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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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