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

Abstract

We give a construction of the universal enveloping $$A_\\infty $$\n \n A\n ∞\n \n algebra of a given $$L_\\infty $$\n \n L\n ∞\n \n algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new $$A_\\infty $$\n \n A\n ∞\n \n model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.