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

Ezra Getzler

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

K-Theory and Homology 2025-07-09 v4

Authors: Ezra Getzler

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 $UL$ .

Keywords

lie algebra cohomology hochschild cohomology graded algebras

Cite

@article{arxiv.2405.14798,
  title  = {Koszul duality and the Poincar\'e-Birkhoff-Witt theorem},
  author = {Ezra Getzler},
  journal= {arXiv preprint arXiv:2405.14798},
  year   = {2025}
}

Comments

31 pages. New appendix with summary of a different contracting homotopy constructed by Dippell et al. arXiv:2410.15903. Several errors corrected

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

Abstract

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