Poisson Nijenhuis Structures Research Articles

Dirac structures and Nijenhuis operators

We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson–Nijenhuis structures. We study several properties of the “Dirac–Nijenhuis” structures thus obtained, including their connection with holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case.

Nijenhuis tensor and invariant polynomials

We discuss the diagonalization problem of the Nijenhuis tensor in a class of Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. We study its action on the ring of invariant polynomials of a Thimm chain of subalgebras. The existence of ϕ-minimal representations defines a suitable basis of invariant polynomials that completely solves the diagonalization problem. We prove that such representations exist in the classical cases AIII, BDI, DIII and CI, and do not exist in the exceptional cases EIII and EVII. We discuss a second general construction that in these two cases computes partially the spectrum and hints at a different behavior with respect to the classical cases.

Pre-Lie analogues of Poisson-Nijenhuis structures and Maurer–Cartan equations

In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce [Formula: see text]-structures on bimodules over pre-Lie algebras. We show that an [Formula: see text]-structure gives rise to a hierarchy of pairwise compatible [Formula: see text]-operators. We study solutions of the strong Maurer–Cartan equation on the twilled pre-Lie algebra associated to an [Formula: see text]-operator, which gives rise to a pair of [Formula: see text]-structures which are naturally in duality. We show that KVN-structures and HN-structures on a pre-Lie algebra [Formula: see text] are corresponding to [Formula: see text]-structures on the bimodule [Formula: see text], and KVB-structures are corresponding to solutions of the strong Maurer–Cartan equation on a twilled pre-Lie algebra associated to an [Formula: see text]-matrix.

Applications of Nijenhuis geometry: non-degenerate singular points of Poisson–Nijenhuis structures

We study and completely describe pairs of compatible Poisson structures near singular points of the recursion operator satisfying natural non-degeneracy condition.

Koszul–Vinberg structures and compatible structures on left-symmetric algebroids

In this paper, we introduce the notion of Koszul–Vinberg–Nijenhuis (KVN) structures on a left-symmetric algebroid as analogues of Poisson–Nijenhuis structures on a Lie algebroid, and show that a KVN-structure gives rise to a hierarchy of Koszul–Vinberg structures. We introduce the notions of [Formula: see text]-structures, pseudo-Hessian–Nijenhuis structures and complementary symmetric [Formula: see text]-tensors for Koszul–Vinberg structures on left-symmetric algebroids, which are analogues of [Formula: see text]-structures, symplectic-Nijenhuis structures and complementary [Formula: see text]-forms for Poisson structures. We also study the relationships between these various structures.

Poisson-Nijenhuis Groupoids

We define multiplicative Poisson-Nijenhuis structures on a Lie groupoid which extends the notion of symplectic-Nijenhuis groupoid introduced by Sti\'e23non and Xu. We also introduce a special class of Lie bialgebroid structure on a Lie algebroid $A$, called P-N Lie bialgebroid, which defines a hierarchy of compatible Lie bialgebroid structures on $A$. We show that under some topological assumption on the groupoid, there is a one-to-one correspondence between multiplicative Poisson-Nijenhuis structures on a Lie groupoid and P-N Lie bialgebroid structures on the corresponding Lie algebroid.

Invariant Poisson–Nijenhuis structures on Lie groups and classification

We study right-invariant (respectively, left-invariant) Poisson–Nijenhuis structures ([Formula: see text]-[Formula: see text]) on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra [Formula: see text]. We show that [Formula: see text]-[Formula: see text] structures can be used to find compatible solutions of the classical Yang–Baxter equation (CYBE). Conversely, two compatible [Formula: see text]-matrices from which one is invertible determine an [Formula: see text]-[Formula: see text] structure. We classify, up to a natural equivalence, all [Formula: see text]-matrices and all [Formula: see text]-[Formula: see text] structures with invertible [Formula: see text] on four-dimensional symplectic real Lie algebras. The result is applied to show that a number of dynamical systems which can be constructed by [Formula: see text]-matrices on a phase space whose symmetry group is Lie group a [Formula: see text], can be specifically determined.

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

ABSTRACTIntroducing Nijenhuis forms on L∞-algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into an L∞-algebra. Then we show that Nijenhuis forms on L∞-algebras also give a short and efficient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.

Poisson–Nijenhuis structures on quiver path algebras

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].

Complete integrability from Poisson–Nijenhuis structures on compact hermitian symmetric spaces

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 tensor are a choice of action variables. Our proof relies on an explicit formula for the contravariant connection defined on vector bundles that are Poisson with respect to the Bruhat-Poisson structure.

Multiplicative integrable models from Poisson–Nijenhuis structures

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].

Canonical Poisson–Nijenhuis structures on higher order tangent bundles

Let $M$ be a smooth manifold of dimension $m>0$, and denote by $S_{\rm can}$ the canonical Nijenhuis tensor on $TM$. Let $\varPi $ be a Poisson bivector on $M$ and $\varPi ^{T}$ the complete lift of $\varPi $ on $TM$. In a previous paper, we have shown th

Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

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 algebroid $T^{0,1}X\bowtie A^{1,0}$. In the case when $(X,\pi)$ is a holomorphic Poisson manifold and $A=(T^*X)_\pi$, such an elliptic Lie algebroid coincides with the Dirac structure corresponding to the associated generalized complex structure of the holomorphic Poisson manifold.

Compatible Structures on Lie Algebroids and Monge-Ampère Operators

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between PN-, PΩ- and ΩN-structures. We then show that the non-degenerate Monge-Ampere structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampere operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold.

Poisson-Nijenhuis structures on oriented 3D-Manifolds

We provide necessary and sufficient conditions on a (l,l)-tensor N on an oriented 3D-Poisson manifold ( M, Ω, π ) such that ( M, π, N ) is a Poisson-Nijenhuis manifold. We give a scheme for obtaining Poisson-Nijenhuis structures and also give some examples on ℝ 3 and the Poisson Lie group SU (2) as an application.

Variational Poisson-Nijenhuis structures for partial differential equations

We explore variational Poisson-Nijenhuis structures on nonlinear partial differential equations and establish relations between the Schouten and Nijenhuis brackets on the initial equation and the Lie bracket of symmetries on its natural extensions (coverings). This approach allows constructing a framework for the theory of nonlocal structures.

Dirac–Nijenhuis structures

In this paper we study the concept of Dirac–Nijenhuis structures. We consider them to be a pair where D is a Dirac structure, defined with respect to a Lie bialgebroid (A, A*), and is a Nijenhuis operator which defines a deformation of the Lie algebroid structure of D in a compatible way. The transformation can be considered to deform also the double of the Lie bialgebroid, which leads to the concept of Dirac–Nijenhuis structures of type I, or to not affect it, leading to Dirac–Nijenhuis structures of type II. We prove that the concept contains Poisson–Nijenhuis structures as a particular case and provides another example for the case of Kahler manifolds.

A class of Poisson–Nijenhuis structures on a tangent bundle

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 are known from previous studies on the cotangent bundle, in the case that Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.

Courant algebroid and Lie bialgebroid contractions

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17B62, 53C15, 53D17.

Reduction of Jacobi–Nijenhuis manifolds

A reduction theorem for Jacobi–Nijenhuis manifolds is established and its relation with the reduction of homogeneous Poisson–Nijenhuis structures is shown. Reduction under Lie group actions is also studied.