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 integrability. In this note, we discuss the relevance of the notion of Poisson quasi-Nijenhuis manifold in the context of finite-dimensional integrable systems. Generically (as we show by an example with $3$ degrees of freedom) the Poisson quasi-Nijenhuis structure is largely too general to ensure Liouville integrability of a system. However, we prove that the closed (or periodic) $n$-particle Toda lattice can be framed in such a geometrical structure, and its well-known integrals of the motion can be obtained as spectral invariants of a "quasi-Nijenhuis recursion operator", that is, a tensor field $N$ of type $(1,1)$ defined on the phase space of the lattice. This example and some of its generalizations are used to understand whether one can define in a reasonable sense a notion of {\em involutive\} Poisson quasi-Nijenhuis manifold. A geometrical link between the open (or non periodic) and the closed Toda systems is also framed in the context of a general scheme connecting Poisson quasi-Nijenhuis and Poisson-Nijenhuis manifolds.
We present a result which allows us to deform a Poisson-Nijenhuis manifold into a Poisson quasi-Nijenhuis manifold by means of a closed 2-form. Under an additional assumption, the deformed structure is also Poisson-Nijenhuis. We apply this result to show that the canonical Poisson-Nijenhuis structure on R^2n gives rise to both the Poisson-Nijenhuis structure of the open (or non periodic) n-particle Toda lattice, introduced by Das and Okubo [6], and the Poisson quasi-Nijenhuis structure of the closed (or periodic) n-particle Toda lattice, described in our recent work [7].
We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.
We explore variational Poisson-Nijenhuis structures on nonlinear PDEs and establish relations between Schouten and Nijenhuis brackets on the initial equation with the Lie bracket of symmetries on its natural extensions (coverings). This approach allows to construct a framework for the theory of nonlocal structures.
We show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie algebroids is motivated by the geometrical study of some reduced bi-Hamiltonian systems. An explicit example of reduction of a Poisson-Nijenhuis Lie algebroid is also provided.
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 study of the Calogero-Moser and Gibbons-Hermsen integrable systems. In the former case, we give a new interpretation of the bihamiltonian reduction performed in [3].
In this paper, we introduce right-invariant (similarly, left-invariant) Poisson-Nijenhuis Structures on Lie groupoids and their infinitesimal counterparts as called $(Λ, \mathbf{n})-$structures. We present a mutual correspondence between $(Λ,\mathbf{n})-$structures on Lie algebroids with Poisson-Nijenhuis structures $(Π, \mathbf{N})$ on their Lie groupoids under some conditions. Also, we will construct Poisson-Nijenhuis pair groupoids as an example from the invariance's point of view.
The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.
An involutivity theorem for a class of Poisson quasi-Nijenhuis manifolds
Universidade Federal Rural de Pernambuco（伯南布哥联邦农村大学）；Università di Milano-Bicocca（米兰比可卡大学）；Universidade de São Paulo（圣保罗大学）；Università di Bergamo（贝加莫大学）；INFN, Sezione di Milano-Bicocca（意大利国家核物理研究所米兰比可卡分部）
We revisit our earlier work on the AKSZ formulation of topological sigma model on generalized complex manifolds, or Hitchin model. We show that the target space geometry geometry implied by the BV master equations is Poisson--quasi--Nijenhuis geometry recently introduced and studied by Stiénon and Xu (in the untwisted case). Poisson--quasi--Nijenhuis geometry is more general than generalized complex geometry and comprises it as a particular case. Next, we show how gauging and reduction can be implemented in the Hitchin model. We find that the geometry resulting form the BV master equation is c…
We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis…
We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that $r$-$n$ structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two compatible r-matrices from which one is invertible determine an $r$-$n$ structure. We classify, up to a natural equivalence, all $r$-matrices and all $r$-$n$ structures with invertible $r$ on four-dimensional symplectic real Lie algebras. The result i…
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 Lie-algebras carry the structure of a Poisson-Nijenhuis manifold.
We study {\em right-invariant (resp., left-invariant) Poisson quasi-Nijenhuis structures} on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called {\em r-qn structures} on the corresponding Lie algebra $\mathfrak g$. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all $r$-$qn$ structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between $r$-$qn$ structures and the generalized complex structures on the Li…
Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis structure from a given type (1,1) tensor field J on Q. It is argued that the complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to co-tangent manifold of Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which…
We discuss the role of Poisson-Nijenhuis geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces and studied in [arXiv:1503.07339].
