Poisson Actions Research Articles

Applications of Nijenhuis Geometry V: Geodesic Equivalence and Finite-Dimensional Reductions of Integrable Quasilinear Systems

We describe all metrics geodesically compatible with a gl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{gl}$$\\end{document}-regular Nijenhuis operator L. The set of such metrics is large enough so that a generic local curve γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} to be a g-geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g-geodesics is naturally equivalent to the Poisson action of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} on the cotangent bundle generated by the integrals coming from geodesic compatibility.

Poisson structure and action–angle variables for the Hirota equation

Poisson structure and action–angle variables for the Hirota equation

On the Poisson structure and action–angle variables for the Wadati–Konno–Ichikawa equation

In this work, we employ the inverse scattering approach to investigate the Poisson structure and action–angle variables for the Wadati–Konno–Ichikawa equation. The Poisson brackets for the scattering data are deduced. Consequently, the action–angle variables are expressed in terms of the scattering data. In addition, we also discuss on the conservation laws and the spectral parameter expression of Hamiltonian for this equation.

Universal deformation formula, formality and actions

In this paper we provide a quantization via formality of Poisson actions of a triangular Lie algebra $(\mathfrak{g},r)$ on a smooth manifold $M$. Using the formality of $E$-polydifferential operators for Lie algebroids $E$ over $M$, we obtain a deformation quantization of $M$ together with a quantum group $\mathscr{U}{\rho\hbar}(\mathfrak{g})$ and a map of associated DGLA's. This motivates a definition of quantum action in terms of $L\_\infty$-morphisms which generalizes the well-known definition given by Drinfeld.

Krieger’s type for ergodic non-singular Poisson actions of non-(T) locally compact groups

AbstractIt is shown that each locally compact second countable non-(T) group G admits non-strongly ergodic weakly mixing IDPFT Poisson actions of any possible Krieger type. These actions are amenable if and only if G is amenable. If G has the Haagerup property, then (and only then) these actions can be chosen of 0-type. If G is amenable, then G admits weakly mixing Bernoulli actions of arbitrary Krieger type.

Poisson-geometric Analogues of Kitaev Models

We define Poisson-geometric analogues of Kitaev’s lattice models. They are obtained from a Kitaev model on an embedded graph Gamma by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double {mathcal {H}}(G). Each vertex (face) of Gamma defines a Poisson action of G (of G^*) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group D(G). We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of D(G). We construct an isomorphism of Poisson D(G)-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly’s Poisson structure for the graph Gamma and the Poisson-Lie group D(G). This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat D(G)-bundles on an oriented surface with boundary constructed from Gamma .

The Entropic Dynamics of Quantum Scalar Fields Coupled to Gravity

Entropic dynamics (ED) are a general framework for constructing indeterministic dynamical models based on entropic methods. ED have been used to derive or reconstruct both non-relativistic quantum mechanics and quantum field theory in curved space-time. Here we propose a model for a quantum scalar field propagating in dynamical space-time. The approach rests on a few key ingredients: (1) Rather than modelling the dynamics of the fields, ED models the dynamics of their probabilities. (2) In accordance with the standard entropic methods of inference, the dynamics are dictated by information encoded in constraints. (3) The choice of the physically relevant constraints is dictated by principles of symmetry and invariance. The first of such principle imposes the preservation of a symplectic structure which leads to a Hamiltonian formalism with its attendant Poisson brackets and action principle. The second symmetry principle is foliation invariance, which, following earlier work by Hojman, Kuchař, and Teitelboim, is implemented as a requirement of path independence. The result is a hybrid ED model that approaches quantum field theory in one limit and classical general relativity in another, but is not fully described by either. A particularly significant prediction of this ED model is that the coupling of quantum fields to gravity implies violations of the quantum superposition principle.

Weakly homoclinic groups of ergodic actions

This work contains the following results: the trajectory fullness of the homoclinic groups, their connections with factors, K-property, weak multiple mixing; the ergodicity of the weakly homoclinic group for Gauss and Poisson actions; the triviality of the weakly homoclinic group for classes of rank-1 actions. Some open problems are discussed.

On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action

Let be the space of bilinear forms on with defining matrices endowed with a quadratic Poisson structure of reflection equation type. The paper begins with a short description of previous studies of the structure, and then this structure is extended to systems of bilinear forms whose dynamics is governed by the natural action of the Poisson–Lie group on . A classification is given of all possible quadratic brackets on preserving the Poisson property of the action, thus endowing with the structure of a Poisson homogeneous space. Besides the product Poisson structure on , there are two other (mutually dual) structures, which (unlike the product Poisson structure) admit reductions by the Dirac procedure to a space of bilinear forms with block upper triangular defining matrices. Further generalisations of this construction are considered, to triples with the Poisson action , and it is shown that then acquires the structure of a Poisson symmetric space. Generalisations to chains of transformations and to the quantum and quantum affine algebras are investigated, as well as the relations between constructions of Poisson symmetric spaces and the Poisson groupoid. Bibliography: 30 titles.

