Characterizations of Lie Higher Derivations on Triangular Algebras

Wenhui Lin ()
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Wenhui Lin: China Agricultural University

Indian Journal of Pure and Applied Mathematics, 2020, vol. 51, issue 1, 77-104

Abstract: Abstract In this paper we mainly characterize Lie higher derivations on triangular algebras by local action. Let $$\mathcal{T} = \begin{bmatrix}\mathcal{A} & \mathcal{M} \\0 & \mathcal {B} \end{bmatrix}$$T=[AM0B] be a triangular algebra over a commutative ring $$\mathcal{R}$$R and $$\mathcal{Z} (\mathcal{T})$$Z(T) be the center of $$\mathcal{T}$$T. Under some mild conditions on $$\mathcal{T}$$T, we prove that if a family $$\Delta = \{\delta_n\}_{n=0}^\infty$$Δ={δn}n=0∞ of $$\mathcal{R}$$R-linear mappings on $$\mathcal{T}$$T satisfies the condition $$\delta_n([X, Y]) = \sum_{i+j=n}[\delta_i(X), \delta_j(Y)]$$δn([X,Y])=∑i+j=n[δi(X),δj(Y)] for any $$X, Y \in \mathcal{T}$$X,Y∈T with XY = 0 (resp. XY = P, where P is a fixed nontrivial idempotent of $$\mathcal{T}$$T), then there exist a higher derivation $$D = \{d_n\}_{n=0}^\infty$$D={dn}n=0∞ and an $$\mathcal{R}$$R-linear mapping $$\tau_n : \mathcal{T} \rightarrow \mathcal{Z}(\mathcal{T})$$τn:T→Z(T) vanishing on commutators [X, Y] with XY = 0 (resp. XY = P) such that $$\delta_n(X)=d_n(X)+\tau_n(X)$$δn(X)=dn(X)+τn(X) for all $$X \in \mathcal{T}$$X∈T.

Keywords: Lie higher derivable mapping; triangular algebra; 16W25; 15A78 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s13226-020-0386-8

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