Lie Algebra generated by locally nilpotent derivations on Danielewski surfaces

Abstract

We give a full description of the Lie algebra generated by locally nilpotent derivations (short LNDs) on smooth Danielewski surfaces Dp given by xy=p(z) . In case deg(p)≥3 it turns out to be not the whole Lie algebra VF ω alg (Dp) of volume preserving algebraic vector fields, thus answering a question posed by Lind and the first author. Also we show algebraic volume density property (short AVDP) for a certain homology plane, a homogeneous space of the form SL₂ (C)/N , where N is the normalizer of the maximal torus and another related example.