Globalizations of infinitesimal actions on supermanifolds

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 $λ:\mathfrak g\rightarrow \mathrm{Vec}(\mathcal M)$ 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 $λ$ and find necessary and sufficient conditions for the existence of a globalization in the sense of Palais.