Jacobi Structures Research Articles

Jacobi Structures on Affine Bundles

We study affine Jacobi structures (brackets) on an affine bundle π : A → M, i.e. Jacobi brackets that close on affine functions. We prove that if the rank of A is non–zero, there is a one–toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle \( A^{ + } = \cup _{{p \in M}} {\text{Aff}}{\left( {A_{p} ,\mathbb{R}} \right)} \) of affine functionals. In the case rank A = 0, it is shown that there is a one–to–one correspondence between affine Jacobi structures on A and local Lie algebras on A +. Some examples and applications, also for the linear case, are discussed. For a special type of affine Jacobi structures which are canonically exhibited (strongly–affine or affine–homogeneous Jacobi structures) over a real vector space of finite dimension, we describe the leaves of its characteristic foliation as the orbits of an affine representation. These affine Jacobi structures can be viewed as an analog of the Kostant–Arnold–Liouville linear Poisson structure on the dual space of a real finite–dimensional Lie algebra.

Integration of Dirac–Jacobi structures

We study precontact groupoids whose infinitesimal counterparts are Dirac-Jacobi structures. These geometric objects generalize contact groupoids. We also explain the relationship between precontact groupoids and homogeneous presymplectic groupoids. Finally, we present some examples of precontact groupoids.

Jacobi structures in R3

The most general Jacobi brackets in R3 are constructed after solving the equations imposed by the Jacobi identity. Two classes of Jacobi brackets were identified, according to the rank of the Jacobi structures. The associated Hamiltonian vector fields are also constructed.

Foliation-coupling Dirac structures

We extend the notion of “coupling with a foliation” from Poisson to Dirac structures and get the corresponding generalization of the Vorobjev characterization of coupling Poisson structures [Yu.M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, Lie algebroids and related topics in differential geometry, Banach Center Publ., Polish Acad. Sci. (Warsaw) 54 (2001) 249–274; I. Vaisman, Coupling Poisson and Jacobi structures, Int. J. Geom. Meth. Mod. Phys. 1 (5) (2004) 607–637]. We show that any Dirac structure is coupling with the fibers of a tubular neighborhood of an embedded presymplectic leaf, give new proofs of the results of Dufour and Wade [J.-P. Dufour, A. Wade, On the local structure of Dirac manifolds. arXiv:math.SG/0405257] on the transversal Poisson structure, and compute the Vorobjev structure of the total space of a normal bundle of the leaf. Finally, we use the coupling condition along a submanifold, instead of a foliation, in order to discuss submanifolds of a Dirac manifold which have differentiable, induced Dirac structures. In particular, we get an invariant that reminds the second fundamental form of a submanifold of a Riemannian manifold.

COUPLING POISSON AND JACOBI STRUCTURES ON FOLIATED MANIFOLDS

Let M be a differentiable manifold endowed with a foliation ℱ. A Poisson structure P on M is ℱ-coupling if ♯P(ann(Tℱ)) is a normal bundle of the foliation. This notion extends Sternberg's coupling symplectic form of a particle in a Yang–Mills field [11]. In the present paper we extend Vorobiev's theory of coupling Poisson structures [16] from fiber bundles to foliated manifolds and give simpler proofs of Vorobiev's existence and equivalence theorems of coupling Poisson structures on duals of kernels of transitive Lie algebroids over symplectic manifolds. We then discuss the extension of the coupling condition to Jacobi structures on foliated manifolds.

Nambu–Jacobi structures and Jacobi algebroids

We describe a Nambu?Jacobi structure as a 'Nambu?Poisson' structure on a certain Jacobi algebroid. It is shown that the matched pair of Leibniz algebroids for a Nambu?Jacobi structure is the Leibniz algebroid associated with this 'Nambu?Poisson' structure. We also see that a different Leibniz algebroid is associated with a Nambu?Jacobi structure, which is a natural generalization of the Lie algebroid associated with a Jacobi structure.

AV-differential geometry: Poisson and Jacobi structures

Based on ideas of W.M. Tulczyjew, a geometric framework for a frame-independent formulation of different problems in analytical mechanics is developed. In this approach affine bundles replace vector bundles of the standard description and functions are replaced by sections of certain affine line bundles called AV-bundles. Categorial constructions for affine and special affine bundles as well as natural analogs of Lie algebroid structures on affine bundles (Lie affgebroids) are investigated. One discovers certain Lie algebroids and Lie affgebroids canonically associated with an AV-bundle which are closely related to affine analogs of Poisson and Jacobi structures. Homology and cohomology of the latter are canonically defined. The developed concepts are applied in solving some problems of frame-independent geometric description of mechanical systems.

