Strongly Ad-Nilpotent Elements of the Lie Algebra of Upper Triangular Matrices

Abstract

In this paper, the strongly ad-nilpotent elements of the Lie algebra $\mathfrak{t}\left(n,ℂ\right)$ of upper triangular complex matrices are studied. We prove that all the nilpotent matrices in $\mathfrak{t}\left(n,ℂ\right)$ are strongly ad-nilpotent if and only if $n\le 6$ . Additionally, we prove that all the elements $\exp \left(\text{ad\hspace{0.17em}}x\right)$ , $x$ strongly ad-nilpotent generate the inner automorphism group $\text{Int\hspace{0.17em}}t\left(n,ℂ\right)$ .