相关论文: Infinite hierarchies of Poisson structures for int…

We establish a new representation of the infinite hierarchy of Pois- son brackets (PB) for the open Toda lattice in terms of its spectral curve. For the classical Poisson bracket (PB) we give a representation in the form of a contour…

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…

In a previous work we have introduced the concept of quasi-integrable quantum system. In the present one we determine sufficient conditions under which, given an integrable classical system, it is possible to construct a quasi-integrable…

In this paper, we consider Hamiltonian structures of hydrodynamic type and some of their generalizations. In particular, we discuss the questions concerning the structure and special forms of the corresponding Poisson brackets and the…

In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a…

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 introduce and study suitable Poisson structures for four dimensional maps derived as lifts and specific periodic reductions of integrable lattice equations. These maps are Poisson with respect to these structures and the corresponding…

We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in n generators. Explicit formulas for Casimir elements of this elliptic Poisson…

We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We also give explicit constructions of Lie groupoids…

We describe a method for computing Casimir invariants that is applicable to both finite and infinite-dimensional Poisson brackets. We apply the method to various finite and infinite-dimensional examples, including a Poisson bracket…

We study complex projective surfaces admitting a Poisson structure. We prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface.

We consider the questions connected with the Hamiltonian properties of the Whitham equations in case of several spatial dimensions. An essential point of our approach here is a connection of the Hamiltonian structure of the Whitham system…

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 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 develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we construct Gibbons-Tsarev systems associated to moduli space of algebraic…

We consider a special class of linear and quadratic Poisson brackets related to ODE systems with matrix variables. We investigate general properties of such brackets, present an example of a compatible pair of quadratic and linear brackets…

Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail.

We give a normal form for families of 3-dimensional Poisson structures. This allows us to classify singularities with nonzero 1-jet and typical bifurcations. The Appendix contains corollaries on classification of families of integrable…

In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattice, whose continuum limit is the AKNS hierarchy. In contrast with other differential-difference versions of the AKNS…

We formulate some conjectures that relates semisimple Frobenius manifolds, their spectral curves and integrable hierarchies.