Homotopy Poisson algebras, Maurer-Cartan elements and Dirac structures of CLWX 2-algebroids
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Publication:6332344
DOI10.4171/JNCG/398zbMATH Open1508.53084arXiv2001.01355MaRDI QIDQ6332344
Publication date: 5 January 2020
Abstract: 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 CLWX 2-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.
Poisson manifolds; Poisson groupoids and algebroids (53D17) Poisson algebras (17B63) Bialgebras (16T10)
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