Poisson Structure Research Articles

Multi-derivations of Poisson algebras

The purpose of this paper is to study multi-derivations of Poisson algebras. First, we show that the space of multi-derivations of a Lie algebra [Formula: see text] with the Nijenhuis–Richardson bracket is a differential graded Lie algebra. Then we introduce the notion of a multi-derivation of a Poisson algebra, and show that the space of multi-derivations of a Poisson algebra with the Nijenhuis–Richardson bracket is also a graded Lie algebra. Finally, we study multi-derivations of three concrete Poisson algebras coming from linear Poisson manifolds, symplectic manifolds and Poisson manifolds with vanishing first cohomology groups. We establish the relationship between multi-derivations of a Lie algebra [Formula: see text] and multi-derivations of the linear Poisson manifold [Formula: see text], and show that a two-derivation of a connected symplectic manifold [Formula: see text] is [Formula: see text], where [Formula: see text] is the Poisson bivector field induced by the symplectic structure [Formula: see text] and [Formula: see text] is a real number. We also prove that a two-derivation of a connected Poisson manifold [Formula: see text] with trivial Poisson cohomology group [Formula: see text] is [Formula: see text], where [Formula: see text] is a Casimir function.

Corner Structure of Four-Dimensional General Relativity in the Coframe Formalism

AbstractThis note describes a local Poisson structure (up to homotopy) associated with corners in four-dimensional gravity in the coframe (Palatini–Cartan) formalism. This is achieved through the use of the BFV formalism. The corner structure contains in particular an Atiyah algebroid that couples the internal symmetries to diffeomorphisms. The relation with BF theory is also described.

On Lie-Rinehart-Poisson algebras structures

We define the Schouten-Nijenhuis bracket on the algebra of the module of Kahler differentials. We give the main features of Poisson manifolds by using the universal property of derivation. We prove the equivalence between a Lie-Rinehart algebra structure and a Poisson structure and we recover Lichnerowicz's notion of Poisson manifold. We show that a symplectic Lie-Rinehart algebra structure induce a nondegenerate Poisson structure and conversely.

A Poisson cokriging method for bivariate count data

Bivariate spatially correlated count data appear naturally in several domains such as ecology, economy and epidemiology. Current methods for analysing such data lack simplicity, interpretability and computational awareness. This paper introduces Poisson cokriging, a bivariate geostatistical technique to model and predict spatially correlated count variables. Our method exploits classical geostatistical theory and the bivariate Poisson distribution to propose an adaptation of cokriging when the underlying process follows a bivariate Poisson structure. A simulation study and a real data application using counts from two mosquito-borne diseases in Colombia showed that our model successfully performs spatial predictions at unobserved locations under different settings. We demonstrate the competitive convenience of Poisson cokriging in filtering rates and modelling highly variant population sizes against traditional geostatistical methods. We conclude that Poisson cokriging improves prediction accuracy and reduces variance prediction errors in comparison with ordinary cokriging.

A new look at Lie algebras

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.

Lagrangian reduction of forced discrete mechanical systems

In this paper we propose a process of Lagrangian reduction and reconstruction for symmetric discrete-time mechanical systems acted on by external forces, where the symmetry group action on the configuration manifold turns it into a principal bundle. We analyze the evolution of momentum maps and Poisson structures under different conditions.

Green Hyperbolic Complexes on Lorentzian Manifolds

We develop a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which cover many examples of derived critical loci for gauge-theoretic quadratic action functionals in Lorentzian signature. We define Green hyperbolic complexes through a generalization of retarded and advanced Green’s operators, called retarded and advanced Green’s homotopies, which are shown to be unique up to a contractible space of choices. We prove homological generalizations of the most relevant features of Green hyperbolic operators, namely that (1) the retarded-minus-advanced cochain map is a quasi-isomorphism, (2) a differential pairing (generalizing the usual fiber-wise metric) on a Green hyperbolic complex leads to covariant and fixed-time Poisson structures and (3) the retarded-minus-advanced cochain map is compatible with these Poisson structures up to homotopy.

On the Poisson structure and action-angle variables for the complex modified Korteweg-de Vries equation

In this paper, we employ the inverse scattering approach to study the Poisson structure and action-angle variables of the complex modified Korteweg-de Vries (cmKdV) equation. We first derive the cmKdV equation via the principle of variation. Then, we successfully obtain the Poisson brackets for the scattering data of the equation. Furthermore, the action-angle variables are expressed in terms of the scattering data. Interestingly, our results show that the coordinate expression and the spectral parameter expression of the Hamiltonian can be related by the conservation laws.

