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Generalized higher derivations on Lie ideals of triangular algebras. (English) Zbl 1390.16039
Summary: Let $$\mathfrak{A}=\begin{pmatrix}\mathcal{A}&\mathcal{M}\\ &\mathcal{B}\end{pmatrix}$$ be the triangular algebra consisting of unital algebras $$\mathcal{A}$$ and $$\mathcal{B}$$ over a commutative ring $$R$$ with identity 1 and $$\mathcal{M}$$ be a unital $$\mathcal{(A,B)}$$-bimodule. An additive subgroup $$\mathfrak{L}$$ of $$\mathfrak{A}$$ is said to be a Lie ideal of $$\mathfrak{A}$$ if $$[\mathfrak{L},\mathfrak {A}]\subseteq\mathfrak{L}$$. A non-central square closed Lie ideal $$\mathfrak{L}$$ of $$\mathfrak{A}$$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $$\mathfrak{A}$$, every generalized Jordan triple higher derivation of $$\mathfrak{L}$$ into $$\mathfrak{A}$$ is a generalized higher derivation of $$\mathfrak{L}$$ into $$\mathfrak{A}$$.
##### MSC:
 16W25 Derivations, actions of Lie algebras 47L35 Nest algebras, CSL algebras
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