Poisson Structure Research Articles

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.

Completely integrable replicator dynamics associated to competitive networks.

The replicator equationsare a family of ordinary differential equationsthat arise in evolutionary game theory, and are closely related to Lotka-Volterra. We produce an infinite family of replicator equationswhich are Liouville-Arnold integrable. We show this by explicitly providing conserved quantities and a Poisson structure. As a corollary, we classify all tournament replicators up to dimension 6 and most of dimension 7. As an application, we show that Fig.1 of Allesina and Levine [Proc. Natl. Acad. Sci. USA 108, 5638 (2011)10.1073/pnas.1014428108] produces quasiperiodic dynamics.

English

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted $\mathsf{fGC}_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree $1$. The ambition of the present work is to generalise Kontsevich's construction to graded symplectic manifolds of arbitrary degree $n\geq1$. The corresponding graph model is given by the full Kontsevich graph complex $\mathsf{fGC}_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex $\mathsf{fGC}_{d}$ is shown to act via $\mathsf{Lie}_\infty$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichm\"{u}ller algebra $\mathfrak{grt}_1\simeq H^0(\mathsf{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented.

Poisson cohomology of 3D Lie algebras

We compute the Poisson cohomology associated with several three dimensional Lie algebras. Together with existing results and the classification of three dimensional Lie algebras, this provides the Poisson cohomology of all linear Poisson structures in dimension 3.

Quasi Poisson structures, weakly quasi Hamiltonian structures, and Poisson geometry of various moduli spaces

Let G be a Lie group and g its Lie algebra. The aim here is to develop a theory of quasi Poisson structures relative to a not necessarily non-degenerate Ad-invariant symmetric 2-tensor in g⊗g and one of general not necessarily non-degenerate quasi Hamiltonian structures relative to a not necessarily non-degenerate Ad-invariant symmetric bilinear form on g, a quasi Poisson structure being given by a skew bracket of two variables such that suitable data defined in terms of G as symmetry group involving the 2-tensor in the tensor square of the Lie algebra g measure how that bracket fails to satisfy the Jacobi identity. The present approach involves a novel concept of momentum mapping and yields, in the non-degenerate case, a bijective correspondence, in terms of explicit algebraic expressions, between non-degenerate quasi Poisson structures and non-degenerate quasi Hamiltonian structures. The new theory applies to various not necessarily non-singular moduli spaces and yields thereupon, via reduction with respect to an appropriately defined momentum mapping, not necessarily non-degenerate ordinary Poisson structures arising from the data including the symmetric 2-tensor in g⊗g. Among these moduli spaces are representation spaces, possibly twisted, of the fundamental group of a Riemann surface, possibly punctured, and moduli spaces of semistable holomorphic vector bundles as well as Higgs bundle moduli spaces. In the non-degenerate case, such a Poisson structure comes down to a stratified symplectic one of the kind explored in the literature and recovers, e.g., the symplectic part of a Kähler structure introduced by Narasimhan and Seshadri for moduli spaces of stable holomorphic vector bundles on a curve. In the algebraic setting, these moduli spaces arise as not necessarily non-singular affine not necessarily non-degenerate Poisson varieties. Along the way, a side result is an explicit equivalence between extended moduli spaces and quasi Hamiltonian spaces independently of gauge theory.

Hamiltonian facets of classical gauge theories on E-manifolds

Manifolds with boundary, with corners, b-manifolds and foliations model configuration spaces for particles moving under constraints and can be described as E-manifolds. E-manifolds were introduced in Nest and Tsygan (2001 Asian J. Math. 5 599–635) and investigated in depth in Miranda and Scott (2021 Rev. Mat. Iberoam. 37 1207–24). In this article we explore their physical facets by extending gauge theories to the E-category. Singularities in the configuration space of a classical particle can be described in several new scenarios unveiling their Hamiltonian aspects on an E-symplectic manifold. Following the scheme inaugurated in Weinstein (1978 Lett. Math. Phys. 2 417–20), we show the existence of a universal model for a particle interacting with an E-gauge field. In addition, we generalise the description of phase spaces in Yang–Mills theory as Poisson manifolds and their minimal coupling procedure, as shown in Montgomery (1986 PhD Thesis University of California, Berkeley), for base manifolds endowed with an E-structure. In particular, the reduction at coadjoint orbits and the shifting trick are extended to this framework. We show that Wong’s equations, which describe the interaction of a particle with a Yang–Mills field, become Hamiltonian in the E-setting. We formulate the electromagnetic gauge in a Minkowski space relating it to the proper time foliation and we see that our main theorem describes the minimal coupling in physical models such as the compactified black hole.

On 2nd-stage quantization of quantum cluster algebras

Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the 2nd-stage quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its 2nd-stage quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses a mutually alternating quantum cluster algebra such that their 2nd-stage quantization can be essentially the same.As an example, we give the 2nd-stage quantized cluster algebra Ap,q(SL(2)) of FunC(SLq(2)) in §7.1 and show that it is a non-trivial 2nd-stage quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. As another example, we present a class of quantum cluster algebras with coefficients which possess a non-trivial 2nd-stage quantization. In particular we obtain a class of quantum cluster algebras from surfaces with coefficients which possess non-trivial 2nd-stage quantization.Finally, we prove that the compatible Poisson structure of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the 2nd-stage quantization of a quantum cluster algebra without coefficients is in fact trivial.

Asymptotic dynamics of Hamiltonian polymatrix replicators

In a previous paper (Alishah et al 2019 Nonlinearity 33 469) we have studied flows defined on polytopes, presenting a method to encapsulate its asymptotic dynamics along the edge-vertex heteroclinic network. Using this result we study here the Hamiltonian character of the asymptotic dynamics of conservative polymatrix replicators. Our main result states that for such conservative polymatrix replicator systems the map describing its asymptotic dynamics is Hamiltonian with respect to some appropriate Poisson structure.

Fuzzy hyperspheres via confining potentials and energy cutoffs

We simplify and complete the construction of fully O(D)-equivariant fuzzy spheres , for all dimensions , initiated in Fiore and Pisacane (2018 J. Geom. Phys. 132 423–51). This is based on imposing a suitable energy cutoff on a quantum particle in subject to a confining potential well V(r) with a very sharp minimum on the sphere of radius r = 1; the cutoff and the depth of the well diverge with . As a result, the noncommutative Cartesian coordinates generate the whole algebra of observables on the Hilbert space ; applying polynomials in the to any we recover the whole . The commutators of the are proportional to the angular momentum components, as in Snyder noncommutative spaces. , as carrier space of a reducible representation of O(D), is isomorphic to the space of harmonic homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative) , which carries an irreducible representation of . Moreover, is isomorphic to . We resp. interpret , as fuzzy deformations of the space of (square integrable) functions on S d and of the associated algebra of observables, because they resp. go to as Λ diverges (with fixed). With suitable , in the same limit goes to the (algebra of functions on the) Poisson manifold ; more formally, yields a fuzzy quantization of a coadjoint orbit of that goes to the classical phase space .

On real forms of Dubrovin-Ugaglia Poisson brackets

Dubrovin and Ugaglia introduced a holomorphic Poisson structure on the space of semisimple Frobenius manifolds, which is identified with the entries of an upper triangular matrix with ones on the diagonal. In this paper, we compute explicitly the real forms of the Dubrovin-Ugaglia Poisson structures, arising from the real forms of the Lie algebra son(C).