In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce ON-structures on bimodules over pre-Lie algebras. We show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. We study solutions of the strong Maurer-Cartan equation on the twilled pre-Lie algebra associated to an O-operator, which gives rise to a pair of ON-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra g are corresponding to ON-structures on the bimodule $(\mathfrak g^*;\mathrm{ad}^*,-R^*)$, and $KVΩ$-structures …
We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $Π_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $Π_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebra…
In this paper, for a Jacobi algebroid $A$, by introducing the notion of Jacobi quasi-Nijenhuis algebroids, which is a generalization of Poisson quasi-Nijenhuis manifolds introduced by Stiénon and Xu, we study generalized complex structures on the Courant-Jacobi algebroid $A\oplus A^*$, which unifies generalized complex (contact) structures on an even(odd)-dimensional manifold.
The Schouten-Nijenhuis bracket is generalized for the superspace case and for the Poisson brackets of opposite Grassmann parities. Quite a number of generalizations for the differential analog of the special Yang-Baxter equations is also proposed.
In this paper, we introduce the notion of Koszul-Vinberg-Nijenhuis structures on a left-symmetric algebroid as analogues of Poisson-Nijenhuis structures on a Lie algebroid, and show that a Koszul-Vinberg-Nijenhuis structure gives rise to a hierarchy of Koszul-Vinberg structures. We introduce the notions of ${\rm KVΩ}$-structures, pseudo-Hessian-Nijenhuis structures and complementary symmetric $2$-tensors for Koszul-Vinberg structures on left-symmetric algebroids, which are analogues of ${\rm PΩ}$-structures, symplectic-Nijenhuis structures and complementary $2$-forms for Poisson structures. We…
We connect two a priori unrelated topics, theory of geodesically equivalent metrics in differential geometry, and theory of compatible infinite dimensional Poisson brackets of hydrodynamic type in mathematical physics. Namely, we prove that a pair of geodesically equivalent metrics such that one is flat produces a pair of such brackets. We construct Casimirs for these brackets and the corresponding commuting flows. There are two ways to produce a large family of compatible Poisson structures from a pair of geodesically equivalent metrics one of which is flat. One of these families is $(n+1)(n+…
In this note we construct a canonical lifting of arbitrary Poisson structures on a manifold to its algbera of densities. Using this construction we proceed to classify all extensions of a fixed structure on the original manifold to its algebra of densities. The question is analogous to the problem studied by H.M.Khudaverdian and Th.Voronov for odd Poisson structures and differential operators. Although the questions are similar the results are distinctly marked, namely in the case of even Poisson structures there always exists a lift which is naturally defined. The proof of this result bears a…
We discuss the relationship between the multiple Hamiltonian structures of the generalized Toda lattices and that of the generalized Volterra lattices. We use a symmtery approach for Poisson structures that generalizes the Poisson involution theorem.
After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds. A review of the spinor approach to the modular class concludes the paper.
An algebra isomorphism between algebras of matrices and difference operators is used to investigate the discrete integrable hierarchy. We find local and non-local families of R-matrix solutions to the modified Yang-Baxter equation. The three R-theoretic Poisson structures and the Suris quadratic bracket are derived. The resulting family of bi-Poisson structures include a seminal discrete bi-Poisson structure of Kupershmidt at a special value.
It is well known that the compatible linear and quadratic Poisson brackets of the full symmetric and of the standard open Toda lattices are restrictions of linear and quadratic $r$-matrix Poisson brackets on the associative algebra ${\mathrm{gl}}(n,{\mathbb{R}})$. We here show that the quadratic bracket on ${\mathrm{gl}}(n,{\mathbb{R}})$, corresponding to the $r$-matrix defined by the splitting of ${\mathrm{gl}}(n,{\mathbb{R}})$ into the direct sum of the upper triangular and orthogonal Lie subalgebras, descends by Poisson reduction from a quadratic Poisson structure on the cotangent bundle $T…
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 integral of some special Abelian differential (meromorphic one-form) on the spectral curve. All higher brackets of the infinite hierarchy are obtained by multiplication of the one-form by a power of the spectral parameter.
A family of Poisson structures, parametrised by an arbitrary odd periodic function $ϕ$, is defined on the space $\cW$ of twisted polygons in $\RR^ν$. Poisson reductions with respect to two Poisson group actions on $\cW$ are described. The $ν=2$ and $ν=3$ cases are discussed in detail and the general $ν$ case in less detail. Amongst the Poisson structures arising in examples are to be found the lattice Virasoro structure, the second Toda lattice structure and some extended Toda lattice structures. A general result is proved showing that, for any $ν$, to certain concrete choices of $ϕ$ there cor…
For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral curves and sheaves supported on them; (c) Symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson…
In this paper we prove superintegrability of Hamiltonian systems generated by functions on $K\backslash G/K$, restriced to a symplectic leaf of the Poisson variety $G/K$, where $G$ is a simple Lie group with the standard Poisson Lie structure, $K$ is the subgroup of fixed points with respect to the Cartan involution.
