Supergrassmannians as Homogeneous Superspaces

TL;DR

This paper demonstrates that the super Lie group GL(m|n) acts transitively on supergrassmannian G_{k|l}(m|n), extending the classical homogeneous space concept into the supergeometry setting.

Contribution

It explicitly constructs the transitive action of GL(m|n) on supergrassmannians using the functor of points approach, providing a concrete supergeometric example.

Findings

GL(m|n) acts transitively on G_{k|l}(m|n)

Construction via functor of points approach

Gluing local actions to define the global action

Abstract

A homogeneous space is a manifold on which a Lie group acts transitively. Super generalization of this concept is also studied in [2] and [4]. In this paper we explicitly show that super Lie group GL(m|n) acts transitively on supergrassmannian G_{k|l}(m|n). In this regard, by using functor of point approach, this action is constructed by gluing local actions.