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).
We introduce and study the notion of contact dual pair adopting a line bundle approach to contact and Jacobi geometry. A contact dual pair is a pair of Jacobi morphisms defined on the same contact manifold and satisfying a certain orthogonality condi
tion. Contact groupoids and contact reduction are the main sources of examples. Among other properties, we prove the Characteristic Leaf Correspondence Theorem for contact dual pairs which parallels the analogous result of Weinstein for symplectic dual pairs.
In this thesis, we study the deformation problem of coisotropic submanifolds in Jacobi manifolds. In particular we attach two algebraic invariants to any coisotropic submanifold $S$ in a Jacobi manifold, namely the $L_infty[1]$-algebra and the BFV-co
mplex of $S$. Our construction generalizes and unifies analogous constructions in symplectic, Poisson, and locally conformal symplectic geometry. As a new special case we also attach an $L_infty[1]$-algebra and a BFV-complex to any coisotropic submanifold in a contact manifold. The $L_infty[1]$-algebra of $S$ controls the formal coisotropic deformation problem of $S$, even under Hamiltonian equivalence. The BFV-complex of $S$ controls the non-formal coisotropic deformation problem of $S$, even under both Hamiltonian and Jacobi equivalence. In view of these results, we exhibit, in the contact setting, two examples of coisotropic submanifolds whose coisotropic deformation problem is obstructed.
VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. A
dditionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation groupoid/algebroid.
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generaliz
ation of Legendrian submanifolds for what concerns deformation and moduli theory. Indeed, being integral coisotropic is proved to be a rigid condition, and moreover the integral coisotropic deformation problem is unobstructed with discrete moduli space.
We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s. settings. The BFV-complex of a coisotropic submanifold $S$ co
ntrols the coisotropic deformation problem of $S$ under both Hamiltonian and Jacobi equivalence.
