Triangular algebras with nonlinear higher Lie n-derivation by local actions

This paper was devoted to the study of the so-called nonlinear higher Lie n-derivation of triangular algebras $ \mathcal{T} $, where $ n $ is a nonnegative integer greater than two. Under some mild conditions, we proved that every nonlinear higher Lie n-derivation by local actions on the triangular algebras is of a standard form. As an application, we gave a characterization of higher Lie $ n $-derivation by local actions on upper triangular matrix algebras, block upper triangular matrix algebras and nest algebras, respectively.