319 research outputs found
Jacobi Structures in
The most general Jacobi brackets in 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
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
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
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
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
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