Non-commutative formality implies commutative and Lie formality

TL;DR

This paper establishes equivalences between different notions of formality in dg Lie and algebra contexts, with implications for rational homotopy theory, revealing deep connections between algebraic structures.

Contribution

It proves that formality of dg Lie algebras is equivalent to the formality of their universal enveloping algebras, and that commutative dg algebra formality is consistent across algebraic frameworks.

Findings

dg Lie algebra formality iff universal enveloping algebra is formal

commutative dg algebra formality consistent across frameworks

applications to rational homotopy theory

Abstract

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.