Research ArticleNo Access

Nonlinear Lie triple derivations by local actions on triangular algebras

Xingpeng Zhao

Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P. R. China

E-mail Address: zhaoxingpeng@tyut.edu.cn

https://doi.org/10.1142/S0219498823500597Cited by:7 (Source: Crossref)

Abstract

Let $𝒰$ be a triangular algebra over a commutative ring $ℛ$ . In this paper, under some mild conditions on $𝒰$ , we prove that if $δ : 𝒰 \to 𝒰$ is a nonlinear map satisfying

δ ([[U, V], W]) = [[δ (U), V], W] + [[U, δ (V)], W] + [[U, V], δ (W)] <math display="block" altimg="eq-00005.gif"><mrow><mi>δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mo class="MathClass-open" stretchy="false">[</mo><mo class="MathClass-open" stretchy="false">[</mo><mi>U</mi><mo class="MathClass-punc">,</mo><mi>V</mi><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-punc">,</mo><mi>W</mi><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">=</mo><mo class="MathClass-open" stretchy="false">[</mo><mo class="MathClass-open" stretchy="false">[</mo><mi>δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>U</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">,</mo><mi>V</mi><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-punc">,</mo><mi>W</mi><mo class="MathClass-close" stretchy="false">]</mo><mo stretchy="false" class="MathClass-bin">+</mo><mo class="MathClass-open" stretchy="false">[</mo><mo class="MathClass-open" stretchy="false">[</mo><mi>U</mi><mo class="MathClass-punc">,</mo><mi>δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-punc">,</mo><mi>W</mi><mo class="MathClass-close" stretchy="false">]</mo><mo stretchy="false" class="MathClass-bin">+</mo><mo class="MathClass-open" stretchy="false">[</mo><mo class="MathClass-open" stretchy="false">[</mo><mi>U</mi><mo class="MathClass-punc">,</mo><mi>V</mi><mo class="MathClass-close" stretchy="false">]</mo><mo class="MathClass-punc">,</mo><mi>δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>W</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-close" stretchy="false">]</mo></mrow></math>

for any

$U, V, W \in 𝒰$ with

$U V W = 0$ . Then

$δ$ is almost additive on

$𝒰$ , that is,

δ (U + V) - δ (U) - δ (V) \in 𝒵 (𝒰) . <math display="block" altimg="eq-00010.gif"><mrow><mi>δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>U</mi><mo stretchy="false" class="MathClass-bin">+</mo><mi>V</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">-</mo><mi>δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>U</mi><mo class="MathClass-close" stretchy="false">)</mo><mo stretchy="false" class="MathClass-bin">-</mo><mi>δ</mi><mo class="MathClass-open" stretchy="false">(</mo><mi>V</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-rel">\in</mo><mi mathvariant="cal">𝒵</mi><mo class="MathClass-open" stretchy="false">(</mo><mi mathvariant="cal">𝒰</mi><mo class="MathClass-close" stretchy="false">)</mo><mo class="MathClass-punc">.</mo></mrow></math>

Moreover, there exist an additive derivation

$d$ of

$𝒰$ and a nonlinear map

$τ : 𝒰 \to 𝒵 (𝒰)$ such that

$δ (U) = d (U) + τ (U)$ for

$U \in 𝒰$ , where

$τ ([[U, V], W]) = 0$ for any

$U, V, W \in 𝒰$ with

$U V W = 0$ .

Communicated by S. K. Jain

Keywords:

AMSC: 16W25, 47B47

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