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 $\mathfrak{A}.$
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