New counterexamples show the Kontsevich tetrahedral flows don't preserve Poisson bi-vectors.
problem Tetrahedral flows don't preserve Poisson bi-vectors.
method Used explicit counterexamples and balance analysis.
result The flow only preserves Poisson bi-vectors with a balance of 1:6.
The flow preserves Poisson bi-vectors if the monomials are balanced.
problem Infinitesimal preservation of Poisson bi-vectors by the Kontsevich tetrahedral flow.
method Explicit proof using Kontsevich graphs and balance ratio 1:6.
result The flow preserves Poisson bi-vectors if and only if the monomials are balanced.
Graph complex acts on Poisson bi-vectors, producing universal cocycles.
problem Understanding the action of graph complex on Poisson bi-vectors.
method Using Lie derivatives and graph cocycles, the graph complex acts on Poisson bi-vectors.
result A uniform construction of universal cocycles for homogeneous Poisson bi-vectors.
Classifies star log symplectic structures on surfaces.
problem Classifying star log symplectic structures on compact surfaces.
method Classifies bi-vectors with degeneracy loci modeled by lines intersecting at a point.
result Computes Poisson cohomology and discusses relationships with second cohomology.
Graph morphism maps Poisson cocycles to symmetries, revealing factorization through Jacobi identity.
problem Mapping graph cocycles to symmetries of Poisson structures.
method Kontsevich graph orientation morphism and differential consequences of Jacobi identity.
result Existence of factorization through differential consequences of Jacobi identity.
Poisson and symplectic structures discussed in lecture notes.
problem Exploring Poisson and symplectic structures in mathematics.
method Presentation of Poisson and symplectic structures, group actions, moment maps, and phase space reduction.
result Comprehensive review of Poisson and symplectic structures, group actions, and reduction.
Given a (m−2)-form $\zw$ and a volume form $\zW$ on a m-manifold one defines a bi-vector $\zL$ by setting $\zL(\za,\zb)={\frac {\za\zex\zb\zex\zw} {\zW}}$ for any 1-forms $\za,\zb$. In this way, locally, a Poisson pair, or bi-Hamiltonian structure, $(\zL,\zL_1 )$ is always represented by a couple of (m−2)-forms…
Geometrically proves twisted Poincaré duality for orientable Poisson manifolds.
problem Establishing twisted Poincaré duality for Poisson manifolds.
method Geometrically reinterprets algebraic constructions of twisted Poisson modules and Poisson chain complexes.
result Explicit chain isomorphism between Poisson cochain and chain complexes with coefficients in Poisson modules.
The abstract semiclassicalises quantum group principal bundles to Poisson geometry.
problem Semiclassicalising quantum group principal bundles to Poisson geometry.
method The theory is developed for Poisson manifolds with Poisson-compatible contravariant connections, and for Poisson-Lie groups with bicovariant Poisson-compatible contravariant connections.
result The construction of the Poisson level of the q-Hopf fibration and the spin connection on a principal bundle. We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifold…
Reformulates Fock-Rosly Poisson structure using quasi-triangular r-matrices.
problem Defining Fock-Rosly Poisson structure on moduli spaces.
method Using Lie algebra actions and quasi-triangular r-matrices.
result Shows Fock-Rosly structure as mixed product Poisson structure.
New Poisson structures on algebras linked to derivatives.
problem Understanding Poisson structures on trivial extension algebras.
method Introducing a correspondence between Poisson structures and derivatives, and formulating results in terms of Lie algebroids.
result One-to-one correspondence between Poisson structures and data involving derivatives.
New Lie groups found for Poisson diffeomorphisms.
problem Finding Lie group structures on Poisson diffeomorphism groups.
method Using Poisson groupoids, develop Lie group structures.
result Poisson diffeomorphism groups of various Poisson manifolds are regular Lie groups.
