The $L_\infty$ structure of Free Differential Algebras and $d=11$ Supergravity

Abstract: We discuss free differential algebras (FDA's), a generalization of the Cartan-Maurer equations for the group manifold vielbein, appropriate for theories containing $p$-forms ($p >1$). Their dual formulation is an extension of Lie algebras, called $L_\infty$ algebras, and we illustrate this duality in a simple example. Finally we review the FDA structure and the dual $L_\infty$ structure of $d=11$ supergravity.