Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids

Abstract

We study Maurer–Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson \(\mathfrak g\)-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term \(L_\infty \)-algebra is a homotopy Poisson manifold of degree \(n-1\), we obtain a Courant algebroid from a 2-term \(L_\infty \)-algebra \(\mathfrak g\) via the degree 2 symplectic NQ-manifold \(T^*[2]\mathfrak g^*[1]\). By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie-quasi-Poisson groupoid from a 2-term \(L_\infty \)-algebra, which is proposed to be the geometric structure on the dual of a Lie 2-algebra. These results lead to a construction of a new 2-term \(L_\infty \)-algebra from a given one, which could produce many interesting examples.