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.     

On a poisson reduction for Gel'fand-Zakharevich manifolds

We formulate and discuss a reduction theorem for Poisson pencils associated with a class of integrable systems, defined on bi-Hamiltonian manifolds, recently studied by Gel'fand and Zakharevich. The reduction procedure is suggested by the bi-Hamiltonian approach to the separation of variables problem.     

The Dirac spectrum of Bieberbach manifolds

The Dirac spectra and the eta invariants of three-dimensional Bieberbach manifolds are computed. Compact connected three-dimensional spin manifolds admitting parallel non-vanishing spinors are identified as flat tori.     

Quantization of Poisson Families and of Twisted Poisson Structures

We describe a deformation quantization of a modification of Poisson geometry by a closed 3-form. Under suitable conditions, it gives rise to a stack of algebras. The basic object used for this aim is a kind of families of Poisson structures given by a Maurer–Cartan equation; they are easily quantized using Kontsevich's formality theorem. We conclude with a section on quantization of complex manifolds.     

A Lagrangian for Hamiltonian vector fields on singular Poisson manifolds

On a manifold equipped with a bivector field, we introduce for every Hamiltonian a Lagrangian on paths valued in the cotangent space whose stationary points project onto Hamiltonian vector fields. We show that the remaining components of those stationary points tell whether the bivector field is Poisson or at least defines an integrable distribution—a class of bivector fields generalizing twisted Poisson structures that we study in detail.     

Trigonometric Dynamical <Emphasis Type="Italic">r</Emphasis>-Matrices Over Poisson Lie Base

Let be a finite dimensional complex Lie algebra and a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group Exp with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra . Let be an open domain parameterizing a neighborhood of the identity in Exp by the exponential map. We present dynamical r-matrices with values in over the Poisson Lie base manifold .*This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center Group Theoretic Methods in the study of Algebraic Varieties of the Israel Science foundation, and by the RFBR grant no. 03-01-00593.     

Linearity of the Transverse Poisson Structure to a Coadjoint Orbit

We prove a simple formula for the transverse Poisson structure to a coadjoint orbit (in the dual of a Lie algebra ) and use it in examples such as and . We also give a sufficient condition on the isotropy subalgebra of so that the transverse Poisson structureto the coadjoint orbit of is linear.     

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.     

The Universal Generating Function of Analytical Poisson Structures

Generating functions of Poisson structures are special functions which induce, on open subsets of , a Poisson structure together with the local symplectic groupoid integrating it. In a previous paper by A. S. Cattaneo, G. Felder and the author, a universal generating function was provided in terms of a formal power series coming from Kontsevich star product. The present article proves that this universal generating function converges for analytical Poisson structures and shows that the induced local symplectic groupoid coincides with the phase space of Karasev–Maslov Mathematics Subject Classification 58H05 (53D05).     

Poisson Actions up to Homotopy and their Quantization

Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmetries up to homotopy of the quantized algebras of functions, using the formality theorem of Kontsevich.Supported in part by the Swiss National Science Foundation     

Analytic k-Linearizability of Some Resonant Poisson Structures

We consider analytic Poisson tensors P whose associated Lie algebra at a singular point is resonant (in the sense of J. P. Dufour (J. Differential Geom. 30 (1990), 415–428)). We give sufficient conditions on the k-jet of P at the given point so that P is analytically linearizable.     

Volumes of compact manifolds

We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius condition for spheres. For this aim we first describe the manifold with some parameters, set up a metric, which induces a volume element, and perform the integration for the adequate range of the parameters; in most cases our manifolds will be either spheres or (twisted) products of spheres, or quotients of spheres (homogeneous spaces).Our results should be useful in several physical instances, as instanton calculations, propagators in curved spaces, sigma models, geometric scattering in homogeneous manifolds, density matrices for entangled states, etc. Some flag manifolds have also appeared recently as exceptional holonomy manifolds; the volumes of compact Einstein manifolds appear in string theory.     

Spin structures, spinors and the Dirac operator: real versus complex manifolds

We describe the relation between spin-structures, spinors and the Dirac operator on a (real) manifold and the analogous definitions in complex holomorphic terms. This may be useful for physicists interested in the algebraic geometric approach to superstrings.     

Local structure of Jacobi–Nijenhuis manifolds

After a brief review on the basic notions and the principal results concerning the Jacobi manifolds, the relationship between homogeneous Poisson manifolds and conformal Jacobi manifolds, and also the compatible Jacobi manifolds, we give a generalization of some of these results needed for the contents of this paper. We introduce the notion of Jacobi–Nijenhuis structure and we study the relation between Jacobi–Nijenhuis manifolds and homogeneous Poisson–Nijenhuis manifolds. We present a local classification of homogeneous Poisson–Nijenhuis manifolds and we establish some local models of Jacobi–Nijenhuis manifolds.     

Generalized results on the role of new-time transformations in finite-dimensional Poisson systems

The problem of characterizing all new-time transformations preserving the Poisson structure of a finite-dimensional Poisson system is completely solved in a constructive way. As a corollary, this leads to a broad generalization of previously known results. Examples are given.