Nijenhuis Structures Research Articles

Integrable sigma model with generalized [formula omitted] structure, Yang-Baxter sigma model with generalized complex structure and multi-Yang-Baxter sigma model

We construct an integrable sigma model with a generalized F structure, which involves a generalized Nijenhuis structure J satisfying J3=−J. Utilizing the expression of the generalized complex structure on the metric Lie group manifold G in terms of operator relations on its Lie algebra g, we formulate a Yang-Baxter sigma model with a generalized complex structure. Additionally, we present multi-Yang-Baxter sigma models featuring two and three compatible Nijenhuis structures. Examples for each of these models are provided.

Super Kupershmidt operators and Kupershmidt-Nijenhuis structures on Malcev superalgebras

In this paper, we focus on the super Kupershmidt operators (also called super -operators or relative Rota-Baxter operators) of a Malcev superalgebra. We first consider the relationship between Nijenhuis operators and super Kupershimidt operators. Then we discuss Kupershmidt-(dual-)Nijenhuis structures on a Malcev superalgebra. Moreover, we study the deformations of Nijenhuis elements and obtain a deformation theory of super Kupershmidt operators on a Malcev superalgebra. Communicated by Alberto Elduque

Integrating Nijenhuis structures

A Nijenhuis operator on a manifold is a (1, 1) tensor whose Nijenhuis torsion vanishes. A Nijenhuis operator {mathcal {N}} determines a Lie algebroid that knows everything about {mathcal {N}}. In this sense, a Nijenhuis operator is an infinitesimal object. In this paper, we identify its global counterpart. Namely, we characterize Lie groupoids integrating the Lie algebroid of a Nijenhuis operator. We illustrate our integration result in various examples, including that of a linear Nijenhuis operator on a vector space or, which is equivalent, a pre-Lie algebra structure.

Conformal r-matrix-Nijenhuis structures, symplectic-Nijenhuis structures, and ON-structures

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an LCMod pair), which lead to the notions of the Nijenhuis operator on the Lie conformal algebra and the Nijenhuis structure on the LCMod pair, respectively. Then, by adding compatibility conditions between Nijenhuis structures and O-operators, we introduce the notion of an ON-structure on an LCMod pair and show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. In particular, we show that compatible O-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of the conformal r-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on Lie conformal algebras and study their relations.

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.

Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra

The aim of this paper is to study infinitesimal deformations of a Malcev algebra with a representation and introduce the notion of Nijenhuis pair, which gives a trivial deformation of a Malcev algebra with a representation. We introduce the notion of Kupershmidt-(dual-)Nijenhuis structure on a Malcev algebra with a representation. Furthermore, we show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we establish a deformation theory of Kupershmidt operators in consistence with the general principles of deformation theories and introduce the notion of Nijenhuis elements.

Cohomologies of [formula omitted] pairs and compatible structures on Poisson algebras

In this paper, first we give the cohomology theory of a Poisson algebra with a module (called a PoiMod pair) and study the linear deformation theory of a PoiMod pair. We introduce the notion of a Nijenhuis structure on a PoiMod pair, which gives a trivial linear deformation. Then by adding compatibility conditions between Nijenhuis structures and O-operators, we introduce the notion of an ON-structure on a PoiMod pair and show that an ON-structure gives rise to a hierarchy of pairwise compatible O-operators. Finally we introduce the notions of PN- and ΩN-structures on Poisson algebras and study their relations.

Cohomologies and relative Rota-Baxter-Nijenhuis structures of 3-LieRep pairs

In this paper, first we give the cohomology theory of a 3-Lie algebra with a representation (called a 3- pair) and study the linear deformation theory of a 3- pair. We introduce the notion of a Nijenhuis structure on a 3- pair, which generates a trivial linear deformation. Then by adding compatibility conditions between Nijenhuis structures and relative Rota-Baxter operators, we introduce the notion of a relative Rota-Baxter-Nijenhuis structure on a 3- pair. Finally we construct relative Rota-Baxter-Nijenhuis structures on 3- pairs from those on pairs through certain linear functions.

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.

Kupershmidt-(dual-)Nijenhuis structure on the alternative algebra with a representation

The aim of this paper is to study Kupershmidt-(dual-)Nijenhuis structures on alternative algebras with representations. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of an alternative algebra with a representation. We introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on an alternative algebra with a representation. Furthermore, we verify that Kupershmidt operators and Kupershmidt-(dual-)Nijenhuis structures can give rise to each other under some conditions. Finally, we study the notions of Rota–Baxter–Nijenhuis structures and alternative [Formula: see text]-matrix-Nijenhuis structures. Meanwhile, we investigate the relation between them.

Holomorphic Jacobi manifolds

In this paper, we develop holomorphic Jacobi structures. Holomorphic Jacobi manifolds are in one-to-one correspondence with certain homogeneous holomorphic Poisson manifolds. Furthermore, holomorphic Poisson manifolds can be looked at as special cases of holomorphic Jacobi manifolds. We show that holomorphic Jacobi structures yield a much richer framework than that of holomorphic Poisson structures. We also discuss the relationship between holomorphic Jacobi structures, generalized contact bundles and Jacobi–Nijenhuis structures.

Nijenhuis operators on pre-Lie algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation

In this paper, first we study infinitesimal deformations of a Lie algebra with a representation and introduce the notion of a Nijenhuis pair, which gives a trivial deformation of a Lie algebra with a representation. Then we introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on a Lie algebra with a representation, which is a generalization of the r − n structure (r-matrix-Nijenhuis structure) introduced by Ravanpak, Rezaei-Aghdam, and Haghighatdoost. We show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we define a Rota-Baxter-Nijenhuis structure to be a Kupershmidt-Nijenhuis structure on a Lie algebra with respect to the adjoint representation and study the relation between Rota-Baxter-Nijenhuis structures and r-matrix-Nijenhuis structures.

Pseudo-Poisson Nijenhuis Manifolds

We introduce the notion of pseudo-Poisson Nijenhuis manifolds. These manifolds are generalizations of Poisson Nijenhuis manifolds by Magri and Morosi \cite{MM}. We show that any pseudo-Poisson Nijenhuis manifold has an associated quasi-Lie bialgebroid as in the case of Poisson quasi-Nijenhuis manifolds by Sti$\acute{\mathrm{e}}$non and Xu \cite{SX}. Hence, since a quasi-Lie bialgebroid has an associated Courant algebroid, we have new materials to construct Courant algebroids. In the "nondegenerate" case, we show that the conditions of pseudo-Poisson Nijenhuis structures can be reduced. Therefore we can provide lots of non-trivial examples of pseudo-Poisson Nijenhuis manifolds.

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.

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].

A bi-Hamiltonian system on the Grassmannian

Considering the recent result that the Poisson–Nijenhuis geometry corresponds to the quantization of the symplectic groupoid integrating a Poisson manifold, we discuss the Poisson–Nijenhuis structure on the Grassmannian defined by the compatible Kirillov–Kostant–Souriau and Bruhat–Poisson structures. The eigenvalues of the Nijenhuis tensor are Gelfand–Tsetlin variables, which, as was proved, are also in involution with respect to the Bruhat–Poisson structure. Moreover, we show that the Stiefel bundle on the Grassmannian admits a bi-Hamiltonian structure.

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].