Hopf algebras, cyclic cohomology and the transverse index theory

Abstract

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.

54 pages

Cited by in corpus (152)

• Hopf Algebras, Renormalization and Noncommutative Geometry
• A Short Survey of Noncommutative Geometry
• Quantum Groups and Noncommutative Geometry
• Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology
• Type III and spectral triples
• Ward identities and combinatorics of rainbow tensor models
• Insertion and Elimination: the doubly infinite Lie algebra of Feynman graphs
• Combinatorics of Rooted Trees and Hopf Algebras
• From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
• Combinatorial Hopf algebras
• Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory
• Renormalization as a functor on bialgebras
• Differentiable cyclic cohomology and Hopf algebraic structures in transverse geometry
• Combinatorics of (perturbative) quantum field theory
• (Co)cyclic (co)homology of bialgebroids: An approach via (co)monads
• Towards cohomology of renormalization: bigrading the combinatorial Hopf algebra of rooted trees
• On the Connes-Kreimer construction of Hopf Algebras
• Hopf Algebras in General and in Combinatorial Physics: a practical introduction
• Bogolubov's Recursion and Integrability of Effective Actions
• Background independent geometry and Hopf cyclic cohomology
• Local index formula and twisted spectral triples
• Position-dependent noncommutative products: classical construction and field theory
• New mathematical structures in renormalizable quantum field theories
• Cyclic homology of Hopf Galois extensions and Hopf algebras
• Spin group and almost commutative geometry
• Connes-Kreimer-Epstein-Glaser Renormalization
• Noncommutative renormalization for massless QED
• A bivariant Chern character for families of spectral triples
• Very Basic Noncommutative Geometry
• Cyclic cohomology and Hopf algebras
• Algebraic Birkhoff decomposition and its applications
• Fractals in Noncommutative Geometry
• Noncommutative Geometry and Conformal Geometry. I. Local Index Formula and Conformal Invariants
• Geometry of Quantum Spheres
• Stochastic expansions and Hopf algebras
• On the antipode of Kreimer's Hopf algebra
• The eta invariant and equivariant index of transversally elliptic operators
• Index theory for basic Dirac operators on Riemannian foliations
• A walk in the noncommutative garden
• On the deformation groupoid of the inhomogeneous pseudo-differential Calculus
• Batalin-Vilkovisky algebras and cyclic cohomology of Hopf algebras
• Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq(2)
• The quantum group of a preregular multilinear form
• NCS Systems over Differential Operator Algebras and the Grossman-Larson Hopf Algebras of Labeled Rooted Trees
• Bicrossproduct approach to the Connes-Moscovici Hopf algebra
• Cyclic Homology and Group Actions
• Spectral action beyond the standard model
• Feynman diagrams and polylogarithms: shuffles and pentagons
• The Hochschild cohomology ring of the standard Podles quantum sphere
• Cyclic Cohomology and Hopf Symmetry
• Renormalization automated by Hopf algebra
• An equivariant index theorem for hypoelliptic operators
• Monoidal Categories, 2-Traces, and Cyclic Cohomology
• Transverse noncommutative geometry of foliations
• Hopf algebras in dynamical systems theory
• On von Neumann spectral triples
• On the Connes-Moscovici Hopf algebra associated to the diffeomorphisms of a manifold
• Relating the Connes-Kreimer and Grossman-Larson Hopf algebras built on rooted trees
• Lie-Hopf algebras and their Hopf cyclic cohomology
• Cyclic Cohomology, Hopf Algebras and the Modular Theory
• Cross Product Bialgebras - Part I
• Twisted Homology of Quantum SL(2) - Part II
• Noncommutative geometry and motives: the thermodynamics of endomotives
• BRS Cohomology And The Chern Character In Non-Commutative Geometry
• Invariant Cyclic Homology
• Newton series and extended derivation relations for multiple $L$-values
• The Godbillon-Vey invariant in equivariant $KK$-theory
• The Holonomy Groupoids of Singularly Foliated Bundles
• On quantum groups associated to a pair of preregular forms
• Equivariant Algebraic Index Theorem
• Rankin-Cohen Brackets and the Hopf Algebra of Transverse Geometry
• SAYD modules over Lie-Hopf algebras
• Bivariant Hopf cyclic cohomology
• On Feynman graphs as elements of a Hopf algebra
• Hopf Modules and Noncommutative Differential Geometry
