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

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 $L_{\mathrm{\infty }}$ algebras, and we illustrate this duality in a simple example. Finally we review the FDA structure and the dual $L_{\mathrm{\infty }}$ structure of $d=11$ supergravity.