Integrating infinitesimal (super) actions

Abstract

In this paper we generalize some results of Richard Palais to the case of Lie supergroups and Lie superalgebras. More precisely, let $G$ be a Lie supergroup, $\mathfrak g$ its Lie superalgebra and let $\rho$ be an infinitesimal action (a representation) of $\mathfrak g$ on a supermanifold $M$. We will show that there always exists a local (smooth left) action of $G$ on $M$ such that $\rho$ is the map that associates the fundamental vector field on $M$ to an algebra element (we will say that the action integrates $\rho$). We also show that if $\rho$ is univalent, then there exists a unique maximal local action of $G$ on $M$ integrating $\rho$. And finally we show that if $G$ is simply connected and all (smooth, even) vector fields $\rho(X)$ are complete then there exists a global (smooth left) action of $G$ on $M$ integrating $\rho$. Omitting all references to the super setting will turn our proofs into variations of those of Palais.