Koszul duality and the Poincar\'e-Birkhoff-Witt theorem

TL;DR

This paper provides an explicit proof that the universal enveloping algebra of a differential graded Lie algebra is Koszul, using homotopy theory and homological perturbation techniques.

Contribution

It introduces a new explicit contracting homotopy to establish Koszulity of universal enveloping algebras for differential graded Lie algebras.

Findings

Proves the Koszul property of $UL$ explicitly

Constructs a contracting homotopy from $ ext{cobar}(CL)$ to $UL$

Utilizes homotopy and homological perturbation theory techniques

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 $UL$.