On solvability of commutator equations in Lie algebras

Abstract

We prove that every commutator equation is solvable over the Heisenberg Lie algebra in the case of arbitrary field in a bigger Lie algebra of upper niltriangular matrices over the same field. It is shown that every member of the descending central series of the Lie ring (algebra) of the upper niltriangular matrices is the image of a commutator one-variable function defined on this Lie ring (algebra).