JournalsjncgVol. 15, No. 1pp. 147–193

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

  • Jiefeng Liu

    Northeast Normal University, Changchun, China
  • Yunhe Sheng

    Jilin University, Changchun, China
Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids cover
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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 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.

Cite this article

Jiefeng Liu, Yunhe Sheng, Homotopy Poisson algebras, Maurer–Cartan elements and Dirac structures of CLWX 2-algebroids. J. Noncommut. Geom. 15 (2021), no. 1, pp. 147–193

DOI 10.4171/JNCG/398