W $$ \mathcal{W} $$ algebras are L∞ algebras

Abstract It is shown that the closure of the infinitesimal symmetry transformations underlying classical W $$ \mathcal{W} $$ algebras give rise to L∞ algebras with in general field dependent gauge parameters. Therefore, the class of well understood W $$ \mathcal{W} $$ algebras provides highly nontrivial examples of such strong homotopy Lie algebras. We develop the general formalism for this correspondence and apply it explicitly to the classical W 3 $$ {\mathcal{W}}_3 $$ algebra