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

## TL;DR

This paper develops a new $A__$ algebra construction for $L__$ algebras and extends the Milnor-Moore theorem to rational homotopy theory, linking higher homotopy and homology products.

## Contribution

It introduces an alternative universal enveloping $A__$ algebra for $L__$ algebras and generalizes the Milnor-Moore theorem to higher homotopy contexts.

## Key findings

- New $A__$ model for rational homotopy types
- Relationship between Whitehead and Massey products
- Extended Milnor-Moore theorem in rational homotopy

## Abstract

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