Cyclic cohomology of Hopf algebras

We give a construction of the Connes–Moscovici cyclic cohomology for any Hopf algebra equipped with a character (solving the technical problem raised in Connes and Moscovici (Comm. Math. Phys. 198 (1998) 199–246)). Furthermore, we introduce a non-commutative Weil complex, which connects the work of Gelfand and Smirnov with cyclic cohomology. We show how the Weil complex arises naturally when looking at Hopf algebra actions and invariant higher traces, to give a non-commutative version of the usual Chern–Weil theory.     

Symplectic resolutions for nilpotent orbits (II)

Based on our previous work, Fu (Invent. Math. 151 (2003) 167–186), we prove that, given any two projective symplectic resolutions Z₁ and Z₂ of a nilpotent orbit closure in a complex simple Lie algebra of classical type, Z₁ is deformation equivalent to Z₂. This provides support for a ‘folklore’ conjecture on symplectic resolutions for symplectic singularities. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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 non-orientable 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.     

On the Hopf Algebraic Structure of Lie Group Integrators

A commutative but not cocommutative graded Hopf algebra H_N, based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees H_C, developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that H_N is naturally obtained from a universal object in a category of noncommutative derivations and, in particular, it forms a foundation for the study of numerical integrators based on noncommutative Lie group actions on a manifold. Recursive and nonrecursive definitions of the coproduct and the antipode are derived. The relationship between H_N and four other Hopf algebras is discussed. The dual of H_N is a Hopf algebra of Grossman and Larson based on ordered rooted trees. The Hopf algebra H_C of Butcher, Connes, and Kreimer is identified as a proper Hopf subalgebra of H_N using the image of a tree symmetrization operator. The Hopf algebraic structure of the shuffle algebra H_Sh is obtained from H_N by a quotient construction. The Hopf algebra H_P of ordered trees by Foissy differs from H_N in the definition of the product (noncommutative concatenation for H_P and shuffle for H_N) and the definitions of the coproduct and the antipode, however, these are related through the tree symmetrization operator.     

Some Bialgebroids Constructed by Kadison and Connes–Moscovici are Isomorphic

We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.      

Dirac submanifolds and Poisson involutions

Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects.In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U₊ (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.     

On the quasi-derivation relation for multiple zeta values

Recently, Masanobu Kaneko introduced a conjecture on an extension of the derivation relation for multiple zeta values. The goal of the present paper is to present a proof of the conjecture by reducing it to a class of relations for multiple zeta values studied by Kawashima. In addition, some algebraic aspects of the quasi-derivation operator on Q〈x,y〉, which was defined by modeling a Hopf algebra developed by Connes and Moscovici, will be presented.     

Cup products in Hopf-cyclic cohomology

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. To cite this article: M. Khalkhali, B. Rangipour, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Fedosov quantization in positive characteristic

We study the problem of deformation quantization for (algebraic) symplectic manifolds over a base field of positive characteristic. We prove a reasonably complete classification theorem for one class of such quantizations; in the course of doing it, we also introduce a notion of a restricted Poisson algebra - the Poisson analog of the standard notion of a restricted Lie algebra - and we prove a version of the Darboux Theorem valid in the positive characteristic setting.

     

Co-Poisson structures on polynomial Hopf algebras

The Hopf dual H° of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are characterized. Some correspondences between co-Poisson and Poisson structures are also established.     

Realization of Poisson enveloping algebra

For a Poisson algebra, the category of Poisson modules is equivalent to the module category of its Poisson enveloping algebra, where the Poisson enveloping algebra is an associative one. In this article, for a Poisson structure on a polynomial algebra S, we first construct a Poisson algebra R, then prove that the Poisson enveloping algebra of S is isomorphic to the specialization of the quantized universal enveloping algebra of R, and therefore, is a deformation quantization of R.     

The local index formula in semifinite Von Neumann algebras I: Spectral flow

We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is ‘almost’ a (b,B)-cocycle in the cyclic cohomology of A.     

The local index formula in semifinite von Neumann algebras II: The even case

We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.     

Cyclic cocycles on deformation quantizations and higher index theorems

We construct a nontrivial cyclic cocycle on the Weyl algebra of a symplectic vector space. Using this cyclic cocycle we construct an explicit, local, quasi-isomorphism from the complex of differential forms on a symplectic manifold to the complex of cyclic cochains of any formal deformation quantization thereof. We give a new proof of Nest-Tsygan's algebraic higher index theorem by computing the pairing between such cyclic cocycles and the K-theory of the formal deformation quantization. Furthermore, we extend this approach to derive an algebraic higher index theorem on a symplectic orbifold. As an application, we obtain the analytic higher index theorem of Connes-Moscovici and its extension to orbifolds.     

The cohomology of the cotangent bundle of Heisenberg groups

Given a parabolic subalgebra g₁×n of a semisimple Lie algebra, Kostant (Ann. Math. 1963) and Griffiths (Acta Math. 1963) independently computed the g₁ invariants in the cohomology group of n with exterior adjoint coefficients. By a theorem of Bott (Ann. Math. 1957), this is the cohomology of the associated compact homogeneous space with coefficients in the sheaf of local holomorphic forms. In this paper we determine explicitly the full module structure, over the symplectic group, of the cohomology group of the Heisenberg Lie algebra with exterior adjoint coefficients. This is the cohomology of the cotangent bundle of the Heisenberg group.     

Les espaces Lp d'une algèbre de von Neumann définies par la derivée spatiale

This work answers a question raised by A. Connes (on the spatial theory of von Neumann algebras, preprint, Inst. Hautes Études Sci., France) and generalizes for a general von Neumann algebra the theory of non-commutative integration of J. Dixmier (Bull. Soc. Math. France81 (1953)) and I. Segal (Ann. of Math.57 (1973)).     

Hopf-cyclic homology and cohomology with coefficients

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter–Drinfeld compatibility condition leading to the concept of a stable anti-Yetter–Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra. To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Cyclic Homology of Hopf Algebras

A cyclic cohomology theory adapted to Hopf algebras has been introduced recently by Connes and Moscovici. In this paper, we consider this object in the homological framework, in the spirit of Loday and Quillen and Karoubi's work on the cyclic homology of associative algebras. In the case of group algebras, we interpret the decomposition of the classical cyclic homology of a group algebra in terms of this homology. We also compute both cyclic homologies for truncated quiver algebras.     

La construction bar d'une algèbre comme algèbre de Hopf E-infini

We prove that the bar construction of an E_∞ algebra forms an E_∞ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt–Eccles operad. (The surjection operad and the Barratt–Eccles operad are classical E_∞ operads.) To cite this article: B. Fresse, C. R. Acad. Sci. Paris, Ser. I 337 (2003).