Algebraic (Volume) Density Property for Affine Homogeneous Spaces

TL;DR

This paper proves algebraic density properties for vector fields on affine homogeneous spaces, showing they are generated by complete fields, with implications for volume-preserving cases and new criteria involving module generating pairs.

Contribution

It establishes the algebraic (volume) density property for affine homogeneous spaces, including cases with invariant volume forms, introducing new criteria based on module generating pairs.

Findings

Vector fields on most affine homogeneous spaces are generated by complete algebraic vector fields.

The space of divergence-free algebraic vector fields coincides with those generated by divergence-free complete fields.

New criteria for algebraic (volume) density property using module generating pairs are developed.

Abstract

Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $\omega$. We prove that the space of all divergence-free (with respect to $\omega$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus). The proof of these results requires new criteria for algebraic (volume) density property based on so called module generating pairs.