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 tn,ℂ of upper triangular complex matrices are studied. We prove that all the nilpotent matrices in tn,ℂ are strongly ad-nilpotent if and only if n≤6. Additionally, we prove that all the elements expad x, x strongly ad-nilpotent generate the inner automorphism group Int tn,ℂ.