Hamiltonian structure of rational isomonodromic deformation systems

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space L*gl(r) of the loop algebra Lgl(r). Nonautonomous isomonodromic counterparts of isospectral systems generated by spectral invariants are obtained by identifying deformation parameters as Casimir elements on the phase space. These are shown to coincide with higher Birkhoff invariants determining local asymptotics near to irregular singular points, together with the pole loci. Pairs consisting of Birkhoff invariants, together with the corresponding dual spectral invariant Hamiltonians, appear as “mirror images” matching, at each pole, the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve with the corresponding positive power terms in the analytic part. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves may be identified as the orbits of a free Abelian local group action. The projection of infinitesimal isomonodromic deformation vector fields to the quotient manifold under this action gives commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci.

Generalized Sasakian structures from a Poisson geometry viewpoint

Generalized Sasakian structures from a Poisson geometry viewpoint

Quantization of ($$-1$$)-Shifted Derived Poisson Manifolds

Quantization of ($$-1$$)-Shifted Derived Poisson Manifolds

A new look into the concept of separable system of Classical Mechanics

The theory of integrable systems of Classical Mechanics has been deeply affected, in the second half of the last century, by the discovery of the integrability of the Korteweg–de Vries equation. New ideas appeared at that time (such as, for instance, the concepts of Lax and bihamiltonian representations, and the notions of Poisson manifold, Poisson pencil, and recursion operator) which have changed our way of seeing the geometry of integrable systems. They also changed our way of seeing the geometry of separable systems. The paper is an attempt to explain why and in what form the idea of recursion operator leads to reshape the definition of separable system, and to discover a new set of separability conditions called “Kowalevski separability conditions”.

Transposed Poisson Structures on Generalized Witt Algebras and Block Lie Algebras

We describe transposed Poisson structures on generalized Witt algebras W(A,V,langle cdot ,cdot rangle ) and Block Lie algebras L(A, g, f) over a field F of characteristic zero, where langle cdot ,cdot rangle and f are non-degenerate. More specifically, if dim (V)>1, then all the transposed Poisson algebra structures on W(A,V,langle cdot ,cdot rangle ) are trivial; and if dim (V)=1, then such structures are, up to isomorphism, mutations of the group algebra structure on FA. The transposed Poisson algebra structures on L(A, g, f) are in a one-to-one correspondence with commutative and associative multiplications defined on a complement of the square of L(A, g, f) with values in the center of L(A, g, f). In particular, all of them are usual Poisson structures on L(A, g, f). This generalizes earlier results about transposed Poisson structures on Block Lie algebras mathcal {B}(q).

Integrable Multi-Hamiltonian Systems from Reduction of an Extended Quasi-Poisson Double of {text {U}}(n)

We construct a master dynamical system on a {text {U}}(n) quasi-Poisson manifold, {mathcal {M}}_d, built from the double {text {U}}(n) times {text {U}}(n) and dge 2 open balls in mathbb {C}^n, whose quasi-Poisson structures are obtained from T^* mathbb {R}^n by exponentiation. A pencil of quasi-Poisson bivectors P_{underline{z}} is defined on {mathcal {M}}_d that depends on d(d-1)/2 arbitrary real parameters and gives rise to pairwise compatible Poisson brackets on the {text {U}}(n)-invariant functions. The master system on {mathcal {M}}_d is a quasi-Poisson analogue of the degenerate integrable system of free motion on the extended cotangent bundle T^*!{text {U}}(n) times mathbb {C}^{ntimes d}. Its commuting Hamiltonians are pullbacks of the class functions on one of the {text {U}}(n) factors. We prove that the master system descends to a degenerate integrable system on a dense open subset of the smooth component of the quotient space {mathcal {M}}_d/{text {U}}(n) associated with the principal orbit type. Any reduced Hamiltonian arising from a class function generates the same flow via any of the compatible Poisson structures stemming from the bivectors P_{underline{z}}. The restrictions of the reduced system on minimal symplectic leaves parameterized by generic elements of the center of {text {U}}(n) provide a new real form of the complex, trigonometric spin Ruijsenaars–Schneider model of Krichever and Zabrodin. This generalizes the derivation of the compactified trigonometric RS model found previously in the d=1 case.

