Algebraic (volume) density property for affine homogeneous spaces

Kaliman, Shulim; Kutzschebauch, Frank (2017). Algebraic (volume) density property for affine homogeneous spaces. Mathematische Annalen, 367(3-4), pp. 1311-1332. Springer 10.1007/s00208-016-1451-9

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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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