We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices…
The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.
problem Analyzing Poisson-flat connections with logarithmic poles.
method Defining an Euler-Poisson principal part and residue theory, establishing a Poisson Poincaré-Dulac theorem.
result Any logarithmic Poisson-flat connection is holomorphically gauge equivalent to a pure Euler-Poisson normal form.
The first cohomology of Poisson algebras is described and conditions for its vanishing are established.
problem Understanding the first cohomology of Poisson algebras.
method Description and mapping of first cohomology to intrinsic cohomologies of Poisson submanifolds, formulation of vanishing conditions.
result Necessary and sufficient conditions for the vanishing of the first cohomology of infinitesimal Poisson algebras are derived.
The paper normalizes Poisson saturation of coregular submanifolds.
problem Normalizing the Poisson saturation of coregular submanifolds.
method Normal form construction and Poisson geometry analysis.
result Local Poisson saturation of coregular submanifolds is an embedded Poisson submanifold with a normal form.
Develops reduction method for strong Dirac maps.
problem Generalizing Poisson momentum maps.
method General procedure for reduction along strong Dirac maps.
result Recover and introduce new Poisson, quasi-Poisson, and Dirac reduced structures.
A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.
problem Defining a Poisson bracket on the space of Poisson structures.
method Constructing a Poisson bracket on P(M) depending on a volume form, and defining invariant of Poisson structures. result Invariant of Poisson structures detects unimodularity and related Poisson bracket for symplectic structures.
The study introduces a new equivalence for Poisson modules on complex projective varieties.
problem Understanding the structure of Poisson modules on complex projective varieties with mild singularities.
method Introducing a weak concept of Morita equivalence in the birational context for Poisson modules.
result Poisson modules are classified into three types based on their properties.
Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.
problem Homotopy Poisson algebra models for reduced spaces.
method Cattaneo-Zambon compatibility and regularity conditions, equivariant map, homotopy Poisson algebra.
result Derivation of homotopy Poisson algebra generalizing classical BFV algebra.
Study Poisson algebras for Hamiltonian systems linearization.
problem Linearize dynamics along Poisson submanifolds.
method Use contravariant derivative to characterize Poisson algebras.
result Infinitesimal Poisson algebras provide a framework for Hamiltonization.
The paper studies Poisson vector fields and tensor deformations.
problem Understanding Poisson vector fields and tensor deformations.
method Proving Poisson properties of lifts and describing infinitesimal deformations.
result Infinitesimal deformations of Poisson tensors have been described.
Gaussian surrogates improve Poisson imaging performance at low doses.
problem Improving Poisson imaging performance at low doses.
method Analysis of Poisson and Gaussian surrogate reconstruction objectives under Poisson noise.
result Gaussian surrogates can achieve MSE comparable to Poisson MAP at low doses.
This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson proc…
Cohomology of 'book' Lie algebra Poisson structure computed.
problem Computing the cohomology of a specific Lie algebra structure.
method Direct computation of cohomology for the given Lie algebra structure.
result Explicit formula for Poisson cohomology of the 'book' Lie algebra.
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. O…
Python toolkit for symbolic Poisson geometry calculations.
problem Computing Poisson-Nijenhuis structures on manifolds.
method Symbolic algorithms implemented in Python.
result Examples of gauge transformations and parametric bivector fields.
Study calculates Poisson cohomology of broken Lefschetz fibrations.
problem Computing Poisson cohomology of broken Lefschetz fibrations.
method Calculates cohomology at fold and Lefschetz singularities, adapting techniques for Sklyanin algebra.
result Compact formulas for Poisson coboundary operator in 4 dimensions.
Local Poisson groupoids over mixed product Poisson structures defined and applied.
problem Defining and studying Poisson structures on groupoids.
method Using a local Lagrangian bisection in a double symplectic groupoid to twist a direct product of Poisson groupoids.
result Proving Gu,u is a Poisson groupoid over Ou. Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.
