Affinity criterion for the quotient of an algebraic group by a one-dimensional subgroup

TL;DR

This paper establishes a criterion for when the quotient of an affine algebraic group by a one-dimensional unipotent subgroup is affine, based on subgroup containment in reductive subgroups.

Contribution

It provides a necessary and sufficient condition for the affineness of the quotient space involving subgroup containment properties.

Findings

The quotient is affine iff the subgroup is not in any reductive subgroup.

Characterization of affineness in terms of subgroup containment.

Clarifies the structure of quotients by unipotent subgroups.

Abstract

In this work we show that the homogeneous space of an affine algebraic group $G$ by a one-dimensional unipotent subgroup $H$ is affine if and only if the subgroup is not contained in any reductive subgroup of $G$.