Formal Poisson Cohomology of Quadratic Poisson Structures

We compute the formal Poisson cohomology of quadratic Poisson structures. We first recall that, generically, quadratic Poisson structures are diagonalizable. Then we compute the formal cohomology of diagonal Poisson structures.     

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     

Jet Coordinates for Local BRST Cohomology

The construction of appropriate jet space coordinates for calculating local BRST cohomology groups is discussed. The relation to tensor calculus is briefly reviewed too.     

Cohomological obstructions to the equivariant extension of closed invariant forms

We study under what condition a closed invariant form on a manifold with a group action admits an equivariant extension. We derive a sequence of obstructions in the cohomology groups of the Lie algebra with coefficients in appropriate modules. We illustrate the result with two specific examples. We then discuss when such cohomological obstructions vanish. Finally, we compare our analysis with the spectral sequence point of view.     

A Homotopy in the Usual Cochain Complex of Free Lie Algebras

In this Letter, we construct a natural contracting homotopy in the usual cochain complex of free Lie algebras. As a consequence, we prove that the triple cohomology of Lie algebras coincides with a slightly different form of the standard cohomology theory.     

Some Remarks on the Cohomology of Krichever–Novikov Algebras

We calculate the continuous cohomology of the Lie algebra of meromorphic vector fields on a compact Riemann surface from the cohomology of the holomorphic vector fields on the open Riemann surface pointed in the poles. This cohomology has been given by Kawazumi. Our result shows the Feigin–Novikov conjecture.     

Coeffective and de Rham cohomologies of symplectic manifolds

We discuss the relation of the coeffective cohomology of a symplectic manifold with the topology of the manifold. A bound for the coeffective numbers is obtained. The lower bound is got for compact Kähler manifolds, and the upper one for non-compact exact symplectic manifolds. A Nomizu's type theorem for the coeffective cohomology is proved. Finally, the behaviour of the coeffective cohomology under deformations is studied.     

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.     

Twisted noncommutative equivariant cohomology: Weil and Cartan models

We propose Weil and Cartan models for the equivariant cohomology of noncommutative spaces which carry a covariant action of Drinfel’d twisted symmetries. The construction is suggested by the noncommutative Weil algebra of Alekseev and Meinrenken (2000) [5]; we show how to implement a Drinfel’d twist of their models in order to take into account the noncommutativity of the spaces we are acting on. We also provide basic examples and properties of the twisted noncommutative equivariant cohomology.     

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.     

Irreducible Freedman‐Townsend vertex and Hamiltonian BRST cohomology

The irreducible Freedman‐Townsend vertex is derived by means of the Hamiltonian deformation procedure based on local BRST cohomology.     

Homogeneous generalized holomorphic structures

Homogeneous generalized holomorphic structures in the context of homogeneous principal fiber bundles are investigated. They are characterized in terms of Lie algebra data, and the generalized Dolbeault cohomology groups associated to a homogeneous generalized holomorphic vector bundle are identified with certain relative Lie algebra cohomology groups. We also provide some examples, using generalized flag manifolds.     

A New Cohomology Theory Associated to Deformations of Lie Algebra Morphisms

We introduce a new cohomology theory related to deformations of Lie algebra morphisms. This notion involves simultaneous deformations of two Lie algebras and a homomorphism between them.This revised version was published online in March 2005 with corrections to the cover date.     

BRST analysis of general mechanical systems

We study the groups of local BRST cohomology associated to the general systems of ordinary differential equations, not necessarily Lagrangian or Hamiltonian. Starting with the involutive normal form of the equations, we explicitly compute certain cohomology groups having clear physical meaning. These include the groups of global symmetries, conservation laws and Lagrange structures. It is shown that the space of integrable Lagrange structures is naturally isomorphic to the space of weak Poisson brackets. The last fact allows one to establish a direct link between the path-integral quantization of general not necessarily variational dynamics by means of Lagrange structures and the deformation quantization of weak Poisson brackets.     

On the computation of the Lichnerowicz–Jacobi cohomology

Lichnerowicz–Jacobi cohomology of Jacobi manifolds is reviewed. The use of the associated Lie algebroid allows to prove that the Lichnerowicz–Jacobi cohomology is invariant under conformal changes of the Jacobi structure. We also compute the Lichnerowicz–Jacobi cohomology for a large variety of examples.     

On Some Second Hochschild Cohomology Spaces for Algebras of Functions on a Manifold

We show that the second Hochschild cohomology space for the space of smooth functions on a manifold corresponding to cochains defined by continuous operators is the same as the one corresponding to differentiable operators, i.e. is given by the space of skewsymmetric contravariant 2-tensors on the manifold. We do this using a coboundary construction due to Omori, Maeda and Yoshioka.     

Unitons of Harmonic Maps from R2 to U(p,q)

We calculate a second cohomology class which determines a deformation quantization up to equivalence for a deformation quantization with separation of variables on a Kähler manifold, following P. Deligne.     

A higher twist in string theory

Considering the gauge field and its dual in heterotic string theory as a unified field, we show that the equations of motion at the rational level contain a twisted differential with a novel degree seven twist. This generalizes the usual degree three twist that lifts to twisted K-theory and raises the natural question of whether at the integral level the abelianized gauge fields belong to a generalized cohomology theory. Some remarks on possible such extension are given.