Let ${𝒰}$ be a 2-torsion free unital generalized matrix algebra, and $\varphi :{𝒰}\to {𝒰}$ be a linear map satisfying $$S,T\in {𝒰},ST=0\Rightarrow \varphi ([S,T])=[\varphi (S),T]=[S,\varphi (T)].$$ In this paper, we study the structure of $\varphi$ and under some mild conditions on ${𝒰}$ we present the necessary and sufficient conditions for $\varphi$ to be in terms of centralizers. We then provide the characterizations of Lie centralizers on ${𝒰}$ and our results generalize some of the previous results. We also refer to some applications of our results for triangular algebras and some operator algebras.

中文翻译：

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广义矩阵代数上零积的李集中器

让${𝒰}$是一个 2-torsion free unit 广义矩阵代数，并且$\phi ：{𝒰}\to {𝒰}$是满足的线性映射$$\mathrm{小号},吨\in {𝒰},\mathrm{小号}吨=0\Rightarrow \phi ([\mathrm{小号},吨])=[\phi (\mathrm{小号}),吨]=[\mathrm{小号},\phi (吨)].$$在本文中，我们研究了结构$\phi$在一些温和的条件下${𝒰}$我们提出了必要和充分条件$\phi$就集中器而言。然后我们提供了李集中器的特征${𝒰}$我们的结果概括了之前的一些结果。我们还提到了我们的结果在三角代数和一些算子代数中的一些应用。