Related papers: Multiplicative integrable models from Poisson-Nije…

The notion of Poisson quasi-Nijenhuis manifold generalizes that of Poisson-Nijenhuis manifold. The relevance of the latter in the theory of completely integrable systems is well established since the birth of the bi-Hamiltonian approach to…

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…

A new method to construct Hamiltonian functions in involution is presented. We show that on left-symmetric algebras a Nijenhuis-tensor is given in a natural manner by the usual right-multiplication. Furthermore we prove that symplectic…

This is the second paper of a series dedicated to the study of Poisson structures of compact types (PMCTs). In this paper, we focus on regular PMCTs, exhibiting a rich transverse geometry. We show that their leaf spaces are integral affine…

We introduce poly-symplectic groupoids, which are natural extensions of symplectic groupoids to the context of poly-symplectic geometry, and define poly-Poisson structures as their infinitesimal counterparts. We present equivalent…

We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's…

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…

Any multiplicative quiver variety is endowed with a Poisson structure constructed by Van den Bergh through reduction from a Hamiltonian quasi-Poisson structure. The smooth locus carries a corresponding symplectic form defined by Yamakawa…

While the construction of symplectic integrators for Hamiltonian dynamics is well understood, an analogous general theory for Poisson integrators is still lacking. The main challenge lies in overcoming the singular and non-linear geometric…

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…

In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results…

We consider the problem of the symplectic realization of a Poisson-Nijenhuis manifold. By applying a new technique developed by M. Crainic and I. Marcut for the study of the above problem in the case of a Poisson manifold, we establish the…

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…

In this paper, we introduce right-invariant (similarly, left-invariant) Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called $(\Lambda , \mathbf{n})-$structures. We present a mutual correspondence…

We show that various notions of integrability for Poisson brackets are all equivalent, and we give the precise obstructions to integrating Poisson manifolds. We describe the integration as a symplectic quotient, in the spirit of the Poisson…

We generalize Poisson-Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the…

We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows…

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an $h$-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension $2n\geq 10$…

A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is proposed. Each of the subalgebras provides a complete set of integrals in involution with respect to the Sklyanin bracket. Using different representations of the…

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be…