Action of vectorial Lie superalgebras on some split supermanifolds

TL;DR

This paper explores how vectorial Lie superalgebras act on a special class of supermanifolds called 'curved' super Grassmannians, revealing their structure as derivations of these geometric objects.

Contribution

It introduces the action of vectorial Lie superalgebras on curved super Grassmannians, expanding understanding of their geometric and algebraic properties.

Findings

Realization of Lie superalgebras as derivations of structure sheaves

Description of 'curved' super Grassmannians as geometric objects

Connection between superalgebra actions and supergeometry

Abstract

The "curved" super Grassmannian is the supervariety of subsupervarieties of purely odd dimension $k$ in a~supervariety of purely odd dimension $n$, unlike the "usual" super Grassmannian which is the supervariety of linear subsuperspacies of purely odd dimension $k$ in a~superspace of purely odd dimension $n$. The Lie superalgebras of all and Hamiltonian vector fields on the superpoint are realized as Lie superalgebras of derivations of the structure sheaves of certain "curved" super Grassmannians,