Finite-dimensionality and solvability after replacing semisimple generators by minimal tori

Determine whether, for the Lie subalgebra h = ⟨a1, …, ak, b1, …, bl⟩ of Der(O(X)) generated by locally nilpotent derivations a1, …, ak and semisimple derivations b1, …, bl, the Lie subalgebra ⟨a1, …, ak, Lie(T_min(b1)), …, Lie(T_min(bl))⟩ is finite-dimensional (respectively, solvable) whenever h is finite-dimensional (respectively, solvable), where T_min(bi) denotes the minimal algebraic torus in Aut(X) whose Lie algebra contains bi.

Background

The paper studies solvable Lie subalgebras of derivations on affine varieties, focusing on local finiteness and finite-dimensionality. A key reduction writes a solvable subalgebra h as generated by locally nilpotent and semisimple derivations.

Corollary cor:lf-fd shows that if each semisimple generator bi is already algebraic, i.e., lies in the Lie algebra of a one-dimensional torus, then h is locally finite if and only if h is finite-dimensional. The subsequent remark raises an unresolved issue about stability of finiteness (and solvability) when each semisimple bi is replaced by the entire Lie algebra of its minimal containing algebraic torus T_min(bi).

Resolving this would clarify how passing from individual semisimple generators to their minimal ambient tori affects structural properties (finite-dimensionality and solvability) of the generated Lie algebra.

References

However, for \mathfrak h in eq:gener, it is not known a priori whether the Lie subalgebra \langle a_1,\ldots, a_k, (T_{\min}(b_1)), \ldots, (T_{\min}(b_l))\rangle_\mathfrak{k} is finite-dimensional (resp. solvable) if \mathfrak h is finite-dimensional (resp. solvable).

— Locally finite solvable Lie algebras of derivations  (2604.02864 - Zaidenberg, 3 Apr 2026) in Remark following Corollary cor:lf-fd, Section 2 (General affine varieties)