The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular r-Matrix

We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs of Poisson actions, which were studied by J.-H. Lu and the author. The Fock-Rosly Poisson structure corresponds to the quasi-Poisson structure studied by Massuyeau, Turaev, Li-Bland, and Severa under an equivalence of categories between Poisson and quasi-Poisson spaces.

DOUBLE BRUHAT CELLS AND SYMPLECTIC GROUPOIDS

Let G be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure π st determined by a pair of opposite Borel subgroups (B, B_). We prove that for each υ in the Weyl group W of G, the double Bruhat cell G υ,υ = BυB Ω B_υB_ in G, together with the Poisson structure π st, is naturally a Poisson groupoid over the Bruhat cell BυB/B in the flag variety G/B. Correspondingly, every symplectic leaf of π st in G υ,υ is a symplectic groupoid over BυB/B. For u, υ ϵ W, we show that the double Bruhat cell (G u,υ , π st) has a naturally defined left Poisson action by the Poisson groupoid (G u,υ , π st) and a right Poisson action by the Poisson groupoid (G u,υ , π st), and the two actions commute. Restricting to symplectic leaves of π st, one obtains commuting left and right Poisson actions on symplectic leaves in G u,υ by symplectic leaves in G u,u and G υ,υ as symplectic groupoids.

DIRAC ACTIONS AND LU’S LIE ALGEBROID

Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form a matched pair. We also give a full classification of Dirac actions for which the base manifold is a homogeneous space H/K, obtaining a generalization of Drinfeld’s classification for the Poisson Lie group case.

Directional recurrence and directional rigidity for infinite measure preserving actions of nilpotent lattices

Let$\unicode[STIX]{x1D6E4}$be a lattice in a simply connected nilpotent Lie group$G$. Given an infinite measure-preserving action$T$of$\unicode[STIX]{x1D6E4}$and a ‘direction’ in$G$(i.e. an element$\unicode[STIX]{x1D703}$of the projective space$P(\mathfrak{g})$of the Lie algebra$\mathfrak{g}$of$G$), some notions of recurrence and rigidity for$T$along$\unicode[STIX]{x1D703}$are introduced. It is shown that the set of recurrent directions${\mathcal{R}}(T)$and the set of rigid directions for$T$are both$G_{\unicode[STIX]{x1D6FF}}$. In the case where$G=\mathbb{R}^{d}$and$\unicode[STIX]{x1D6E4}=\mathbb{Z}^{d}$, we prove that (a) for each$G_{\unicode[STIX]{x1D6FF}}$-subset$\unicode[STIX]{x1D6E5}$of$P(\mathfrak{g})$and a countable subset$D\subset \unicode[STIX]{x1D6E5}$, there is a rank-one action$T$such that$D\subset {\mathcal{R}}(T)\subset \unicode[STIX]{x1D6E5}$and (b)${\mathcal{R}}(T)=P(\mathfrak{g})$for a generic infinite measure-preserving action$T$of$\unicode[STIX]{x1D6E4}$. This partly answers a question from a recent paper by Johnson and Şahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where$G$is the Heisenberg group$H_{3}(\mathbb{R})$and$\unicode[STIX]{x1D6E4}=H_{3}(\mathbb{Z})$, a rank-one$\unicode[STIX]{x1D6E4}$-action$T$is constructed for which${\mathcal{R}}(T)$is not invariant under the natural ‘adjoint’$G$-action.

Tangent Dirac Structures and Poisson Dirac Submanifolds

The local equations that characterize the submanifolds N of a Dirac manifold M is an isotropic (coisotropic) submanifold of TM endowed with the tangent Dirac structure. In the Poisson case which is a result of Xu: the submanifold N has a normal bundle which is a coisotropic submanifold of TM with the tangent Poisson structure if and only if N is a Dirac submanifold. In this paper we have proved a theorem in the general Poisson case that the fixed point set MG has a natural induced Poisson structure that implies a Poisson-Dirac submanifolds, where G×M?M be a proper Poisson action. DOI: http://dx.doi.org/10.3329/dujs.v62i1.21955 Dhaka Univ. J. Sci. 62(1): 21-24, 2014 (January)

On homotopy Poisson actions and reduction of symplectic Q-manifolds

We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.