New type of manifolds derived from Poisson structures.
problem Generalizing Poisson Nijenhuis manifolds.
method Introducing pseudo-Poisson Nijenhuis manifolds and showing their properties.
result Found new materials to construct Courant algebroids.
Study of Poisson homeomorphisms and rigidity of coisotropic submanifolds.
problem Rigidity and non-rigidity phenomena in Poisson geometry.
method Study of Poisson homeomorphisms, use of clean intersection points, and analysis of characteristic partitions.
result Poisson homeomorphisms preserve symplectic foliations and coisotropic submanifolds are flexible.
In this paper, we first study the Poisson reductions of controlled Hamiltonian (CH) system and symmetric CH system by controllability distributions. These reductions are the extension of Poisson reductions by distribution for Poisson manifolds to that for phase spaces of CH systems with external force and control. We g…
Study of symplectic and Poisson reduction, proposing Poisson implosion.
problem Understanding and generalizing symplectic reduction to Poisson manifolds.
method Recalled and reviewed symplectic and Poisson reduction, proved cross-section theorem for Poisson manifolds.
result Generalized Guillemin-Sternberg theorem for Poisson manifolds, identified Poisson transversals.
Study KMS measures in Poisson geometry, focusing on b-Poisson manifolds.
problem Characterize KMS measures in Poisson geometry.
method Generalize symplectic results to b-Poisson manifolds. result Complete characterization of KMS measures on b-Poisson manifolds. Local formulas for Poisson structures on wrinkled fibrations are derived.
problem Understanding Poisson structures on specific geometric fibrations.
method Local formulæ for Poisson bivectors and symplectic forms on wrinkled fibrations.
result Local formulas for Poisson structures on wrinkled fibrations are derived.
We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group G on a Poisson manifold M, we find an explicit description of the lifted hamiltonian act…
Abstract: Geometrically describes Poisson cohomology groups around symplectic leaves.
problem Understanding the first Poisson cohomology groups around symplectic leaves.
method Splitting theorems for infinitesimal automorphisms of coupling Poisson structures.
result Derives criteria for vanishing of first Poisson cohomology groups.
Symplectic groupoids create Poisson integrators for complex systems.
problem Creating efficient integrators for non-linear Poisson structures.
method Recursive solutions of Hamilton-Jacobi equation, interpreted as Lagrangian bisections.
result Constructs Poisson integrators using symplectic groupoids.
Study deformations of holomorphic Poisson maps, extending Horikawa's work.
problem Deforming holomorphic Poisson maps.
method Algebraic approach using functors of Artin rings.
result Identification of first-order deformations and obstructions.
We provide a quasi-Poisson version of the Drinfeld's correspondence between Poisson homogeneous spaces and Lagrangian subalgebras.
We associate a homotopy Poisson-n algebra to any higher symplectic structure, which generalizes the common symplectic Poisson algebra of smooth functions. This provides robust n-plectic prequantum data for most approaches to quantization. UPDATE: It has been brought to my attention that the exterior product does not cl…
New Poisson structures defined on surface moduli spaces.
problem Generalizing Poisson brackets to quasi-surfaces.
method Defined quasi-Poisson brackets on quasi-surfaces.
result New Poisson structures on quasi-surfaces.
This paper studies Poisson structures defined by divisor ideals.
problem Understanding Poisson structures with degeneracy captured by divisor ideals.
method Developed a framework using divisor ideals and Lie algebroids.
result Effective methods for studying Poisson structures of divisor-type.
New Poisson structure found on near-symplectic manifolds with specific properties.
problem Understanding Poisson structures on near-symplectic manifolds.
method Defined a singular Poisson structure on a near-symplectic 4-manifold's tubular neighbourhood, computed its cohomology.
result Smooth Poisson cohomology of the structure depends on the modular vector field and is finite-dimensional.
Uses Dirac geometry to prove Poisson geometry results.
problem Classical results in Poisson geometry.
method Dirac geometry techniques.
result Proofs of Poisson geometry results.