Related papers: Compatible Poisson Structures of Toda Type Discret…
This paper is mainly a review of the multi--Hamiltonian nature of Toda and generalized Toda lattices corresponding to the classical simple Lie groups but it includes also some new results. The areas investigated include master symmetries,…
We compute the Poisson bracket relations for the monodromy matrix of the auxiliary linear problem. If the basic Poisson bracket relations of the model contain derivatives, this computation leads to a peculiar type of symmetry breaking which…
A family of solutions of the Jacobi PDEs is investigated. This family is $n$-dimensional, of arbitrary nonlinearity and can be globally analyzed (thus improving the usual local scope of Darboux theorem). As an outcome of this analysis it is…
We classify all fundamental integrable spin chains with two-dimensional local Hilbert space which have regular R-matrices of difference form. This means that the R-matrix underlying the integrable structures is of the form R(u,v)=R(u-v) and…
We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified(or additional) terms because of a…
We propose realizations of the Poisson structures for the Lax representations of three integrable $n$-body peakon equations, Camassa--Holm, Degasperis--Procesi and Novikov. The Poisson structures derived from the integrability structures of…
In this paper, we show that the twisted Poincar\'e duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson…
We study Poisson symmetric spaces of group type with Cartan subalgebra "adapted" to the Lie cobracket.
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…
The Poisson structure arising in the Hamiltonian approach to the rational Gaudin model looks very similar to the so-called modified Reflection Equation Algebra. Motivated by this analogy, we realize a braiding of the mentioned 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…
In this paper we define a family of systems which have similarities with the Toda lattice. We construct two Lax pair representations and the associate Poisson structures for these systems. These systems lie between the classical Toda…
In this paper we classify all four dimensional real Lie bialgebras of symplectic type. The classical r- matrices for these Lie bialgebras and Poisson structures on all of the related four dimensional Poisson-Lie groups are also obtained.…
The dispersionless KP and Toda hierarchies possess an underlying twistorial structure. A twistorial approach is partly implemented by the method of Riemann-Hilbert problem. This is however still short of clarifying geometric ingredients of…
A new Lax representation for the Bogoyavlensky lattice is found, its $r$--matrix interpretation is elaborated. The $r$--matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of associative…
We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of…
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 investigate the bi-Hamiltonian structures associated with constrained dispersionless modified KP hierarchies which are constructed from truncations of the Lax operator of the dispersionless modified KP hierarchy. After transforming their…
A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with…
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized…