Lie, associative and commutative quasi-isomorphism
Ricardo Campos, Dan Petersen, Daniel Robert-Nicoud, Felix Wierstra
TL;DR
The paper solves a long-standing question in characteristic $0$ rational homotopy theory by proving that two commutative dg algebras are quasi-isomorphic as commutative algebras if and only if they are so as associative algebras, implying that rational homotopy type can be detected from the associative dg cochains. It develops a Koszul-dual framework via operads $ extsf{Lie}$, $ extsf{Ass}$, and $ extsf{Com}$, using PBW decompositions to produce a chain-level splitting of Hochschild into Harrison cochains and to transfer $ extsf{A}_ oight ext{infty}$-isotopies to $ extsf{C}_ oight ext{infty}$-isotopies. Theorem B provides a dual result for homotopy complete dg Lie algebras: if $U rak g$ and $U rak h$ are quasi-isomorphic, then their homotopy completions are quasi-isomorphic; in the nilpotent arithmetic this recovers isomorphism of the Lie algebras themselves. The paper blends bar–cobar techniques, completed cobar constructions, and $ extsf{P}_ oight ext{infty}$-coalgebra formalism to unify deformation theory with rational homotopy, producing a robust framework with broad operadic applicability and informing the recovery of algebraic structures from enveloping algebras.
Abstract
Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational homotopy theory, showing that the rational homotopy type of a space is determined by its associative dg algebra of rational cochains. We also show a Koszul dual statement, under an additional completeness hypothesis: two homotopy complete dg Lie algebras whose universal enveloping algebras are quasi-isomorphic as associative dg algebras must themselves be quasi-isomorphic. The latter result applies in particular to nilpotent Lie algebras (not differential graded), in which case it says that two nilpotent Lie algebras whose universal enveloping algebras are isomorphic as associative algebras must be isomorphic.