Related papers: Integrable Hamiltonian systems on the symplectic r…

We construct integrable Hamiltonian systems with Lie bialgebras $({\bf g} , {\bf \tilde{g}})$ of the bi-symplectic type for which the Poisson-Lie groups ${\bf G}$ play the role of the phase spaces, and their dual Lie groups ${\bf {\tilde…

We construct a hierarchy of integrable systems whose Poisson structure corresponds to the BMS$_{3}$ algebra, and then discuss its description in terms of the Riemannian geometry of locally flat spacetimes in three dimensions. The analysis…

Phase space of a characteristic Hamiltonian system is a symplectic leaf of a factorizable Poisson Lie group. Its Hamiltonian is a restriction to the symplectic leaf of a function on the group which is invariant with respect to conjugations.…

Constrained Hamiltonian systems fall into the realm of presymplectic geometry. We show, however, that also Poisson geometry is of use in this context. For the case that the constraints form a closed algebra, there are two natural Poisson…

Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost…

A symplectic integration of a Poisson manifold $(M,\Lambda)$ is a symplectic groupoid $(\Gamma,\eta)$ which realizes the given Poisson manifold, i.e. such that the space of units $\Gamma_0$ with the induced Poisson structure $\Lambda_0$ is…

The solutions to the Euler-Poisson equations are geodesic lines of $SO(3)$ manifold with the metric determined by the inertia tensor. However, the Poisson structure on the corresponding symplectic leaf does not depend on the inertia tensor.…

For the integrable system on $e(3,2)$ found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These…

Using tools from Dirac geometry and through an explicit construction, we show that every Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which…

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and…

We study metric-compatible Poisson structures in the semi-classical limit of noncommutative emergent gravity. Space-time is realized as quantized symplectic submanifold embedded in R^D, whose effective metric depends on the embedding as…

We show that the $m$-dimensional Euler--Manakov top on $so^*(m)$ can be represented as a Poisson reduction of an integrable Hamiltonian system on a symplectic extended Stiefel variety $\bar{\cal V}(k,m)$, and present its Lax representation…

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold…

For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral…

We classify all the quadratic Poisson structures on $so^*(4)$ and $e^*(3)$, which have the same foliation by symplectic leaves as the canonical Lie-Poisson tensors. The separated variables for the some of the corresponding bi-integrable…

Symplectic and Poisson structures of certain moduli spaces/Huebschmann,J./ Abstract: Let $\pi$ be the fundamental group of a closed surface and $G$ a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a…

We study the Poisson geometrical formulation of quantum mechanics for finite dimensional mixed and pure states. Equivalently, we show that quantum mechanics can be understood in the language of classical mechanics. We review the symplectic…

Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in…

We propose an exactly solvable Hamiltonian for topological phases in $3+1$ dimensions utilising ideas from higher lattice gauge theory, where the gauge symmetry is given by a finite 2-group. We explicitly show that the model is a…

We consider higher generalizations of both a (twisted) Poisson structure and the equivariant condition of a momentum map on a symplectic manifold. On a Lie algebroid over a (pre-)symplectic and (pre-)multisymplectic manifold, we introduce a…