仿射齐性空间的代数（体）密度性质
Algebraic (Volume) Density Property for Affine Homogeneous Spaces
摘要：
设$X$是$\C$上线性代数群$G$的连通仿射齐性空间. (1)如果$X$不同于直线或环面，则证明了$X$上的所有代数向量场的空间与$X$上的完备代数向量场生成的李代数重合. (2)假设$X$具有$G$不变体积形式$ω$。证明了$X$上所有关于$ω$的无发散代数向量场的空间与$X$上无发散完全代数向量场生成的李代数重合（包括$X$为直线或环面的情形）.这些结果的证明需要基于模生成对的代数（体）密度性质的新判据。
英文摘要：
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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