${\cal W}$ algebras are L$_\infty$ algebras

Abstract

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