Derived Poincaré–Birkhoff–Witt theorems: with an appendix by Vladimir Dotsenko

Anton Khoroshkin, Pedro Tamaroff

Research output: Contribution to journalArticlepeer-review


We propose a new general formalism that allows us to study Poincaré–Birkhoff–Witt type phenomena for universal enveloping algebras in the differential graded context. Using it, we prove a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras. In particular, our results imply that all the previously known constructions of universal envelopes of L-algebras (due to Baranovsky, Lada and Markl, and Moreno-Fernández) represent the same object of the homotopy category of differential graded associative algebras. We also extend Quillen’s classical quasi-isomorphism C⟶ BU from differential graded Lie algebras to L-algebras; this confirms a conjecture of Moreno-Fernández.

Original languageEnglish
Article number15
JournalLetters in Mathematical Physics
Issue number1
StatePublished - Feb 2023

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature B.V.


  • Algebraic operads
  • Homotopical algebra
  • L -algebras
  • Universal enveloping algebras

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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