Algebraic density property of homogeneous spaces

TL;DR

This paper proves that under certain conditions, the Lie algebra generated by integrable algebraic vector fields on affine homogeneous spaces equals all algebraic vector fields, extending known density properties.

Contribution

It establishes the algebraic density property for a broad class of affine homogeneous spaces with specific SL_2-actions, generalizing previous results.

Findings

Lie algebra generated by integrable vector fields equals all algebraic vector fields

The property holds for most homogeneous spaces of the form G/R with G linear and R reductive

Identifies conditions under which the algebraic density property is valid

Abstract

Let $X$ be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that $X$ is equipped with several non-degenerate fixed point free $SL_2$-actions satisfying some mild additional assumption. Then we show that the Lie algebra generated by completely integrable algebraic vector fields on $X$ coincides with the set of all algebraic vector fields. In particular, we show that apart from a few exceptions this fact is true for any homogeneous space of form $G/R$ where $G$ is a linear algebraic group and $R$ is its proper reductive subgroup.