Over a field of characteristic zero we prove two formality conditions. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg algebra. We present some consequences of these theorems in rational homotopy theory.
"Noncommutative formality implies commutative and Lie formality." Algebr. Geom. Topol. 17 (4) 2523 - 2542, 2017. https://doi.org/10.2140/agt.2017.17.2523