Integrating infinitesimal (super) actions

TL;DR

This paper extends Palais' results to Lie supergroups, showing conditions under which infinitesimal actions can be integrated into local or global actions on supermanifolds.

Contribution

It generalizes integration of infinitesimal actions to the super setting, establishing existence and uniqueness of local and global actions under certain conditions.

Findings

Existence of local actions integrating infinitesimal actions.

Uniqueness of maximal local actions for univalent infinitesimal actions.

Global actions exist if the supergroup is simply connected and vector fields are complete.

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.