Vertical Lifts of Higher Order of Multivector Fields and Applications to Poisson Geometry

Vertical Lifts of Higher Order of Multivector Fields and Applications to Poisson Geometry

Integration of generalized complex structures

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a holomorphic stack with a shifted symplectic structure; in other words, a real symplectic groupoid with a compatible complex structure is defined only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. We describe several concrete examples of these integrations. Crucial to our solution are new technical tools, which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids.

A novel compression-torsion coupling metamaterial with tunable Poisson's ratio

In this study, a three-dimensional compression-torsion coupled negative Poisson structure was designed by combining the perforated plate with the inclined rod out of the plane. This structure can realize the selection of the positive and negative Poisson's ratio of the three-dimensional structure by changing the direction of the inclined rod. The absolute value of the Poisson's ratio of the two structures is close to the same but the positive and negative are opposite. Both structures maintain close to the same stress–strain curve. This structural characteristic can realize the selection of an ideal Poisson's ratio without changing the mechanical properties of the structure. Two 3D compression-torsion coupling structures were printed using metal 3D printing technology, and compression tests were carried out in the axial and transverse directions. Then, parametric analysis was carried out under axial compression, including energy absorption and specific energy absorption analyses. The results show that the experimental and finite element results are in good agreement and the finite element simulation is accurate. The Poisson's ratio of this negative Poisson's ratio metamaterial is selectable under both axial and lateral pressure conditions, so the structure has the excellent characteristics of selectable Poisson's ratio in three mutually perpendicular axes. The results show that the absolute value of Poisson's ratio exceeds 1 when the structure is under lateral compression, which indicates that the lateral deformation rate of the structure is greater than the axial deformation rate under this compression form. Finally, two tubular structures with different arrangements were designed by using these two structural cells, and the ball passing test was carried out. The results show that different arrangements will cause the balls to experience different resistances when passing through different positions in the tube. Research shows that the structure has potential applications in energy absorption, biomechanical devices, smart actuators, and devices with special requirements for Poisson's ratio.

Asymptotic dynamics of three dimensional supergravity and higher spin gravity revisited

We reconsider the Hamiltonian reduction of the action for three dimensional AdS supergravity and W3 higher spin AdS gravity in the Chern-Simons formulation under asymptotically anti-de Sitter boundary conditions. We show that the reduction gives two copies of chiral bosons on the boundary. In particular, we take into account the holonomy of the Chern-Simons connection which manifests itself as zero mode of the momentum of the boundary chiral boson. We provide an equivalent formulation of the boundary action which we claim to be the geometric action on symplectic leaves of a (super-)Virasoro or a higher spin WN Poisson manifold in the case of supergravity or higher spin gravity respectively, where the intersection of leaves (given in terms of leaves representatives) can be identified as the bulk holonomy. This concludes the extension to non-linear algebras where the notion of coadjoint representation is not well-defined. The boundary Hamiltonian depends on a choice of boundary conditions and is equivalent to the Schwarzian action for corresponding Brown-Henneaux boundary conditions. We make this connection explicit in the extended supersymmetric case. Moreover, we discuss the geometric action in the case of W3 AdS3 gravity in both mathfrak{sl} (3) highest weight representations based on principal and diagonal mathfrak{sl} (2) embeddings.

Multiplicative Ehresmann connections

We develop the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering the identity, as well as their infinitesimal counterparts. We construct obstructions to the existence of such connections, and we prove existence for several interesting classes of Lie groupoids and Lie algebroids, including all proper Lie groupoids. We show that many notions from the theory of principal bundle connections have analogues in this general setup, including connections 1-forms, curvature 2-forms, Bianchi identity, etc. In [19] we provide a non-trivial application of the results obtained here to construct local models in Poisson geometry and to obtain linearization results around Poisson submanifolds.

The Poisson geometry of Plancherel formulas for triangular groups

In this paper we establish the existence of canonical coordinates for generic co-adjoint orbits on triangular groups. These orbits correspond to a set of full Plancherel measure on the associated dual groups. This generalizes a well-known coordinatization of co-adjoint orbits of a minimal (non-generic) type originally discovered by Flaschka. The latter had strong connections to the classical Toda lattice and its associated Poisson geometry. Our results develop connections with the Full Kostant–Toda lattice and its Poisson geometry. This leads to novel insights relating the details of Plancherel theorems for Borel Lie groups to the invariant theory for Borels and their subgroups. We also discuss some implications for the quantum integrability of the Full Kostant Toda lattice.