Jacobi Structures Research Articles

Integrable Bi-Hamiltonian Systems by Jacobi Structure on Real Three-Dimensional Lie Groups

By Poissonization of Jacobi structures on real three-dimensional Lie groups {textbf{G}} and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on {textbf{G}} otimes {mathbb {R}}.

Normal forms for Dirac–Jacobi bundles and splitting theorems for Jacobi structures

The aim of this paper is to prove a normal form Theorem for Dirac–Jacobi bundles using a recent techniques of Bursztyn, Lima and Meinrenken. As the most important consequence, we can prove the splitting theorems of Jacobi pairs which was proposed by Dazord, Lichnerowicz and Marle. As another application we provide an alternative proof of the splitting theorem of homogeneous Poisson structures.

Weak dual pairs in Dirac–Jacobi geometry

Adopting the omni-Lie algebroid approach to Dirac–Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac–Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. (1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac–Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac–Jacobi transversals. (2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).

Jacobi-Lie Hamiltonian Systems on Real Low-Dimensional Jacobi-Lie Groups and their Lie Symmetries

We study Jacobi-Lie Hamiltonian systems admitting Vessiot-Guldberg Lie algebras of Hamiltonian vector fields related to Jacobi structures on real low-dimensional Jacobi-Lie groups. Also, we find some examples of Jacobi-Lie Hamiltonian systems on real two- and three- dimensional Jacobi-Lie groups. Finally, we present Lie symmetries of Jacobi-Lie Hamiltonian systems on some three-dimensional real Jacobi-Lie groups.

The deformation L∞ algebra of a Dirac–Jacobi structure

We develop the deformation theory of a Dirac–Jacobi structure within a fixed Courant–Jacobi algebroid. Using the description of split Courant–Jacobi algebroids as degree 2 contact NQ manifolds and Voronov's higher derived brackets, each Dirac–Jacobi structure is associated with a cubic L∞ algebra for any choice of a complementary almost Dirac–Jacobi structure. This L∞ algebra governs the deformations of the Dirac–Jacobi structure: there is a one-to-one correspondence between the MC elements of this L∞ algebra and the small deformations of the Dirac–Jacobi structure. Further, by Cattaneo and Schätz's equivalence of higher derived brackets, this L∞ algebra does not depend (up to L∞-isomorphisms) on the choice of the complementary almost Dirac–Jacobi structure. These same ideas apply to get a new proof of the independence of the L∞ algebra of Dirac structure from the choice of a complementary almost Dirac structure (a result proved using other techniques by Gualtieri, Matviichuk and Scott).

Modular classes of Jacobi bundles

This paper is the first part of a dilogy devoted to modular classes of Jacobi structures from the general line bundle perspective as well as their associated Lie algebroids. First, we explain the relationship between Jacobi algebroids and their associated Gerstenhaber-Jacobi algebras. Then, we show that given a Jacobi manifold, there is a differential complex associated to it whose differential operator is similar to the so-called Koszul-Brylinski operator. This allows us to define Jacobi homology for Jacobi bundles. Moreover, we show that there are generating operators for the Gerstenhaber-Jacobi algebra associated to the Atiyah algebroid DL whose sections are derivations of the associated line bundle L.

Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures

We study aspects of two-dimensional nonlinear sigma models with Wess-Zumino term corresponding to a nonclosed 3-form, which may arise upon dimensional reduction in the target space. Our goal in this paper is twofold. In a first part, we investigate the conditions for consistent gauging of sigma models in the presence of a nonclosed 3-form. In the Abelian case, we find that the target of the gauged theory has the structure of a contact Courant algebroid, twisted by a 3-form and two 2-forms. Gauge invariance constrains the theory to (small) Dirac structures of the contact Courant algebroid. In the non-Abelian case, we draw a similar parallel between the gauged sigma model and certain transitive Courant algebroids and their corresponding Dirac structures. In the second part of the paper, we study two-dimensional sigma models related to Jacobi structures. The latter generalise Poisson and contact geometry in the presence of an additional vector field. We demonstrate that one can construct a sigma model whose gauge symmetry is controlled by a Jacobi structure, and moreover we twist the model by a 3-form. This construction is then the analogue of WZW-Poisson structures for Jacobi manifolds.

Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi–Lie systems

Using the adjoint representations of Lie algebras, we classify all Jacobi structures on real two- and three-dimensional Lie groups. Also, we study Jacobi-Lie systems on these real low-dimensional Lie groups. Our results are illustrated through examples of Jacobi-Lie Hamiltonian systems on some real two- and three-dimensional Lie groups.

Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems

С использованием присоединенных представлений алгебр Ли проведена классификация всех структур Якоби на вещественных дву- и трехмерных группах Ли. Кроме того, изучены системы Якоби-Ли на таких вещественных группах Ли с небольшой размерностью. Полученные результаты иллюстрируются примерами гамильтоновых систем Якоби-Ли на некоторых вещественных дву- и трехмерных группах Ли.

A linear algebraic setting for Jacobi structures

We study linear Jacobi structures on vector spaces and their non-degenerate version, the linear contact structures. We touch several aspects, like transitivity, Poissonization, coisotropy, product structures, transversals. We consider the linear version of contact dual pairs and we point out some of their properties.

Contact geometry for simple thermodynamical systems with friction

By means of the Jacobi structure associated with a contact structure, we use the so-called evolution vector field to propose a new characterization of isolated thermodynamical systems with friction, a simple but important class of thermodynamical systems which naturally satisfy the first and second laws of thermodynamics, i.e. total energy preservation of isolated systems and non-decreasing total entropy, respectively. We completely clarify its qualitative dynamics, the underlying geometrical structures and we also show how to apply discrete gradient methods to numerically integrate the evolution equations for these systems.

LOCAL FORMULAS FOR MULTIPLICATIVE FORMS

We provide explicit formulas for integrating multiplicative forms on local Lie groupoids in terms of infinitesimal data. Combined with our previous work [8], which constructs the local Lie groupoid of a Lie algebroid, these formulas produce concrete integrations of several geometric stuctures defined infinitesimally. In particular, we obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis–Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids, respectively.

Jacobi structures with background

Combining the twisted Jacobi structure [Twisted Jacobi manifolds, twisted Dirac–Jacobi structures and quasi-Jacobi bialgebroids, J. Phys. A: Math. Gen. 39(33) (2006) 10449–10475] with that of a Poisson structure with a 3-form background [Poisson geometry with a 3-form background, Prog. Theor. Phys. Suppl. 144 (2001) 145–154], alias twisted Poisson, we propose and analyze a new structure, called Jacobi structure with background. The background is a pair consisting of a [Formula: see text]-form and a [Formula: see text]-form. We describe their characteristic leaves. For twisted contact dual pairs, we show the correspondence of characteristic leaves of the two Jacobi manifolds with background.

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.

Weakly regular Jacobi structures and generalized contact bundles

A Jacobi structure $J$ on a line bundle $L\to M$ is weakly regular if the sharp map $J^\sharp : J^1 L \to DL$ has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5-dimensional nilmanifold is equipped with an invariant generalized contact structure, 2) we show that, unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that do not possess a compatible generalized contact structure.

Holomorphic Jacobi manifolds and holomorphic contact groupoids

This is the second part of a series of two papers dedicated to a systematic study of holomorphic Jacobi structures. In the first part, we introduced and study the concept of a holomorphic Jacobi manifold in a very natural way as well as various tools. In the present paper, we solve the integration problem for holomorphic Jacobi manifolds by proving that they integrate to complex contact groupoids. A crucial tool in our proof is what we call the "homogenization scheme", which allows us to identify holomorphic Jacobi manifolds with homogeneous holomorphic Poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids.

Gauge Transformations of Jacobi Structures and Contact Groupoids

We define gauge transformations of Jacobi structures on a manifold. This is related to gauge transformations of Poisson structures via the Poissonization. Finally, we discuss how the contact structure of a contact groupoid is effected by a gauge transformation of the Jacobi structure on its base.

Compatibility of Riemannian structures and Jacobi structures

We give a notion of compatibility between a Riemannian structure and a Jacobi structure. We prove that in case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, 1/2-Kenmotsu structures and locally conformally Kahler structures.

Dirac–Jacobi bundles

We show that a suitable notion of Dirac-Jacobi structure on a generic line bundle $L$, is provided by Dirac structures in the omni-Lie algebroid of $L$. Dirac-Jacobi structures on line bundles generalize Wade's $\mathcal E^1 (M)$-Dirac structures and unify generic (i.e.~non-necessarily coorientable) precontact distributions, Dirac structures and local Lie algebras with one dimensional fibers in the sense of Kirillov (in particular, Jacobi structures in the sense of Lichnerowicz). We study the main properties of Dirac-Jacobi structures and prove that integrable Dirac-Jacobi structures on line-bundles integrate to (non-necessarily coorientable) precontact groupoids. This puts in a conceptual framework several results already available in literature for $\mathcal E^1 (M)$-Dirac structures.

Covariant Jacobi brackets for test particles

We show that the space of observables of test particles has a natural Jacobi structure which is manifestly invariant under the action of the Poincaré group. Poisson algebras may be obtained by imposing further requirements. A generalization of Peierls procedure is used to extend this Jacobi bracket to the space of time-like geodesics on Minkowski spacetime.