Weak dual pairs in Dirac–Jacobi geometry

Abstract

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).