Strong commutativity preserving maps of strictly triangular matrix Lie algebras

Abstract

Let [Formula: see text] be the Lie algebra consisting of all strictly upper triangular [Formula: see text] matrices over a field [Formula: see text]. An invertible linear map [Formula: see text] on [Formula: see text] is called to be strong commutativity preserving (simply denoted by SCP) if [Formula: see text] for any [Formula: see text]. We show that for [Formula: see text], an invertible linear map [Formula: see text] preserves strong commutativity if and only if there exist [Formula: see text] and a linear function [Formula: see text] with [Formula: see text] such that [Formula: see text], where [Formula: see text] is an inner SCP, [Formula: see text] are extremal SCPs, [Formula: see text] is a central SCP.