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Alan Weinstein - One of the best experts on this subject based on the ideXlab platform.

  • lie Poisson Structure on some Poisson lie groups
    Journal of the American Mathematical Society, 1992
    Co-Authors: Viktor L Ginzburg, Alan Weinstein
    Abstract:

    Poisson Lie groups appeared in the work of Drinfel'd (see, e.g., [Drl, Dr2]) as classical objects corresponding to quantum groups. Going in the other direction, we may say that a Poisson Lie group is a group of symmetries of a phase space that are allowed to "twist," in a certain sense, the symplectic or Poisson Structure. The Poisson Structure on the group controls this twisting in a precise way. Quantizing both the phase space and the symmetries, one may obtain a quantum group acting on a quantum phase space. In recent work, Lu and Ratiu [LR] used so-called standard Poisson Structures on a compact semisimple Lie group K and on its Poisson dual K* in order to give a new proof of the nonlinear convexity theorem of Kostant [Ko]. Their method is analogous to the famous symplectic proof of the linear convexity theorem given by Atiyah [A] and Guillemin and Stemnberg [GS]. The nonlinear convexity theorem, like the linear one, follows from a very general result on convexity of the image of the momentum map [A, GS]; however, in [LR] it is applied not to a coadjoint orbit in t*, but to a symplectic leaf in K* . The main result of this paper is that the standard Poisson Structure on the Poisson dual K* to a compact semisimple Poisson Lie group K is actually isomorphic to the linear one on t* . This theorem seems to be related to some facts in the theory of quantum groups. Namely, (for generic q) the universal enveloping algebra U(t) and its quantum deformation Uq(t) are isomorphic as algebras (though not as coalgebras, of course, since the quantum version is not cocommutative). In particular, there is a bijective correspondence between their representations. A direct connection between our work and its quantum analogues, though, is still to be found. The present work supplies a positive answer to Question 5.1 in [LR] and strongly depends on that paper. To simplify reading, we keep the notations of [LR] wherever possible, on one hand, but give all necessary definitions, on the other. Our work is also related to [Du], in which the nonlinear convexity theorem is reduced to the linear one by a deformation argument not unlike the one that we use in ?5. The paper is organized as follows. In ?2 we define Poisson Lie groups, discuss

Paolo Lorenzoni - One of the best experts on this subject based on the ideXlab platform.

Francesco Bonechi - One of the best experts on this subject based on the ideXlab platform.

  • the Poisson sigma model on closed surfaces
    Journal of High Energy Physics, 2012
    Co-Authors: Francesco Bonechi, Alberto S Cattaneo, Pavel Mnev
    Abstract:

    Using methods of formal geometry, the Poisson sigma model on a closed surface is studied in perturbation theory. The effective action, as a function on vacua, is shown to have no quantum corrections if the surface is a torus or if the Poisson Structure is regular and unimodular (e.g., symplectic). In the case of a Kahler Structure or of a trivial Poisson Structure, the partition function on the torus is shown to be the Euler characteristic of the target; some evidence is given for this to happen more generally. The methods of formal geometry introduced in this paper might be applicable to other sigma models, at least of the AKSZ type.

  • Poisson Sigma Model on the Sphere
    Communications in Mathematical Physics, 2008
    Co-Authors: Francesco Bonechi, Maxim Zabzine
    Abstract:

    We evaluate the path integral of the Poisson sigma model on the sphere and study the correlators of quantum observables. We argue that for the path integral to be well-defined the corresponding Poisson Structure should be unimodular. The construction of the finite dimensional BV theory is presented and we argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kahler manifold we discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson Structure the semiclassical result for the correlators is indeed the full quantum result.

Pavel Mnev - One of the best experts on this subject based on the ideXlab platform.

  • the Poisson sigma model on closed surfaces
    Journal of High Energy Physics, 2012
    Co-Authors: Francesco Bonechi, Alberto S Cattaneo, Pavel Mnev
    Abstract:

    Using methods of formal geometry, the Poisson sigma model on a closed surface is studied in perturbation theory. The effective action, as a function on vacua, is shown to have no quantum corrections if the surface is a torus or if the Poisson Structure is regular and unimodular (e.g., symplectic). In the case of a Kahler Structure or of a trivial Poisson Structure, the partition function on the torus is shown to be the Euler characteristic of the target; some evidence is given for this to happen more generally. The methods of formal geometry introduced in this paper might be applicable to other sigma models, at least of the AKSZ type.

Holger R Dullin - One of the best experts on this subject based on the ideXlab platform.

  • Poisson Structure of the three dimensional euler equations in fourier space
    Journal of Physics A, 2019
    Co-Authors: Holger R Dullin, J D Meiss, Joachim Worthington
    Abstract:

    We derive a simple Poisson Structure in the space of Fourier modes for the vorticity formulation of the Euler equations on a three-dimensional periodic domain. This allows us to analyse the Structure of the Euler equations using a Hamiltonian framework. The Poisson Structure is valid on the divergence free subspace only, and we show that using a projection operator it can be extended to be valid in the full space. We then restrict the simple Poisson Structure to the divergence-free subspace on which the dynamics of the Euler equations take place, reducing the size of the system of ODEs by a third. The projected and the restricted Poisson Structures are shown to have the helicity as a Casimir invariant. We conclude by showing that periodic shear flows in three dimensions are equilibria that correspond to singular points of the projected Poisson Structure, and hence that the usual approach to study their nonlinear stability through the Energy-Casimir method fails.

  • the lie Poisson Structure of the reduced n body problem
    Nonlinearity, 2013
    Co-Authors: Holger R Dullin
    Abstract:

    The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. One novelty of our approach is that we do not fix the centre of mass but rather use a momentum shifting trick to change the kinetic part of the Hamiltonian to arrive at a new, dynamically equivalent Hamiltonian which is easier to reduce. As a result we obtain a reduced system with a Lie–Poisson Structure which is isomorphic to , independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. This splitting allows us to construct a Poisson integrator for the reduced n-body problem which is efficient away from collisions for n = 3. In particular, we could integrate the figure eight orbit in 18 time steps.

  • the lie Poisson Structure of the reduced n body problem
    arXiv: Dynamical Systems, 2012
    Co-Authors: Holger R Dullin
    Abstract:

    The classical n-body problem in d-dimensional space is invariant under the Galilean symmetry group. We reduce by this symmetry group using the method of polynomial invariants. As a result we obtain a reduced system with a Lie-Poisson Structure which is isomorphic to sp(2n-2), independently of d. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced n-body problem using a splitting method.