Locally finite solvable Lie algebras of derivations

TL;DR

This paper investigates conditions under which a solvable Lie algebra of derivations, generated by locally finite subalgebras, is itself locally finite, especially focusing on the affine plane case.

Contribution

It provides criteria for local finiteness of such Lie algebras and confirms the conjecture in the affine plane scenario under certain conditions.

Findings

Criteria for local finiteness of Lie algebras of derivations.

Affirmative answer for the affine plane case under additional assumptions.

Extends understanding of solvable Lie algebras generated by locally finite subalgebras.

Abstract

Let X be an affine variety and L 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 L is itself locally finite. Here we present some criteria for the local finiteness of L. Under some additional assumption, we answer this question in the affirmative in the particular case where X is the affine plane.