# The Super Lie Groups Associated to Odd Involutions

## TL;DR

This paper introduces $
u$-Grassmannians as a new supergeometric generalization of Grassmannians, constructs them via $
u$-domains with odd involutions, and explores their homogeneous superspace structure and associated supergroups.

## Contribution

It presents the novel concept of $
u$-Grassmannians, constructs them using $
u$-domains, and introduces a supergroup linked to the odd involution $
u$, expanding supergeometry frameworks.

## Key findings

- $
u$-Grassmannians are homogeneous superspaces
- Construction of $
u$-domains with odd involutions
- Introduction of a supergroup associated to the odd involution $
u$

## Abstract

A new generalization of Grassmannians in supergeometry, called $\nu-$Grassmannians, are constructed by gluing $\nu-$domains. By a $\nu-$domain, we mean a superdomain with an odd involution say $\nu$ on its structure sheaf, as morphism of modules. Then we show that $\nu-$Grassmannians are homogeneous superspaces. In addition, in the last section, a supergroup associated to the odd involution $\nu$ is introduced.