New Poisson brackets defined for Banach manifolds that can use higher-order derivatives.
problem Constructing Poisson brackets with higher-order derivatives on Banach manifolds.
method Method to construct Poisson brackets on Banach manifolds with dependence on higher-order derivatives.
result Counterexamples to the Leibniz property implying the existence of a Poisson tensor.
We show that L∞-algebroids, understood in terms of Q-manifolds can be described in terms of certain higher Schouten and Poisson structures on graded (super)manifolds. This generalises known constructions for Lie (super)algebras and Lie algebroids.
In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditio…
New algebraic structures extend Courant algebroids to higher multi-Courant algebroids.
problem Extending Courant algebroid structures to higher multi-Courant algebroids.
method Constructing higher geometric versions of algebraic structures defined by Keller and Waldmann.
result Higher multi-Courant algebroids form a Poisson algebra.
We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds. We also provide a generalisation of triangular Lie bialgebroids as higher Poisson and Schouten structures on Lie algebroids.
Reviewing Q-manifolds, modular classes, and applications.
problem Obtaining invariant volumes on Q-manifolds.
method Exploring Q-manifolds, modular classes, and applying to specific examples.
result Applications to L∞-algebroids and higher Poisson manifolds. Paper constructs L∞-algebroids from homotopy Poisson structures.
problem Homotopy Poisson structures and their algebraic properties.
method Introduces thick morphisms and L∞-morphisms. result Establishes an L∞-algebra structure on forms. The paper classifies Poisson structures on toric manifolds and computes their cohomology.
problem Classifying real Poisson structures on complex toric manifolds of type (1,1) and computing their cohomology. method Investigates algebraic and differential Poisson structures, focusing on smooth and generically non-degenerate cases.
result Computes the first two cohomology groups for algebraic Poisson structures.
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical…
The paper introduces a new form on Lie algebroids over multisymplectic manifolds.
problem Higher generalizations of Poisson structures and momentum maps.
method Introducing a compatible E-n-form on Lie algebroids.
result The introduced form satisfies a compatibility condition with Lie algebroid and multisymplectic structures.
New model estimates higher-order interactions in stochastic processes using lower-dimensional projections.
problem Estimating higher-order interaction effects in stochastic processes with limited data.
method Additive Poisson Process (APP) combines information geometry and generalized additive models to model intensity functions in lower dimensions.
result The model can estimate higher-order intensity functions with sparse data.
Authors prove de Rham cohomology of Poisson and Jacobi manifolds is trivial.
problem Understanding algebraic structures on de Rham cohomology of Poisson and Jacobi manifolds.
method Using DG operads and quasi-isomorphisms, they show the de Rham cohomology structure is trivial.
result The de Rham cohomology of Poisson and Jacobi manifolds has no higher structure beyond commutativity.
The paper constructs and generalizes Poisson brackets for Jacobi elliptic functions and higher-dimensional systems.
problem Understanding and generalizing Poisson brackets for Jacobi elliptic functions and higher-dimensional systems.
method Symplectic realization and bi-hamiltonian formulation for constructing and generalizing Poisson brackets.
result The Jacobi identity is satisfied only when the Plücker relations hold for these rank 2 Poisson brackets.
Quantizes the relationship between Koszul and Schouten brackets in Poisson geometry.
problem Quantizing the relationship between Koszul and Schouten brackets in Poisson geometry.
method Employing Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of L∞-algebroids. result Quantizes the L∞-morphism into a single linear operator, a formal Fourier integral operator. 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…
This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that n-Lie algebroid structures correspond to n-ary generalization of Gerstenhaber algebras and are implied by n-ary generalization of linear Poisson structures on the dual bundle. A Nambu-…
In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM⊕∧nT∗M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n+1)-vector fi…
We show how the relation between Poisson brackets and symplectic forms can be extended to the case of inhomogeneous multivector fields and inhomogeneous differential forms (or pseudodifferential forms). In particular we arrive at a notion which is a generalization of a symplectic structure and gives rise to higher Pois…
We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity Δ(e^x)=e^x(Δ(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the high…
Motivated by the quest to understand the analog of non-geometric flux compactification in the context of M-theory, we study higher dimensional analogs of generalized Poisson sigma models and corresponding dual string and p-brane models. We find that higher generalizations of the algebraic structures due to Dorfman, Roy…
We study higher-degree generalizations of symplectic groupoids, referred to as {\em multisymplectic groupoids}. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe "higher'' versions of Poisson structures by identifying the infinitesimal counterparts of mul…
A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an (n+1)-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations o…
Study of higher-order Dirac structures in field theory.
problem Extending multisymplectic structures to higher-order analogues.
method Define and analyze higher Dirac structures as involutive subbundles of TM+∧kTM∗. result Recover higher Poisson structures as infinitesimal counterparts of multisymplectic groupoids.
Introduces a new operator generating higher Koszul brackets on differential forms.
problem Developing a new operator for higher Koszul brackets on differential forms.
method Introducing a formal ℏ-differential operator Δ generating higher Koszul brackets on differential forms. result Established properties of the introduced BV type operator and its inclusion in a one-parameter family.
