  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.
