arXiv subjects

Henri Moscovici

Publications and source records attributed to Henri Moscovici.

29 records · Page 2Linked to original sources

arXiv2006

Type III and spectral triples

We explain how a simple twisting of the notion of spectral triple allows to incorporate type III examples, such as those arising from the transverse geometry of codimension one foliations. Since the twisting of the commutators turns the usual hypertrace constructed out of the Dixmier trace into a twisted trace on the coordinate algebra, one would be tempted to interpret that as a manifestation of twisting at the level of cyclic cohomology, akin to that introduced by the authors in the context of Hopf cyclic cohomology. The main point of this note, besides giving simple natural examples of the general notion and developing the first basic steps of the theory, is to show that contrary to the initial expectations no cohomological twisting is in fact required. The Chern character of finitely summable spectral triples extends to the twisted case, and lands in fact in ordinary (untwisted) cyclic cohomology. The same holds true for the local Hochschild character. The index pairing with ordinary (untwisted) K-theory continues to make sense and the index formula is still given by the pairing of the corresponding Chern characters. This opens the road to extending the local index formula, as well as the analogue of the hypoelliptic construction on the dual system together with the corresponding Thom isomorphism, to the context of twisted spectral triples of type III.

math.OA↗

arXiv2006

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 nonorientable 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.

math.QA↗

arXiv2005

Transgressions of the Godbillon-Vey class and Rademacher functions

We construct, out of modular symbols, 1-traces that are invariant with respect to the actions of the Hopf algebra $\Hc_{1}$ on the crossed product $\Ac_{Q}$ of the algebra of modular forms of all levels by $\GL^+ (2,Q)$ investigated in earlier work. This provides a conceptual explanation for the construction of the Euler cocycle representing the image of the universal Godbillon-Vey class under the characteristic map of noncommutative Chern-Weil theory which we developed in our earlier work. We then refine the construction to produce secondary data by transgression. For the action determined by the Ramanujan connection the transgression takes place within the Euler class and the resulting cocycle coincides with the classical Rademacher function. The actions associated to cusp forms of higher weight produce transgressed cocycles that implement the Eichler-Shimura isomorphism. Finally, the actions corresponding to Eisenstein series give rise by transgression to the Eisenstein cocycle, expressed in terms of higher Dedekind sums and generalized Rademacher functions.

math.QA↗

arXiv2005

Background independent geometry and Hopf cyclic cohomology

This is primarily a survey of the way in which Hopf cyclic cohomology has emerged and evolved, in close relationship with the application of the noncommutative local index formula to transverse index theory on foliations. Being Diff-invariant, the geometric framework that allowed us to treat the `space of leaves' of a general foliation provides a `background independent' set-up for geometry that could be of relevance to the handling of the the background independence problem in quantum gravity. With this potential association in mind, we have added some new material, which complements the original paper and is also meant to facilitate its understanding. Section 2 gives a detailed description of the Hopf algebra that controls the `affine' transverse geometry of codimension $n$ foliations, and Section 5 treats the relative version of Hopf cyclic cohomology in full generality, including the case of Hopf pairs with noncompact isotropy.

math.QA↗

arXiv2003

Modular Hecke Algebras and their Hopf Symmetry

We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with values in Eisenstein series of weight 2, which when coupled with the `period' cocycle yields a representative of the Euler class.

math.QA↗

arXiv2003

Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry

We settle in this paper a question left open in our paper ``Modular Hecke algebras and their Hopf symmetry'', by showing how to extend the Rankin-Cohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation corresponding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin-Cohen brackets give rise to associative deformations.

math.QA↗

arXiv2001

Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry

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 level, by a cochain map constructed out of a fixed torsion-free linear connection. This allows the direct treatment of the index formula for the hypoelliptic signature operator - representing the diffeomorphism invariant transverse fundamental $K$-homology class of an oriented manifold - in the general case, when this operator is constructed by means of an arbitrary coupling connection.

math.DG↗

arXiv2000

Cyclic Cohomology and Hopf 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 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 well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square.

math.OA↗

arXiv1999

Cyclic Cohomology, Hopf Algebras and the Modular Theory

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.

math.QA↗

arXiv1999

Cyclic cohomology and Hopf algebras

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.

math.QA↗

arXiv1998

Hopf algebras, cyclic cohomology and the transverse index theory

We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined and shown to be canonically isomorphic to the Gelfand-Fuks cohomology.

math.DG↗