Kontsevich flow simplified for 2D Poisson structures.
problem Formality conjecture for 2D Poisson structures.
method Universal flow construction and Poisson-cohomology triviality proof.
result For n=2, the flow Γ1 is Poisson-cohomology trivial. The paper proves actions of lattices in higher rank groups have cost one.
problem Fixed price question for higher rank semisimple Lie groups.
method Low intensity Poisson point processes and geometry of Voronoi tessellations.
result Proves all probability measure preserving actions of lattices in higher rank groups have cost one.
In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic groupoids). We prove that for every source simply connected Lie groupoid G with Lie …
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an L∞-algebra to each submanifold. result Controls the deformation theory of Lagrangian NQ-submanifolds using an L∞-algebra. Three perspectives on quantizing magnetic Poisson structures are explored.
problem Quantization of magnetic Poisson structures with non-associativity.
method Deformation quantization, symplectic realization, geometric quantization using bundle gerbes.
result Comparison and contrast of different quantization approaches.
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.
Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.
problem Local radial rigidity of elliptic systems on Riemannian manifolds.
method Reduction to singular ordinary differential equations of Euler type.
result Local uniqueness and existence results for solutions with prescribed initial jets.
The paper connects higher-dimensional mechanics to Lie n-algebroids.
problem Understanding interactions in higher-dimensional gauge systems.
method Comparing BV/BRST formalism with Lie n-algebroids and defining polytorsion.
result Relates topological n-branes to differential geometry on Lie n-algebroids.
Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.
problem Relating Lie bialgebroids to homotopy Poisson structures on supermanifolds.
method Introduces L-infinity bialgebroids and higher Koszul brackets to connect these structures.
result Shows that (TM,T∗M) has an L-infinity bialgebroid structure for homotopy Poisson structures. 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.
Paper improves fraud detection in imbalanced financial data.
problem Detecting fraud in imbalanced financial datasets.
method Uses time-varying Poisson processes for fraud prediction.
result Method outperforms baseline in imbalanced data.
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.
Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles …
We introduce the notion of Poisson quasi-Nijenhuis manifolds generalizing the Poisson-Nijenhuis manifolds of Magri-Morosi. We also investigate the integration problem of Poisson quasi-Nijenhuis manifolds. In particular, we prove that, under some topological assumption, Poisson (quasi)-Nijenhuis manifolds are in one-one…
Quantization of (-1)-shifted derived Poisson manifolds via BV-infinity operators.
problem Quantizing (−1)-shifted derived Poisson manifolds. method Using BV-infinity operators on the space of Berezinian half-densities, proving quantization via lifting of Maurer-Cartan elements.
result Quantization of (−1)-shifted derived Poisson manifolds is equivalent to the vanishing of the second Poisson cohomology group. Let M be a paracompact differentiable manifold, A a local algebra and M^{A} a manifold of infinitely near points on M of kind A. We define the notion of A-Poisson manifold on M^{A}. We show that when M is a Poisson manifold, then M^{A} is an A-Poisson manifold. We also show that if (M,) is a symplectic manifold, the st…
Reduces Poisson manifolds with Hamiltonian Lie algebroids.
problem Handling Poisson manifolds with Hamiltonian Lie algebroids.
method Introducing compatibility of momentum sections and quotienting zero level sets.
result Quotient space of zero level set of compatible momentum section is a Poisson manifold.
New Poisson manifolds created over 2-tori.
problem Creating Poisson manifolds over 2-tori.
method Modified construction using K3 surfaces and strongly integral affine 2-tori.
result New class of Poisson manifolds with 2-torus as leaf space.
The study characterizes and proves properties of 3D Poisson quasi-Nijenhuis manifolds.
problem Characterizing and understanding 3D Poisson quasi-Nijenhuis manifolds.
method Characterization through deformation and application of Haantjes structures.
result Every 3D Poisson quasi-Nijenhuis manifold is a Haantjes manifold.
The paper studies deformations of Poisson brackets in two dimensions, finding non-trivial cohomology groups.
problem Deformations of multidimensional Poisson brackets of hydrodynamic type.
method Cohomology computation of PVAs associated with Poisson brackets at third differential degree.
result Non-trivial third cohomology group indicates non-equivalent infinitesimal deformations.
Generalizes sigma model with Lie algebroid structure and geometric conditions.
problem Consistency of constraints and gauge symmetry in topological sigma models.
method Analysis of geometric conditions and constraints in Hamiltonian and Lagrangian formalisms.
result Identifies universal compatibility condition between Lie algebroid and multi-symplectic structure.
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.
Poisson learning improves graph-based semi-supervised learning at very low label rates.
problem Degeneracy of Laplacian semi-supervised learning at low label rates.
method Replaces label assignment with source and sink placement, solving Poisson equation.
result Provably more stable and informative predictions than Laplacian learning.
Associate stacks to Poisson manifolds for equivalence checking.
problem Equivalence checking of Poisson manifolds.
method Associate stacks to Poisson manifolds using Dirac manifolds and closed 2-forms.
result Two Poisson manifolds are equivalent if their associated stacks are isomorphic.