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 of ${\mathrm{Der}}_{\mathbb{K}}(A)$ 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: every Lie subalgebra of ${\mathrm{Der}}_{\mathbb{K}}(A)$ whose elements are locally nilpotent is solvable; when $A$ is reduced, its derived length is at most $\dim A$, and this bound is sharp. In general, the $(\dim A)$-th term of its derived series is nilpotent, of class bounded in terms of $A$ only. We also prove that every solvable Lie subalgebra of ${\mathrm{Der}}_{\mathbb{K}}(A)$ generated by finitely many locally finite derivations is finite-dimensional and acts locally finitely on $A$. For an affine variety $X$ over an algebraically closed field, we deduce that a subgroup of $\mathrm{Aut}(X)$ 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 gives an answer to a 2005 problem of Popov. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proof uses derivations vanishing to second order along a suitable finite subscheme, on which local finiteness forces local nilpotence.