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Algebraic (volume) density property for affine homogeneous spaces

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Abstract

Let X be a connected affine homogenous space of a linear algebraic group G over \(\mathbb {C}\). (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.

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Notes

  1. That is, \(B_i\) is isomorphic to \(R_i \times \mathbb {C}\) and the action is nothing but a translation on the second factor.

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Authors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33124, USA

    Shulim Kaliman

  2. Mathematisches Institut, Universität Bern, Sidlerstr. 5, 3012, Bern, Switzerland

    Frank Kutzschebauch

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  1. Shulim Kaliman
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  2. Frank Kutzschebauch
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Correspondence to Shulim Kaliman.

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Kaliman, S., Kutzschebauch, F. Algebraic (volume) density property for affine homogeneous spaces. Math. Ann. 367, 1311–1332 (2017). https://doi.org/10.1007/s00208-016-1451-9

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  • DOI: https://doi.org/10.1007/s00208-016-1451-9

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