Homotopy Loday algebras and higher Dorfman brackets

Using the technique of higher derived brackets, we construct a homotopy Loday algebra in the sense of Ammar and Poncin associated to any symplectic 2-manifold. The algebra we obtain accommodates the Dorfman bracket of a Courant algebroid as the binary operation in the hierarchy of operations, and the defect in the symmetry of each operation is measurable in a certain precise sense. We move to call such an algebra a homotopy Dorfman algebra, or a D1-algebra, which leads to the construction of a homotopy Courant algebroid.