Abstract

We present two “Koszul dual” theorems: (A) If two commutative dg algebras in characteristic zero are quasi-isomorphic as dg algebras, then they are also quasi-isomorphic as commutative dg algebras. (B) If two dg Lie algebras in characteristic zero have universal enveloping algebras which are quasi-isomorphic as dg algebras, then the dg Lie algebras are themselves quasi-isomorphic. Theorem B says in particular that two Lie algebras in characteristic zero (with no grading or differential) are isomorphic if and only if their universal enveloping algebras are isomorphic as algebras; even this result is new and answers an open question. Via the work of Quillen and Sullivan, these theorems have immediate consequences in rational homotopy theory: two simply connected spaces have the same rational homotopy type if and only if their algebras of rational cochains are quasi-isomorphic, if and only if the algebras of rational chains on their Moore loop spaces are quasi-isomorphic. (Joint work with R. Campos, D. Robert-Nicoud, F. Wierstra)