In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-algebroids and split Lie 2-bialgebroids. We further introduce the notion of a weak Dirac structure of a CLWX 2-algebroid and show that the graph of a Maurer–Cartan element of the homotopy Poisson algebra of degree 3 associated to a split Lie 2-bialgebroid is a weak Dirac structure. Various examples including the string Lie 2-algebra, split Lie 2-algebroids constructed from integrable distributions and left-symmetric algebroids are given.

中文翻译：

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CLWX 2代数的同伦Poisson代数，Maurer-Cartan元素和Dirac结构

在本文中，我们构建了与分裂的Lie 2-代数相关的3度同伦Poisson代数，从而给出了表征分裂Lie 2-双代数的新方法。我们开发了与分裂李2-代数相关的微积分，并建立了分裂李2-代数的曼宁三元理论。更确切地说，我们给出严格的狄拉克结构的概念，并将分裂李2代数的Manin三元定义为具有两个横向严格狄拉克结构的CLWX2代数。我们表明，Mani三元组在分裂Lie 2-代数和分裂Lie 2-双代数之间是一一对应的。我们进一步介绍了CLWX 2代数的弱Dirac结构的概念，并表明与分裂Lie 2-双代数有关的3级同构Poisson代数的Maurer-Cartan元素图是弱Dirac结构。给出了各种示例，包括字符串Lie 2代数，由可积分布构造的分裂Lie 2代数和左对称代数。