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 $${\mathfrak {A}}$$ .