Koszul duality and the Poincaré–Birkhoff–Witt theorem

Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for A_{\infty} -algebras, we give an explicit proof that the universal enveloping algebra UL of a differential graded Lie algebra L is Koszul, via an explicit contracting homotopy from the cobar construction \Omega CL of the Chevalley–Eilenberg chain coalgebra CL of L to UL . This may be viewed as an extension of the Poincaré–Birkhoff–Witt theorem to L_{\infty} -algebras.