Finitely generated solvable Lie subalgebras of locally finite derivations on the coordinate ring of a quasi-affine variety are locally finite; they are integrable when the field is algebraically closed of characteristic zero and the variety is irreducible and affine.
Locally finite solvable Lie algebras of derivations
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abstract
Let X be an affine variety. The local finiteness of a Lie subalgebra h of Lie(Aut(X)) is equivalent to the existence of an algebraic subgroup G of Aut(X) such that h is contained in Lie(G). Let h be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether h is itself locally finite. After presenting some criteria for the local finiteness of h, we answer this question in the affirmative in the particular case where X is the affine plane.
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Locally finite sets of derivations
Finitely generated solvable Lie subalgebras of locally finite derivations on the coordinate ring of a quasi-affine variety are locally finite; they are integrable when the field is algebraically closed of characteristic zero and the variety is irreducible and affine.