Globalizations of infinitesimal actions on supermanifolds

TL;DR

This paper investigates the conditions under which infinitesimal actions of Lie supergroups on supermanifolds can be integrated into global actions, establishing existence results and criteria for globalization.

Contribution

It proves the existence of local actions inducing given infinitesimal actions and provides necessary and sufficient conditions for their globalization.

Findings

Existence of local $ ext{G}$-actions inducing infinitesimal actions.

Necessary and sufficient conditions for globalization of infinitesimal actions.

Characterization of when a local action extends to a global one.

Abstract

Let $\mathcal G$ be a Lie supergroup with Lie superalgebra $\mathfrak g$, $\mathcal M$ a supermanifold and $\mathrm{Vec}(\mathcal M)$ the set of vector fields on $\mathcal M$. Let $\lambda:\mathfrak g\rightarrow \mathrm{Vec}(\mathcal M)$ be an infinitesimal action, i.e. a homomorphism of Lie superalgebras. We show the existence of a local $\mathcal G$-action on $\mathcal M$ inducing the infinitesimal action $\lambda$ and find necessary and sufficient conditions for the existence of a globalization in the sense of Palais.