Generalized Jordan triple $(\sigma, \tau)$-higher derivation on Triangular algebras

Abstract

Let $\mathcal{R}$ be a commutative ring with unity,  $\mathfrak{A}=Tri(\mathcal{A},\mathcal{M},\mathcal{B})$  be a triangular algebra consisting of unital algebras $\mathcal{A},\mathcal{B}$ and $(\mathcal{A},\mathcal{B})$-bimodule $\mathcal{M}$ which is  faithful  as a left $\mathcal{A}$-module and also as a right $\mathcal{B}$-module. Let $\sigma$ and $\tau$ be two automorphisms of  $\mathfrak{A}.$ A family $\Delta=\{\delta_n\}_{n\in\mathbb{N}}$  of $\mathcal{R}$-linear mappings $\delta_n:\mathfrak{A}\rightarrow\mathfrak{A}$ is said to be a generalized Jordan triple $(\sigma, \tau)$-higher derivation on $\mathfrak{A}$ if  there exists a Jordan triple $(\sigma, \tau)$-higher derivation $\mathfrak{D}=\{d_n\}_{n\in\mathbb{N}}$  on $\mathfrak{A}$  such that $\delta_0=I_{\mathfrak{A}},$ the identity map of $\mathfrak{A}$  and $\delta_n(XYX)=\sum\limits_{i+j+k=n}\delta_i(\sigma^{n-i}(X))d_j(\sigma^{k}\tau^{i}(Y))d_k(\tau^{n-k}(X))$  holds for all $X,Y\in\mathfrak{A}$ and each $n\in\mathbb{N}.$ In this article, we study  generalized Jordan triple $(\sigma, \tau)$-higher derivation on  $\mathfrak{A}$ and prove that every generalized  Jordan triple $(\sigma, \tau)$-higher derivation on $\mathfrak{A}$ is a generalized $(\sigma, \tau)$-higher derivation on $\mathfrak{A}.$