General containment of tangent Lie algebra in derivations with constant divergence

Ascertain whether, for any variety of algebras over a field of characteristic zero and any free algebra A in that variety, the tangent Lie algebra T(Aut(A)) is always contained in the Lie algebra of derivations with constant divergence \~SDer(A).

Background

The paper proves that for several important varieties—Nielsen–Schreier varieties, associative algebras, commutative associative algebras, and metabelian Lie algebras—the tangent algebra T(Aut(A)) is a subalgebra of the derivations with constant divergence ~SDer(A), using universal derivations, Jacobians, and divergence. The inclusion relies on structural properties of the universal enveloping algebra and its radical.

The authors note that this inclusion may depend on the chosen notion of divergence and its quotient by the commutator and radical, and explicitly state that it is an open question whether the same inclusion holds in full generality across arbitrary varieties.