Poisson Structure Research Articles

Comomentum Sections and Poisson Maps in Hamiltonian Lie Algebroids

In a Hamiltonian Lie algebroid over a pre-symplectic manifold and over a Poisson manifold, we introduce a map corresponding to a comomentum map, called a comomentum section. We show that the comomentum section gives a Lie algebroid morphism among Lie algebroids. Moreover, we prove that a momentum section on a Hamiltonian Lie algebroid is a Poisson map between proper Poisson manifolds, which is a generalization that a momentum map is a Poisson map between the symplectic manifold to dual of the Lie algebra. Finally, a momentum section is reinterpreted as a Dirac morphism on Dirac structures.

Buckling and post-buckling behavior of auxetic cellular structures

In this work, experimental tests and numerical simulations are carried out to investigate the buckling behavior and failure modes of auxetic cellular structures and sandwich panels with auxetic cores. Different Poisson's ratios and densities are considered to evaluate the impact of these parameters on the deformation mechanisms under uniaxial compression loading. The numerical analysis is performed using the Riks method, while considering geometric nonlinearity and elastoplastic behavior. The results indicate that negative Poisson's ratio and structure density have a significant influence on the buckling critical stress and the failure mechanisms of cellular structures. Although the inverted honeycomb and the double arrowhead with different Poisson's ratios exhibit similar load capacity, facesheet failure is more pronounced with the conventional inverted honeycomb. This result can be attributed to the dominant effect of the facesheet on the load evolution. The effects of the cell-wall thickness and the facesheet thickness on the buckling load are also discussed based on the finite element model.

Transposed Poisson structures on not-finitely graded Witt-type algebras

Transposed Poisson structures on not-finitely graded Witt-type algebras

Kontsevich graphs act on Nambu–Poisson brackets, I. New identities for Jacobian determinants

Abstract Nambu-determinant brackets on ℝ d ∋ x = (x 1, …, x d ), {f, g} d (x) = ϱ(x) · det (∂(f, g, a 1, …, a d−2) / ∂(x 1, …, x d )), with a i ∈ C ∞ (ℝ d ) and ϱ · ∂ x ∈ X d ( R d ) , are a class of degenerate (rank ⩽ 2) Poisson structures with (non)linear coefficients, e.g., polynomials of arbitrarily high degree. With ‘good’ cocycles in the graph complex, Kontsevich associated universal – for all Poisson bi-vectors P on affine ℝ aff d – elements P ˙ = Q γ ( [ P ] ) ∈ H P 2 ( ℝ aff d ) in the Lichnerowicz–Poisson second cohomology groups; we note that known graph cocycles γ preserve the Nambu–Poisson class {P (ϱ, [a]), and we express, directly from γ, the evolution ϱ ˙ , a ˙ that induces P ˙ . Over all d ⩾ 2 at once, there is no ‘universal’ mechanism for the bi-vector cocycles Q d γ to be trivial, Q d γ = 〚 P , X → d γ ( [ P ] ) 〛 , w.r.t. vector fields defined uniformly for all dimensions d by the same graph formula. While over ℝ2, the graph flows P ˙ = Q 2 D γi ( P ( ϱ ) ) for γ ∈ {γ3, γ5, γ7. …} are trivialised by vector fields X → 2 D γ i = ( d x ∧ d y ) − 1 d d R ( Ham γ i ⁡ ( P ) ) of peculiar shape, we detect that in d ⩾ 3, the 1-vectors from 2D, now with P (ϱ,a 1,… ,a d−2) inside, do not solve the problems Q d ⩾ 3 γ i = 〚 P , X → d ⩾ 3 γ i ( P ( ϱ , [ a ] ) ) 〛 , yet they do yield a good Ansatz where we find solutions X → d = 3 , 4 γ i ( P ( ϱ , [ a ] ) ) . In the study of the step d → d + 1, by adapting the Kontsevich graph calculus to the Nambu–Poisson class of brackets, we discover more identities for the Jacobian determinants within P (ϱ, [a]), i.e. for multivector-valued GL(d)-invariants on ℝ aff d .

On the Lichnerowicz-Poisson cohomology complex and quasi-Poisson algebras

This article explores F-commutative algebra (A,·) with Poisson structure defined by a non-Lie algebra bivector Q. We analyse the Lichnerowicz-Poisson coboundary δ*Q and its implications as a quasi-Poisson structure, where the Schouten bracket does not vanish, complicating cohomology computations. The examined open question is whether the deviating term is a coboundary of Poisson cohomology. Our main objectives are to construct a Poisson subalgebra on which the twisted term δ2Q(Q) vanishes, and to study the existence of Poisson structures P on a Poisson ideal satisfying the condition δ2P(·) = δ2Q(Q).

