Homotopy Loday Algebras and Symplectic $2$-Manifolds

TL;DR

This paper constructs a new class of homotopy Loday algebras, called homotopy Dorfman algebras, from symplectic 2-manifolds using higher derived brackets, linking them to Courant algebroids.

Contribution

It introduces homotopy Dorfman algebras associated to symplectic 2-manifolds, providing a new framework for understanding Courant algebroids in a homotopical setting.

Findings

Constructed homotopy Loday algebra from symplectic 2-manifolds.

Embedded Dorfman bracket within the hierarchy of operations.

Defined a homotopy Courant algebroid as a consequence.

Abstract

Using the technique of higher derived brackets developed by Voronov, we construct a homotopy Loday algebra in the sense of Ammar and Poncin associated to any symplectic $2$-manifold. The algebra we obtain has a particularly nice structure, in that it 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 $D_\infty$-algebra, which leads to the construction of a homotopy Courant algebroid.