We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle $A\to X$ is shown to be equivalent to a matched pair of complex Lie algebroids $(T^{0,1}X,A^{1,0})$, in the sense of Lu. The holomorphic Lie algebroid cohomology of $A$ is isomorphic to the cohomology of the elliptic Lie alge…
The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.
We extend the construction of the relativistic Toda chains as integrable systems on the Poisson submanifolds in Lie groups beyond the case of A-series. For the simply-laced case this is just a direct generalization of the well-known relativistic Toda chains, and we construct explicitly the set of the Ad-invariant integrals of motion on symplectic leaves, which can be described by the Poisson quivers being just the blown up Dynkin diagrams. We also demonstrate how to get the set of "minimal" integrals of motion, using the co-multiplication rules for the corresponding Lie algebras. In the non si…
In this paper we will review a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We will discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the \bih structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda L…
Structure and properties of families of critical points for classes of functions $W(z,\bar{z})$ obeying the elliptic Euler-Poisson-Darboux equation $E(1/2,1/2)$ are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented There are the extended dispersionless Toda/nonlinear Schrödinger hierarchies, the "inverse" hierarchy and equations associated with the real-analytic Eisenstein series $E…
Let H_T=C[T,T^{-1}] be the Hopf algebra of symmetries of a lattice of rank 1, or equivalently, H_T is the group algebra of a free Abelian group with one generator T. We construct conformal algebras, vertex Poisson algebras and vertex algebras with H_T as symmetry. For example, the Hamiltonian structure for the infinite Toda lattice gives rise to an H_T-vertex Poisson structure on a free difference algebra. Examples of H_T-vertex algebras are constructed from representations of a class of infinite dimensional Lie algebras related to H_T in the same way loop algebras are related to the Hopf alge…
Using exact results from the theory of completely integrable systems of the Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann approximation.
We discuss trivial deformations of the canonical Poisson brackets associated with the Toda lattices, relativistic Toda lattices, Henon-Heiles, rational Calogero-Moser and Ruijsenaars-Schneider systems and apply one of these deformations to construct a new trivial family of noncommutative integrable systems.
We study the deep connection between integrable models and Poisson-Lie T-duality working on a finite dimensional example constructed on SL(2,C) and its Iwasawa factors SU(2) and B. We shown the way in which Adler-Kostant-Symes theory and collective dynamics combine to solve the equivalent systems from solving the factorization problem of an exponential curve in SL(2,C). It is shown that the Toda system embraces the dynamics of the systems on SU(2) and B.
We define an integrable hamiltonian system of Toda type associated with the real Lie algebra $\so{p}{q}$. As usual there exists a periodic and a non-periodic version. We construct, using the root space, two Lax pair representations and the associated Poisson tensors. We prove Liouville integrability and examine the multi-hamiltonian structure. The system is a projection of a canonical $A_n$ type Toda lattice via a Flaschka type transformation. It is also obtained via a complex change of variables from the classical Toda lattice.
In this paper, we present a general scheme to construct integrable systems based on realization in the coboundary dynamical Poisson groupoids of Etingof and Varchenko. We also present a factorization method for solving the Hamiltonian flows. To illustrate our scheme and factorization theory, we consider a family of hyperbolic spin Ruijsenaars-Schneider models related to affine Toda field theories and solve the equations of motion in a simple case.
We study completely integrable systems attached to Takiff algebras $\mathfrak{g}_N$, extending open Toda systems of split simple Lie algebras $\mathfrak{g}$. With respect to Darboux coordinates on coadjoint orbits $\mathcal{O}$, the potentials of the hamiltonians are products of polynomial and exponential functions. General solutions for equations of motion for $\mathfrak{g}_N$ are obtained using differential operators called jet transformations. These results are applied to a $3$-body problem based on $\mathfrak{sl}(2)$, and to an extension of soliton solutions for $A_\infty$ to associated Ta…
We describe a class of integrable systems on Poisson submanifolds of the affine Poisson-Lie groups $\widehat{PGL}(N)$, which can be enumerated by cyclically irreducible elements the co-extended affine Weyl groups $(\widehat{W}\times \widehat{W})^\sharp$. Their phase spaces admit cluster coordinates, whereas the integrals of motion are cluster functions. We show, that this class of integrable systems coincides with the constructed by Goncharov and Kenyon out of dimer models on a two-dimensional torus and classified by the Newton polygons. We construct the correspondence between the Weyl group e…
