Non-commutative formality implies commutative and Lie formality

摘要

Over a field of characteristic zero we prove two formality conditions. Weprove that a dg Lie algebra is formal if and only if its universal envelopingalgebra is formal. We also prove that a commutative dg algebra is formal as adg 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.