₂-graded Hochschild cohomology and closed currents in terms of cyclic cocycles (they are particular multilinear forms on Gr(N). An explicit construction of the vector space of closed currents of degree p on Gr(N) is given by using Berezin integration.     

On Hermitian-Holomorphic Classes Related to Uniformization,the Dilogarithm,and the Liouville Action

Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a ech-de Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bundles. We reprove the relation with the determinant of cohomology construction. Furthermore, if we specialize the covering to the one provided by a Kleinian uniformization (thereby allowing possibly disconnected surfaces) the same class can be reinterpreted as the transgression of the regulator class expressed by the Bloch-Wigner dilogarithm.     

Cohomology of Canonical Projection Tilings

We define the cohomology of a tiling as the cocycle cohomology of its associated groupoid and consider this cohomology for the class of tilings which are obtained from a higher dimensional lattice by the canonical projection method in Schlottmann's formulation. We prove the cohomology to be equivalent to a certain cohomology of the lattice. We discuss one of its qualitative features, namely that it provides a topological obstruction for a generic tiling to be substitutional. We develop and demonstrate techniques for the computation of cohomology for tilings of codimension smaller than or equal to 2, presenting explicit formulae. These in turn give computations for the $K$-theory of certain associated non-commutative C ^* algebras. Received: 24 June 1999 / Accepted: 18 October 2001