Koszul duality and the Poincaré–Birkhoff–Witt theorem

Abstract

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.