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Koszul duality and the Poincaré–Birkhoff–Witt theorem

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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.

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This work demonstrates Koszul duality for the universal enveloping algebra in A_infinity-algebras, extending the Poincaré–Birkhoff–Witt theorem.

Ezra Getzler (2025) studied this question.

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This work demonstrates Koszul duality for the universal enveloping algebra in A_infinity-algebras, extending the Poincaré–Birkhoff–Witt theorem.