Nonlinear Generalized Lie Triple Higher Derivation on Triangular Algebras

Abstract

Let ${\mathcal {R}}$ be a commutative ring with unity. A triangular algebra is an algebra of the form ${\mathfrak {A}} = \left[ \begin{array}{cc} {\mathcal {A}} &{} {\mathcal {M}} \\ 0 &{} {\mathcal {B}} \\ \end{array} \right]$ where ${\mathcal {A}}$ and ${\mathcal {B}}$ are unital algebras over ${\mathcal {R}}$ and ${\mathcal {M}}$ is an $({\mathcal {A}},{\mathcal {B}})$-bimodule which is faithful as a left ${\mathcal {A}}$-module as well as a right ${\mathcal {B}}$-module. In this paper, we study nonlinear generalized Lie triple higher derivation on ${\mathfrak {A}}$ and show that under certain assumptions on ${\mathfrak {A}}$, every nonlinear generalized Lie triple higher derivation on ${\mathfrak {A}}$ is of standard form, i.e., each component of a nonlinear generalized Lie triple higher derivation on ${\mathfrak {A}}$ can be expressed as the sum of an additive generalized higher derivation and a nonlinear functional vanishing on all Lie triple products on ${\mathfrak {A}}$.