Koszul duality and the Poincaré–Birkhoff–Witt theorem

Key Points

• The universal enveloping algebra of a differential graded Lie algebra is shown to be Koszul.
• An explicit contracting homotopy is constructed from the cobar construction to the universal enveloping algebra.
• This result provides a new understanding of the Poincaré–Birkhoff–Witt theorem in the context of L_infinity-algebras.
• The application of homological perturbation theory facilitates this proof involving A_infinity-algebras.

Abstract