相关论文: Integrability of Poisson brackets

We study isomorphism classes of symplectic dual pairs P <- S -> P-, where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For…

In recent years methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this note it is shown that the latter method is actually…

For a class of symplectic manifolds, we introduce a functional which assigns a real number to any pair of continuous functions on the manifold. This functional has a number of interesting properties. On the one hand, it is Lipschitz with…

It is known that any Poisson manifold can be embedded into a bigger space which admites a description in terms of the canonical Poisson structure, i.e., Darboux coordinates. Such a procedure is known as a symplectic realization and has a…

Scattering symplectic manifolds are (closed) manifolds with a mildly degenerate Poisson structure. In particular they can be viewed as symplectic structures on a Lie algebroid which is almost everywhere isomorphic to the tangent bundle. In…

We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is…

We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any…

Using a Poisson bracket representation, in 3D, of the Lie algebra $\mathfrak{sl}(2)$, we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras…

In this short survey, we describe our approach for constructing hierarchies of Poisson brackets for classical integrable systems using its' spectral curves.

We consider a class of complex manifolds constructed as multiplicative quiver varieties associated with a cyclic quiver extended by an arbitrary number of arrows starting at a new vertex. Such varieties admit a Poisson structure, which is…

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an…

This note gives an overview on the construction of symplectic groupoids as reduced phase spaces of Poisson sigma models and its generalization in the infinite dimensional setting (before reduction).

We suggest a homotopical description of the Poisson bracket invariants for tuples of closed sets in symplectic manifolds. It implies that these invariants depend only on the union of the sets along with topological data.

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…

Some Poisson structures do admit resolutions by symplectic manifolds of the same dimension. We give examples and simple conditions under which such resolutions can not exist.

Poisson brackets admit infinitesimal symmetries which are encoded using oriented graphs; this construction is due to Kontsevich (1996). We formulate several open problems about combinatorial and topological properties of the graphs…

We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable…

In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding…

We describe the Poisson ideals and attached symplectic geometry of a cluster algebra with compatible Poisson structure. We apply these results to determine the spectrum of a quantum cluster algebra. As an application, we describe the…

We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular…