Related papers: A Poisson bracket on the space of Poisson structur…

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 give a self-contained algebraic description of a formal symplectic groupoid over a Poisson manifold M. To each natural star product on M we then associate a canonical formal symplectic groupoid over M. Finally, we construct a unique…

The Poisson structures for 3D systems possessing one constant of motion can always be constructed from the solution of a linear PDE. When two constants of the motion are available the problem reduces to a quadrature and the structure…

Topological constraints play a key role in the self-organizing processes that create structures in macro systems. In fact, if all possible degrees of freedom are actualized on equal footing without constraint, the state of "equipartition"…

A covariant Poisson bracket and an associated covariant star product in the sense of deformation quantization are defined on the algebra of tensor-valued differential forms on a symplectic manifold, as a generalization of similar structures…

We discover Hamiltonian structure of the complex Monge-Amp`ere equation when written in a first order two-component form. We present Lagrangian and Hamiltonian functions, a symplectic form and the Hamiltonian operator that determines the…

We study a modification of Poisson geometry by a closed 3-form. Just as for ordinary Poisson structures, these "twisted" Poisson structures are conveniently described as Dirac structures in suitable Courant algebroids. The additive group of…

It is argued that renormalisation group flow can be interpreted as being a Hamiltonian vector flow on a phase space which consists of the couplings of the theory and their conjugate \lq\lq momenta", which are the vacuum expectation values…

We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to…

According to the Weinstein splitting theorem, any Poisson manifold is locally, near any given point, a product of a symplectic manifold with another Poisson manifold whose Poisson structure vanishes at the point. Similar splitting results…

We demonstrate the common bihamiltonian nature of several integrable systems. The first one is an elliptic rotator that is an integrable Euler-Arnold top on the complex group GL(N) for any $N$, whose inertia ellipsiod is related to a choice…

We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ to a Poisson manifold $(M,\pi)$ starting from an algebroid bracket of differential forms. We use these brackets to generate Poisson structures…

Connecting ideas of geometric formulation of quantum mechanics with new results in symplectic geometry a new approach to geometrical quantization procedure is proposed. As a first result we verify that the correspondence between "classical"…

We present a class of Poisson structures on trivial extension algebras which generalize some known structures induced by Poisson modules. We show that there exists a one-to-one correspondence between such a class of Poisson structures and…

We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in…

In this paper we study the problem of Hamiltonization of nonholonomic systems from a geometric point of view. We use gauge transformations by 2-forms (in the sense of Severa and Weinstein [29]) to construct different almost Poisson…

In this paper we continue our analysis of the stationary flows of $M$ component, coupled KdV (cKdV) hierarchies and their modifications. We describe the general structure of the $t_1$ and $t_2$ flows, using the case $M=3$ as our main…

The paper investigates the Poisson structures associated with dynamical systems of the heavenly type, focusing on the Mikhalev-Pavlov and Pleba\'nski equation. The dynamical system is represented as a Hamiltonian system on a functional…

We consider an arbitrary Dubrovin-Novikov bracket of degree $k$, namely a homogeneous degree $k$ local Poisson bracket on the loop space of a smooth manifold $M$ of dimension $n$, and show that $k$ connections, defined by explicit linear…

We analyze the symplectic structure on the dressing group in the \shG\, model by calculating explicitly the Poisson bracket $\{g\x g\}$ where $g$ is the \dg\, element which creates a generic one soliton solution from the vacuum. Our result…