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.
Bibliographical noteFunding Information:
This paper saw great progress while the second author was visiting the Higher School of Economics in Moscow. We thank the Higher School of Economics for their hospitality and wonderful working conditions. We thank Vladimir Dotsenko for his constant support and valuable advice during the preparation of these notes, for his insight into PBW theorems for operads that motivated this sequel to the joint work in  and for his careful reading of the manuscript. We thank Alexander Efimov for useful discussions that encouraged us to write Sect. and Ricardo Campos, Daniel Robert-Nicoud and Luis Scoccola for their useful comments and suggestions. We also thank Ben Knudsen for answering some questions about his work  on enveloping -algebras of spectral Lie algebras, and Guillermo Tochi for pointing us to this paper in the first place. Finally, we thank an anonymous referee for useful comments and suggestions that improved the quality of our article. The research of A. Kh. was carried out within the HSE University Basic Research Program and supported in part by the Russian Academic Excellence Project ‘5-100’ and in part by the Simons Foundation.
© 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