Invertible linear maps on Borel subalgebras preserving zero Lie products

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra of rank l over an algebraically closed field of characteristic zero, $\mathfrak{b}$ a Borel subalgebra of $\mathfrak{g}$. An invertible linear map φ on $\mathfrak{b}$ is said preserving zero Lie products in both directions if for $x,y\in \mathfrak{b}$, $[x,y]=0$ if and only if $[\phi (x),\phi (y)]=0$. In this paper, it is shown that an invertible linear map φ on $\mathfrak{b}$ preserving zero Lie products in both directions if and only if it is a composition of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism, which extends the main result in [8] from a linear solvable Lie algebra to far more general cases.