Multiplicative Dirac structures

Abstract

In this paper we introduce multiplicative Dirac structures on Lie groupoids, providing a unified framework to study both multiplicative Poisson bivectors (i.e., Poisson group(oid)s) and multiplicative closed 2-forms (e.g., symplectic groupoids). We prove that for every source simply connected Lie groupoid $G$ with Lie algebroid $AG$, there exists a one-to-one correspondence between multiplicative Dirac structures on $G$ and Dirac structures on $AG$, which are compatible with both the linear and algebroid structures of $AG$. We explain in what sense this extends the integration of Lie bialgebroids to Poisson groupoids carried out in \cite{MX2} and the integration of Dirac manifolds of \cite{BCWZ}. We also explain the connection between multiplicative Dirac structures and higher geometric structures such as $\mathcal{L}\mathcal{A}$-groupoids and $\mathcal{C}\mathcal{A}$-groupoids.

References in corpus (4)

• Tangent Lifts of Poisson and Related Structures
• Dirac Algebroids in Lagrangian and Hamiltonian Mechanics
• Linear and multiplicative 2-forms
• Multiplicative Dirac structures on Lie groups

Cited by in corpus (6)

• Vector bundles over Lie groupoids and algebroids
• Lie theory of multiplicative tensors
• Infinitesimal Automorphisms of VB-groupoids and algebroids
• Glanon groupoids
• Poisson groupoids and moduli spaces of flat bundles over surfaces
• Poisson manifolds and their associated stacks