Pseudogroups via pseudoactions: Unifying local, global, and infinitesimal symmetry

TL;DR

This paper introduces pseudoactions on Lie groupoids as a unifying framework for local, global, and infinitesimal symmetries, establishing a global converse to Lie's third theorem and extending integrability results.

Contribution

It defines pseudoactions as a new concept linking local and global symmetries, and proves a global converse to Lie's third theorem for twisted Lie algebra actions.

Findings

Every twisted Lie algebra action is integrable by a pseudoaction.

Complete twisted Lie algebra actions integrate to twisted Lie group actions.

Provides a generalization of Palais' global integrability theorem.

Abstract

A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a "pseudoaction" on the base manifold $M$. A pseudoaction generates a pseudogroup of transformations of $M$ in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseudoaction, one obtains the action of a Lie algebra on $M$, possibly twisted. A global converse to Lie's third theorem proven here states that every twisted Lie algebra action is integrated by a pseudoaction. When the twisted Lie algebra action is complete it integrates to a twisted Lie group action, according to a generalization of Palais' global integrability theorem.