Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space

Abstract

In this paper we classify linear maps preserving commutativity in both directions on the space N( F ) of strictly upper triangular ( n+1)×( n+1) matrices over a field F . We show that for n⩾3 a linear map ϕ on N( F ) preserves commutativity in both directions if and only if ϕ= ϕ ′+ f where ϕ ′ is a product of standard maps on N( F ) and f is a linear map of N( F ) into its center.