Jacobi Structures in R3\mathbb{R}^3

The most general Jacobi brackets in $\mathbb{R}^3$ 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

On quasi-Jacobi and Jacobi-quasi bialgebroids

We study quasi-Jacobi and Jacobi-quasi bialgebroids and their relationships with twisted Jacobi and quasi Jacobi manifolds. We show that we can construct quasi-Lie bialgebroids from quasi-Jacobi bialgebroids, and conversely, and also that the structures induced on their base manifolds are related via a quasi Poissonization

Jacobi-Nijenhuis algebroids and their modular classes

Jacobi-Nijenhuis algebroids are defined as a natural generalization of Poisson-Nijenhuis algebroids, in the case where there exists a Nijenhuis operator on a Jacobi algebroid which is compatible with it. We study modular classes of Jacobi and Jacobi-Nijenhuis algebroids

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.Comment: 10 pages. Brief changes in the introduction. References update

The Conformal Penrose Limit and the Resolution of the pp-curvature Singularities

We consider the exact solutions of the supergravity theories in various dimensions in which the space-time has the form M_{d} x S^{D-d} where M_{d} is an Einstein space admitting a conformal Killing vector and S^{D-d} is a sphere of an appropriate dimension. We show that, if the cosmological constant of M_{d} is negative and the conformal Killing vector is space-like, then such solutions will have a conformal Penrose limit: M^{(0)}_{d} x S^{D-d} where M^{(0)}_{d} is a generalized d-dimensional AdS plane wave. We study the properties of the limiting solutions and find that M^{(0)}_{d} has 1/4 supersymmetry as well as a Virasoro symmetry. We also describe how the pp-curvature singularity of M^{(0)}_{d} is resolved in the particular case of the D6-branes of D=10 type IIA supergravity theory. This distinguished case provides an interesting generalization of the plane waves in D=11 supergravity theory and suggests a duality between the SU(2) gauged d=8 supergravity of Salam and Sezgin on M^{(0)}_{8} and the d=7 ungauged supergravity theory on its pp-wave boundary.Comment: 20 pages, LaTeX; typos corrected, journal versio

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.Comment: 20 page

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.Comment: 20 pages, a few typos corrected; final version to be published in J. Phys. A: Math. Ge

Differential graded contact geometry and Jacobi structures

We study contact structures on nonnegatively-graded manifolds equipped with homological contact vector fields. In the degree 1 case, we show that there is a one-to-one correspondence between such structures (with fixed contact form) and Jacobi manifolds. This correspondence allows us to reinterpret the Poissonization procedure, taking Jacobi manifolds to Poisson manifolds, as a supergeometric version of symplectization.Comment: 9 pages. v2: Added references, improved proof of Proposition 3.3. v3: Expanded introduction, clarifying remarks, some changes of sign conventions. Main results are unchanged. v4: Final version, implementing changes suggested by referee

Lie algebroid foliations and E1(M){\cal E}^1(M)-Dirac structures

We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid $A$ over $M$ and the leaves of the Lie algebroid foliation on $M$ associated with $A$. Using these results, we show that a ${\cal E}^1(M)$-Dirac structure $L$ induces on every leaf $F$ of its characteristic foliation a ${\cal E}^1(F)$-Dirac structure $L_F$, which comes from a precontact structure or from a locally conformal presymplectic structure on $F$. In addition, we prove that a Dirac structure $\tilde{L}$ on $M\times \R$ can be obtained from $L$ and we discuss the relation between the leaves of the characteristic foliations of $L$ and $\tilde{L}$.Comment: 25 page