Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids
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 $\g$-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 $\g$ via the degree $2$ symplectic NQ-manifold $T^*[2]\g^*[1]$. 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.