Lie higher derivations on triangular algebras:A new perspective

Abstract

In this paper, we focus on the structure of Lie higher derivations on
triangular algebras $\mathcal{T}$ without assuming unity. We prove that Lie higher derivation on every triangular algebra can be decomposed into a sum of a higher derivation, an extreme Lie higher derivation, and a central mapping vanishing on commutators $[x,y]$. As by-products, we use it on some typical algebras: upper triangular matrix algebras over faithful algebras and semiprime algebras, respectively.