Non-Linear Invertible Maps that Preserve Staircase Subalgebras

Abstract

Let $\mathcal{g}$ be the general linear Lie algebra consisting of all $n × n$ matrices over a field $F$ and with the usual bracket operation $[x, y] = xy − yx$. An invertible map $φ : \mathcal{g} → \mathcal{g}$ is said to preserve staircase subalgebras if it maps every staircase subalgebra to some staircase subalgebra of the same dimension. In this paper, we devote to giving an explicit description on the invertible maps on $\mathcal{g}$ that preserve staircase subalgebras.