Super G-spaces

TL;DR

This paper reviews super G-spaces, proves a key theorem relating super Harish-Chandra pairs to super Lie group actions, and establishes the representability of the stability subgroup functor, advancing the theoretical framework of supergeometry.

Contribution

It provides a proof of a theorem connecting super Harish-Chandra pairs and super Lie group actions without using Frobenius theorem, using Koszul realization.

Findings

Proved the theorem relating super Harish-Chandra pairs to super Lie group actions.

Established the representability of the stability subgroup functor.

Provided a proof avoiding Frobenius theorem, using Koszul realization.

Abstract

We review the basic theory of super $G$-spaces. We prove a theorem relating the action of a super Harish-Chandra pair $(G_0, \mathfrak{g})$ on a supermanifold to the action of the corresponding super Lie group $G$. The theorem was stated in [DM99] without proof. The proof given here does not use Frobenius theorem but relies on Koszul realization of the structure sheaf of a super Lie group (see [Kosz83]). We prove the representability of the stability subgroup functor.