• Hopf algebras in noncommutative geometry
• On the Hopf Algebraic Structure of Lie Group Integrators
• The equivariant analytic index for proper groupoid actions
• Generalized Coefficients for Hopf Cyclic Cohomology
• Contractive spectral triples for crossed products
• Cup products in Hopf cyclic cohomology via cyclic modules I
• Quantum Groupoids and their Hopf Cyclic Cohomology
• Cyclic Homology and Quantum Orbits
• Loday--Quillen--Tsygan Theorem for Coalgebras
• Anti-Yetter-Drinfeld Modules for Quasi-Hopf Algebras
• Classification of differentials and Cartan calculus on bicrossproducts
• Equivariant Hopf Galois extensions and Hopf cyclic cohomology
• Localization over complex-analytic groupoids and conformal renormalization
• On the quasi-derivation relation for multiple zeta values
• The localized longitudinal index theorem for Lie groupoids and the van Est map
• On the Topological Interpretation of Gravitational Anomalies
• Cyclic vs mixed homology
• Rankin-Cohen Deformations and Representation Theory
• A Short Survey of Cyclic Cohomology
• Cyclic Cohomology of Corings
• The Gauss-Manin connection and noncommutative tori
• Cycles over DGH-semicategories and pairings in categorical Hopf-cyclic cohomology
• K-homology and index theory on contact manifolds
• Hopf algebroids and twists for quantum projective spaces
• Equivariant Chern classes in Hopf cyclic cohomology
• A characteristic map for compact quantum groups
• Hopf algebra ${\mathcal{K}}_n$ and universal Chern classes
• Index map, $\sigma$-connections, and Connes-Chern character in the setting of twisted spectral triples
• Hopf Galois (Co)Extensions In Noncommutative Geometry
• The weak heat kernel asymptotic expansion and the quantum double suspension
• Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1
• The universal Hopf cyclic theory
• Cyclic Cohomology of Hopf algebras and Hopf Algebroids
• A sharp analysis of the mixing time for random walk on rooted trees
• Lectures on Noncommutative Geometry
• A New Cyclic Module for Hopf Algebras
• Cyclic homology of Hopf algebras
• The interface of noncommutative geometry and physics
• Chern character, Hopf algebras, and BRS cohomology
• Symetries Galoisiennes et Renormalisation
• Introduction to Hopf algebras in renormalization and noncommutative geometry
• Hopf-cyclic homology and cohomology with coefficients
• Noncommutative geometry of foliations
• A Note on Cyclic Duality and Hopf Algebras
• Equivariant cyclic cocycles on the Boutet de Monvel symbol algebra
• Hopf cyclic cohomology in braided monoidal categories
• Para-Hopf algebroids and their cyclic cohomology
• Hopf cyclic cohomology and Hodge theory for proper actions
• Morita base change in Hopf-cyclic (co)homology
• New Coefficients For Hopf Cyclic Cohomology
• Products in Hopf-Cyclic Cohomology
• A note on cohomology for multiplier Hopf algebras
• Bialgebra Cyclic Homology with Coefficients, Part II
• Noncommutative geometry and quantization
• Hopf algebroids and secondary characteristic classes
• Classification of Group Actions Extended to Symplectic Deformation Quantizations
• Hopf algebras of primitive Lie pseudogroups and Hopf cyclic cohomology
• 2-categories and cyclic homology
• Quasi-elementary H-Azumaya algebras arising from generalized (anti) Yetter-Drinfeld modules
• Bialgebra Cyclic Homology with Coefficients, Part I
• A model structure and Hopf-cyclic theory on the category of coequivariant modules over a comodule algebra
• Finite dimensional comodules over the Hopf algebra of rooted trees
• Secondary Characteristic Classes and Cyclic Cohomology of Hopf Algebras
• A pairing between super Lie-Rinehart and periodic cyclic homology
• Generalized (anti) Yetter-Drinfeld modules as components of a braided T-category
• Excision in Hopf cyclic homology
• Gelfand-Fuchs cohomology of invariant formal vector fields
• Turaev bicategories, generalized Yetter-Drinfel`d modules in 2-categories and a Turaev 2-category for bimonads in 2-categories
• Nonlinear Perturbation Theory
• Hopf--Hochschild (co)homology of module algebras
• Uniqueness of pairings in Hopf-cyclic cohomology
• MSc Thesis: Presentation of Certain New Trends in Noncommutative Geometry
• Associative algebra deformations of the Connes-Moscovici's Hopf algebra ${\mathcal{H}}_1$
• Inertial Hopf-cyclic homology
• Hopf-type Cyclic Cohomology via Karoubi Operator
• Forces from noncommutative geometry
• Hopf algebras from poset growth models