Locally Conformal Dirac Structures and Infinitesimal Automorphisms

We study the main properties of locally conformal Dirac bundles, which include Dirac structures on a manifold and locally conformal symplectic manifolds. It is proven that certain locally conformal Dirac bundles induce Jacobi structures on quotient manifolds. Furthermore we show that, given a locally conformal Dirac bundle over a smooth manifold M, there is a Lie homomorphism between a subalgebra of the Lie algebra of infinitesimal automorphisms and the Lie algebra of admissible functions. We also show that Dirac manifolds can be obtained from locally conformal Dirac bundles by using an appropriate covering map. Finally, we extend locally conformal Dirac bundles to the context of Lie algebroids.

Generalized Lie bialgebras and Jacobi structures on Lie groups

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras and we prove that they can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structure. We also propose a method generalizing the Yang-Baxter equation method to obtain generalized Lie bialgebras. Finally, we classify the compact generalized Lie bialgebras.

On characterization of Poisson and Jacobi structures

Abstract We characterize Poisson and Jacobi structures by means of complete lifts of the corresponding tensors: the lifts have to be related to canonical structures by morphisms of corresponding vector bundles. Similar results hold for generalized Poisson and Jacobi structures (canonical structures) associated with Lie algebroids and Jacobi algebroids.

The graded Jacobi algebras and (co)homology

Jacobi algebroids (i.e. `Jacobi versions' of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten's gauging of exterior derivative) is developed. One constructs a corresponding Cartan differential calculus (graded commutative one) in a natural manner. This, in turn, gives canonical generating operators for triangular Jacobi algebroids. One gets, in particular, the Lichnerowicz-Jacobi homology operators associated with classical Jacobi structures. Courant-Jacobi brackets are obtained in a similar way and use to define an abstract notion of a Courant-Jacobi algebroid and Dirac-Jacobi structure. All this offers a new flavour in understanding the Batalin-Vilkovisky formalism.

On the computation of the Lichnerowicz–Jacobi cohomology

Lichnerowicz–Jacobi cohomology of Jacobi manifolds is reviewed. The use of the associated Lie algebroid allows to prove that the Lichnerowicz–Jacobi cohomology is invariant under conformal changes of the Jacobi structure. We also compute the Lichnerowicz–Jacobi cohomology for a large variety of examples.

On a new relation between Jacobi and homogeneous Poisson manifolds

We establish a new, very close, relationship which links Jacobi structures and homogeneous Poisson structures defined on the same manifold and study the characteristic foliations of the related structures. Several examples of this construction are also given.

Jacobi structures revisited

Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as odd Jacobi brackets on the supermanifolds associated with the vector bundles. Jacobi bialgebroids are defined in the same manner. A lifting procedure of elements of this Grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding brackets is developed. This gives the possibility of associating canonically a Lie algebroid with any local Lie algebra in the sense of Kirillov.

Matched pairs of Leibniz algebroids, Nambu–Jacobi structures and modular class

The notion of a matched pair of Leibniz algebroids is introduced and it is shown that a Nambu–Jacobi structure of order n, n>2, over a manifold M defines a matched pair of Leibniz algebroids. As a consequence, one deduces that the vector bundle ⋀ n−1( T ∗M)⊕⋀ n−2( T ∗M)→M is a Leibniz algebroid. Finally, if M is orientable, the modular class of M is defined as a cohomology class of order 1 with respect to this Leibniz algebroid.

Generalized Lie bialgebroids and Jacobi structures

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialgebroid. We also show that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce a duality theorem. Finally, some special classes of generalized Lie bialgebroids are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.

A generalization of Poisson–Nijenhuis structures

We generalize Poisson–Nijenhuis structures. We prove that on a manifold endowed with a Nijenhuis tensor and a Jacobi structure which are compatible, there is a hierarchy of pairwise compatible Jacobi structures. Furthermore, we study the homogeneous Poisson–Nijenhuis structures and their relations with Jacobi structures.

Some linear Jacobi structures on vector bundles

We study Jacobi structures on the dual bundle $A^\ast$ to a vector bundle $A$ such that the Jacobi bracket of linear functions is again linear and the Jacobi bracket of a linear function and the constant function 1 is a basic function. We prove that a Lie algebroid structure on $A$ and a 1-cocycle $\phi \in \Gamma (A^\ast)$ induce a Jacobi structure on $A^\ast$ satisfying the above conditions. Moreover, we show that this correspondence is a bijection. Finally, we discuss some examples and applications.

Conformal Dirac Structures

The Courant bracket defined originally on the sections of a vector bundle TM⊕T*M→M is extended to the direct sum of the 1-jet vector bundle and its dual. The extended bracket allows one to interpret many structures encountered in differential geometry, in terms of Dirac structures. We give here a new approach to conformal Jacobi structures.

Jacobi—Nijenhuis manifolds and compatible Jacobi structures

We propose a definition of Jacobi—Nijenhuis structures, that includes the Poisson—Nijenhuis structures as a particular case. The existence of a hierarchy of compatible Jacobi structures on a Jacobi—Nijenhuis manifold is also obtained.