The higher flows of harmonic maps and an application to the Virasoro action

One characteristic of soliton equations is their infinite number of conservation laws. These conservation laws give rise to a hierarchy of partial differential equations. One example is the AKNS hierarchy. Terng and Uhlenbeck, Poisson Actions and Scattering Theory for Integrable Systems, Surveys in Differential Geometry: Integrable Systems (A supplement to J. Diff. Geom.), 4 (1998) described the conservation laws that give rise to the AKNS hierarchy. This hierarchy can also be realized in a loop group setting, where the equations are called the positive flows. The name arises from the power of λ which generates each flow. Terng (Loop Groups and Integrable Systems, Preliminary notes) also showed that in some sense, the harmonic map equation arises as a −1-flow in this hierarchy. The harmonic map equation also has a hierarchy of negative flows. However, these are not differential equations like the positive flows. It remained to be asked how the positive and negative flows relate and if they commute. The answer to the latter question is in the positive. As for the first question, the two hierarchies are part of one larger hierarchy. This was solved by using a more complex loop group and explicitly determining the loop algebra splitting. This splitting has applications to physics and is applied to an example of Schwarz involving the Virasoro action in Sec. VIII.

Topology of the Liouville foliation on a 2-sphere in the Dullin-Matveev integrable case

The paper is concerned with the study of the topology of the Liouville foliations of the Dullin-Matveev integrable case. The critical point set of the Hamiltonian is found, the types of isoenergy surfaces are calculated, the non-degeneracy conditions are verified, the types of non-degenerate points of the Poisson action are determined, the moment map is investigated and the bifurcation diagram is constructed. A test for the Bott property is verified by numerical simulation. The indices of critical circles, the bifurcation types and the rough molecules are found. The rough Liouville classification of this integrable case is virtually accomplished as a result.Bibliography: 24 titles.

Poisson Action and Formality

Using a formality on a Poisson manifold, we construct a star product and for each Poisson vector field a derivation of this star product. Starting with a Poisson action of a Lie group, we are able under a natural cohomological assumption to define a representation of its Lie algebra in the space of derivations of the star product. Finally, we use these results to define some generically tangential star products on duals of Lie algebra as in [1] but in a more realistic context.

Poisson Actions up to Homotopy and their Quantization

Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmetries up to homotopy of the quantized algebras of functions.

Vanishing of the entropy pseudonorm for certain integrable systems

We introduce the notion of entropy pseudonorm for an action of R and prove that it vanishes for the group actions associated with a big class of integrable Hamiltonian systems. 1. Entropy pseudonorm Let W be a smooth manifold and Φ : (R, +) → Diff(W ) a smooth action on it. Assume there exists a compact Φ-invariant exhaustion of W . Define the following function on R (where htop is the topological entropy): ρΦ(v) = htop(Φ(v)), v ∈ R. This function is a pseudonorm on R (ρΦ(v) is well-defined because with our hypothesis the entropy hd of [Bo] does not depend on the distance function d, homogeneity is standard and the triangle inequality follows from the Hu formula [H]). We call ρΦ the entropy pseudonorm. We will investigate it in the case of the Poisson action corresponding to an integrable Hamiltonian system on a symplectic manifold (W , ω). Namely, let (W , ω) possess pair-wise Poisson commuting functions I1, I2, . . . , In, which are functionally independent almost everywhere. Denote by φi the time τ shift along the Hamiltonian vector field of the function Ii. The maps φi commute and therefore generate the Poisson action of the group (R, +), Φ(τ1, . . . , τn) def = φ1 1 ◦ · · · ◦ φn n : W 2n → W , with the corresponding momentum map Ψ = (I1, . . . , In) : W 2n → R, see [A]. The entropy pseudonorm ρΦ vanishes in the following important cases: − Williamson-Vey-Eliasson-Ito non-degenerate singularities [E, I]; − Taimanov non-degeneracy condition [T]. In the first case vanishing of topological entropy of the Hamiltonian flow was proved in [P2], in the second case in [T]. Since there is nothing special about the Hamiltonian in these situations, it can be changed to any of the integrals and ρΦ ≡ 0 follows. Also in [P1, BP]) vanishing of htop was proven for the cases: − Systems integrable with periodic integrals; − Collectively integrable systems (the definition is in [GS]). It is not difficult to see that in both cases the entropy pseudonorm ρΦ vanishes as well. Note that Liouville integrability does not imply vanishing of topological entropy, see [BT] (more examples in [Bu]). For these examples the entropy pseudonorm is degenerate, but it is possible to construct integrable examples [K] such that ρΦ is a norm. In the present paper we prove vanishing of the entropy pseudonorm for another class of integrable systems. These systems were recently actively studied in mathematical physics in the