Rigidity for Lie algebras of locally finite derivations

Let $A$ be a finitely generated commutative algebra over a field $\mathbb K$ of characteristic zero. We prove that every finitely generated Lie subalgebra $L\subseteq\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally finite on $A$ is finite-dimensional. Consequently, for a Lie subalgebra generated by finitely many locally finite derivations, the following are equivalent: it is finite-dimensional, it acts locally finitely on $A$, and all its elements are locally finite. A key ingredient is a second theorem of independent interest: every Lie subalgebra of $\operatorname{Der}_{\mathbb K}(A)$ whose elements are locally nilpotent is solvable; when $A$ is reduced, its derived length is at most $\dim A$, and this bound is sharp. For an affine variety $X$ over an algebraically closed field, we deduce that a subgroup of $\operatorname{Aut}(X)$ generated by finitely many connected algebraic subgroups is algebraic if and only if every element of the Lie algebra generated by their tangent algebras is locally finite. For two unipotent one-parameter subgroups, this provides an answer to a problem posed by Popov in 2005. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proofs rest on the introduction of a cofinite ideal meeting the closure of every associated point of $\operatorname{Spec} A$: on the subalgebra of derivations vanishing to second order along the corresponding finite subscheme, local finiteness forces local nilpotence. The intersection of that subalgebra with $L$ is therefore solvable by the second theorem, and has finite codimension in $L$; together with local finiteness of the adjoint action, this yields finite-dimensionality.