A neighbourhood theorem for submanifolds in generalized complex geometry

We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B -field equivalent to a holomorphic Poisson structure. This is intimately tied with our second main result, which is a rigidity theorem for generalized complex deformations of holomorphic Poisson structures. Specifically, on a compact manifold with boundary we provide explicit conditions under which any generalized complex perturbation of a holomorphic Poisson structure is B -field equivalent to another holomorphic Poisson structure. The proofs of these results require two analytical tools: Hodge decompositions on almost complex manifolds with boundary, and the Nash–Moser algorithm. As a concrete application of these results, we show that on a four-dimensional generalized complex submanifold which is generically symplectic, a neighbourhood of the entire complex locus is B -field equivalent to a holomorphic Poisson structure. Furthermore, we use the neighbourhood theorem to develop the theory of blowing down submanifolds in generalized complex geometry.

A Comparative Analysis of Drone and Boat Monitoring for the Endangered Bolivian River Dolphin

Accurate population monitoring is essential for effective wildlife conservation. This study compares the effectiveness of drone-based vs traditional boat-based methods for assessing an endangered Bolivian river dolphin (Inia geoffrensis boliviensis; BRD) population in Bolivia. Data were collected using high-resolution video recorded with a DJI Mavic 2 zoom drone and standardized methodologies for boat surveys. Two mixed-effects linear models incorporating Poisson and negative binomial error structures were used to compare counts obtained from drone- and boat-based surveys. Results show that drone-based surveys detected 1.15% more individuals on average than boat-based surveys. Drone counts were higher at sites with larger group sizes, leading to congruent estimates. The aerial perspective that drones offer lets researchers overcome potential challenges from boat-based surveys—for example, difficulties related to confirming individual IDs because of limited visibility due to sun glare. Transitioning from boat- to drone-based surveys offers advantages such as reduced disturbance from placement closer to the animals and improved detection rates. Ethical considerations and responsible flight practices are crucial. Standardizing methodologies and prioritizing ethical research factors are key for successful implementation. Drone-based surveys offer a promising approach to enhance wildlife monitoring and conservation practices. This study is the first of this kind for Bolivia, a national natural heritage site, and contributes to the conservation and knowledge of the BRD.

Transposed Poisson structures on Virasoro-type algebras

We compute 12-derivations on the deformed generalized Heisenberg-Virasoro1 algebras and on not-finitely graded Heisenberg-Virasoro algebras Wˆ(G), W˜(G), and HW˜(G). We classify all transposed Poisson structures on such algebras.

Comparing two different types of stochastic parametrization in geophysical flow

This paper investigates the effects of stochastic variations in bathymetry on the solutions of the thermal quasi-geostrophic (TQG) equations. These stochastic perturbations generate a variety of different types of ensemble spread in the solution behavior whilst also preserving the deterministic Lie–Poisson structure and Casimir conservation laws. We numerically compare the solution sensitivity to another type of structure-preserving stochastic perturbation where instead of bathymetry, the velocity is stochastically perturbed.

Representation theory of the reflection equation algebra II: Theory of shapes

We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the R-matrix associated to the standard q-deformation of GL(N,C) for 0<q<1. We consider the Poisson structure appearing as the classical limit of the R-matrix, and parametrize the symplectic leaves explicitly in terms of a type of matrix we call a shape matrix. We then introduce a quantized version of the shape matrix for the REA, and show that each irreducible representation of the REA has a unique shape.

Supersymmetric Integrable Hamiltonian Systems, Conformal Lie Superalgebras K(1, N = 1, 2, 3), and Their Factorized Semi-Supersymmetric Generalizations

We successively reanalyzed modern Lie-algebraic approaches lying in the background of effective constructions of integrable super-Hamiltonian systems on functional N=1,2,3- supermanifolds, possessing rich supersymmetries and endowed with suitably related compatible Poisson structures. As an application, we describe countable hierarchies of new nonlinear Lax-type integrable N=2,3-semi-supersymmetric dynamical systems and constructed their central extended superconformal Lie superalgebra K(1|3) and its finite-dimensional coadjoint orbits, generated by the related Casimir functionals. Moreover, we generalized these results subject to the suitably factorized super-pseudo-differential Lax-type representations and present the related super-Poisson brackets and compatible suitably factorized Hamiltonian superflows. As an interesting point, we succeeded in the algorithmic construction of integrable super-Hamiltonian factorized systems generated by Casimir invariants of the centrally extended super-pseudo-differential operator Lie superalgebras on the N=1,2,3-supercircle.

