Invertible Linear Maps on the General Linear Lie Algebras Preserving Solvability

Abstract

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