(Re)constructing Lie groupoids from their bisections and applications to prequantisation

Abstract

This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper we investigate the relation of the bisections to a given Lie groupoid, where the second part is about the construction of Lie groupoids from candidates for their bisection Lie groups. The procedure of this second part becomes feasible due to some recent progress in the infinite-dimensional Frobenius theorem, which we heavily exploit. The main application to the prequantisation of (pre)symplectic manifolds comes from an integrability constraint of closed Lie subalgebras to closed Lie subgroups. We characterise this constraint in terms of a modified discreteness conditions on the periods of that manifold.

45 pages, LaTex, this is a sequel to arXiv:1409.1428

References in corpus (7)

• Abelian extensions of infinite-dimensional Lie groups
• Fundamentals of submersions and immersions between infinite-dimensional manifolds
• Universal Central Extension of the Lie Algebra of Hamiltonian Vector Fields
• Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields
• The Frobenius theorem for Banach distributions on infinite-dimensional manifolds and applications in infinite-dimensional Lie theory
• Functorial aspects of the reconstruction of Lie groupoids from their bisections
• Differentiable vectors and unitary representations of Frechet-Lie supergroups

Cited by in corpus (8)

• Lie groupoids of mappings taking values in a Lie groupoid
• Strong topologies for spaces of smooth maps with infinite-dimensional target
• Functorial aspects of the reconstruction of Lie groupoids from their bisections
• Linking Lie groupoid representations and representations of infinite-dimensional Lie groups
• Poisson geometrical aspects of the Tomita-Takesaki modular theory
• Banach-Lie groupoids and generalized inversion
• The Lie group of vertical bisections of a regular Lie groupoid
• Controllability and diffeomorphism groups on manifolds with boundary