Twists of graded Poisson algebras and related properties

We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring A:=k[x1,…,xn] is a graded twist of a unimodular Poisson structure on A, namely, if π is a graded Poisson structure on A, then π has a decompositionπ=πunim+1∑i=1ndeg⁡xiE∧m where E is the Euler derivation, πunim is the unimodular graded Poisson structure on A corresponding to π, and m is the modular derivation of (A,π). This result is a generalization of the same one in the quadratic setting. The rigidity of graded twisting, PH1-minimality, and H-ozoneness are studied. As an application, we compute the Poisson cohomologies of the quadratic Poisson structures on the polynomial ring of three variables when the potential is irreducible, but not necessarily having an isolated singularity.

Symmetry Preservation in Hamiltonian Systems: Simulation and Learning

This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether’s theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of G-invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether’s theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry toward simulating and learning problems.

Cluster structures on braid varieties

We show the existence of cluster A \mathcal {A} -structures and cluster Poisson structures on any braid variety, for any simple Lie group. The construction is achieved via weave calculus and a tropicalization of Lusztig’s coordinates. Several explicit seeds are provided and the quiver and cluster variables are readily computable. We prove that these upper cluster algebras equal their cluster algebras, show local acyclicity, and explicitly determine their DT-transformations as the twist automorphisms of braid varieties. The main result also resolves the conjecture of B. Leclerc [Adv. Math. 300 (2016), pp. 190–228] on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties.

Transposed Poisson structures on solvable Lie algebras with filiform nilradical

Transposed Poisson structures on solvable Lie algebras with filiform nilradical

Erratum to “Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings”

This erratum corrects the false submisson date displayed on the last page of [Enseign. Math. 70, 313–381 (2024)].

Brane mechanics and gapped Lie n-algebroids

We draw a parallel between the BV/BRST formalism for higher-dimensional (≥ 2) Hamiltonian mechanics and higher notions of torsion and basic curvature tensors for generalized connections in specific Lie n-algebroids based on homotopy Poisson structures. The gauge systems we consider include Poisson sigma models in any dimension and “generalised R-flux” deformations thereof, such as models with an (n + 2)-form-twisted R-Poisson target space. Their BV/BRST action includes interaction terms among the fields, ghosts and antifields whose coefficients acquire a geometric meaning by considering twisted Koszul multibrackets that endow the target space with a structure that we call a gapped almost Lie n-algebroid. Studying covariant derivatives along n-forms, we define suitable polytorsion and basic polycurvature tensors and identify them with the interaction coefficients in the gauge theory, thus relating models for topological n-branes to differential geometry on Lie n-algebroids.

Physical Motivation On Deformation Quantisation

We study the Poisson structures associated with deformation quantisation and its non-degradable factor on the Casimir function. We also describe a filtered associative algebra in a quotient space as a Poisson algebra and the automorphism of the Poisson bracket is discussed.

Transposed Poisson structures on Virasoro-type (super)algebras

We explore transposed Poisson structures on the Lie algebra: the deformed twisted Schrödinger-Virasoro algebra D(λ), as well as two Lie superalgebras: the super-BMS3 algebra and the twisted N=1 Schrödinger-Neveu-Schwarz algebra. Initially, we demonstrate the absence of non-trivial transposed Poisson structures on the Lie algebra D(λ) for λ≠1 and provide an example of a transposed Poisson algebra with associative and Lie parts isomorphic to the algebra of triadic extended Laurent polynomials and D(1). Subsequently, we establish that the super-BMS3 algebra possesses non-trivial 12-superderivations but lacks a non-trivial transposed Poisson structure. Finally, we prove that the twisted N=1 Schrödinger-Neveu-Schwarz algebra does not have non-trivial 12-superderivations and thus lacks non-trivial transposed Poisson structures.

On gauge Poisson brackets with prescribed symmetry

Abstract In the context of averaging method, we describe a reconstruction of invariant connection-dependent Poisson structures from canonical actions of compact Lie groups on fibered phase spaces. Some symmetry properties of Wong's type equations are derived from the main results