Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids

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