Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability

Let Mn be the algebra of all n × n complex matrices and gl(n, C) be the general linear Lie algebra, where n ≥ 2. An invertible linear map φ : gl(n, C) → gl(n, C) preserves solvability in both directions if both φ and φ-1 map every solvable Lie subalgebra of gl(n, C) to some solvable Lie subalgebra. In this paper we classify the invertible linear maps preserving solvability on gl(n, C) in both directions. As a sequence, such maps coincide with the invertible linear maps preserving commutativity on Mn in both directions.