Noncommutative formality implies commutative and Lie formality

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.

Non-Formality of Planar Configuration Spaces in Characteristic 2

“…This means that C * (X, R) is connected to its cohomology H * (X, R) by a zig-zag of homomorphisms of differential graded associative R-algebras inducing isomorphisms in cohomology. If R is a field then this property depends only on the characteristic of R. If R is a field of characteristic 0 then the formality in the associative sense is equivalent to the usual commutative formality by a recent result of Saleh [17].…”

The Milnor-Moore theorem for $$L_\infty $$ algebras in rational homotopy theory

“…In presence of a contraction, Lemma 1 gives a straightforward proof of the fact that L is formal as a DGL if, and only if, U L is formal as a DGA. This result was recently proven in [27], and generalized in [8,Thm. B].…”

KoszulA∞-algebras and free loop space homology

“…Building on ideas of Halperin and Stasheff [14], an obstruction theory for formality of dg algebras over fields of characteristic 0 has been described by Saleh [24]. For associative dg algebras, Saleh's obstruction theory is valid over more general ground rings, but since the proof in [24] relies on working in characteristic zero, we need to indicate the necessary modifications.…”

“…For more details about obstructions for formality via Hochschild cohomology, we refer to [Sal17] and the references therein. In loc.…”