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arXiv · 1507.07604

Algebraic (Volume) Density Property for Affine Homogeneous Spaces

Shulim Kaliman·Frank Kutzschebauch
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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 $ω$. We prove that the space of all divergence-free (with respect to $ω$) 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.

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Shulim Kaliman, Frank Kutzschebauch. 2015-07-27. Algebraic (Volume) Density Property for Affine Homogeneous Spaces. https://arxiv.org/abs/1507.07604

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