In this paper we mainly characterize Lie higher derivations on triangular algebras by local action. Let \(\mathcal{T} = \begin{bmatrix}\mathcal{A} & \mathcal{M} \\0 & \mathcal {B} \end{bmatrix}\) be a triangular algebra over a commutative ring \(\mathcal{R}\) and \(\mathcal{Z} (\mathcal{T})\) be the center of \(\mathcal{T}\). Under some mild conditions on \(\mathcal{T}\), we prove that if a family \(\Delta = \{\delta_n\}_{n=0}^\infty\) of \(\mathcal{R}\)-linear mappings on \(\mathcal{T}\) satisfies the condition$$\delta_n([X, Y]) = \sum_{i+j=n}[\delta_i(X), \delta_j(Y)]$$for any \(X, Y \in \mathcal{T}\) with XY = 0 (resp. XY = P, where P is a fixed nontrivial idempotent of \(\mathcal{T}\)), then there exist a higher derivation \(D = \{d_n\}_{n=0}^\infty\) and an \(\mathcal{R}\)-linear mapping \(\tau_n : \mathcal{T} \rightarrow \mathcal{Z}(\mathcal{T})\) vanishing on commutators [X, Y] with XY = 0 (resp. XY = P) such that$$\delta_n(X)=d_n(X)+\tau_n(X)$$for all \(X \in \mathcal{T}\).

中文翻译：

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三角代数上Lie高阶导数的刻画

在本文中，我们主要通过局部作用来刻画三角形代数上的李高阶导数。假设\（\ mathcal {T} = \ begin {bmatrix} \ mathcal {A}＆\ mathcal {M} \\ 0＆\ mathcal {B} \ end {bmatrix} \）是交换环上的三角形代数\ （\ mathcal {R} \）和\（\ mathcal {Z}（\ mathcal {T}）\）是\（\ mathcal {T} \）的中心。下的一些温和的条件\（\ mathcal【T} \），证明了如果一个家庭\（\德尔塔= \ {\ delta_n \} _ {N = 0} ^ \ infty \）的\（\ mathcal {R } \）- \（\ mathcal {T} \）上的线性映射满足条件$$ \ delta_n（[X，Y]）= \ sum_ {i + j = n} [\ delta_i（X），\ delta_j（ Y）] $$任何\（X，Y \ in \ mathcal {T} \）中的XY = 0（分别为XY = P，其中P是\（\ mathcal {T} \）的固定非平等幂，），则存在较高的导数\（D = \ {d_n \} _ {n = 0} ^ \ infty \）和一个\（\ mathcal {R} \）线性映射\（\ tau_n：\ mathcal {T} \ rightarrow \ mathcal {Z }（\ mathcal {T}）\）在XY = 0（分别为XY = P）的换向器[ X，Y ]上消失，使得$$ \ delta_n（X）= d_n（X）+ \ tau_n（X）$ $代表所有\（X \ in \ mathcal {T} \）。