Abstract We give an explicit formula showing how the double Poisson algebra introduced in [14] appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on A ⊕ A ⁎ . Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra A and emphasises the special role of the fourth component of pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces ( Rep n A ) G l n for any associative algebra A.

中文翻译：

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Pre-Calabi-Yau 代数作为非交换泊松结构

摘要 我们给出了一个明确的公式，表明 [14] 中引入的双泊松代数如何作为前卡拉比-丘结构的特定部分出现，即循环不变，相对于自然内在形式，毛勒-嘉当方程的解在 A ⊕ A ⁎ 上。描述了该解的具体部分，与双泊松代数结构一一对应。结果适用于任何结合代数 A，并强调了前卡拉比-丘结构的第四个组成部分在这方面的特殊作用。因此，对于任何关联代数 A，我们有适当的 pre-Calabi-Yau 结构在表示空间 ( Rep n A ) G ln 上引入泊松括号。