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. This proposes a new A_infty 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.
Highlights
The main goal of this paper is to construct a universal enveloping A∞ algebra for a given L∞ algebra, alternative to the already existing versions [3,15], and to study some consequences of such a structure in rational homotopy theory.Let L be an L∞ algebra
The original motivation for introducing the envelope we present was for extending the classical Milnor-Moore theorem [24] to L∞ algebras in the rational setting
An A∞ morphism f : A → B is a family of linear maps fk : A⊗k → B of degree k − 1 such that the following equation holds for every i ≥ 1: (−1)r+st fr+1+t id⊗r ⊗ms ⊗ id⊗t i=r +s+t s≥1 r,t≥0
Summary
The main goal of this paper is to construct a universal enveloping A∞ algebra for a given L∞ algebra, alternative to the already existing versions [3,15], and to study some consequences of such a structure in rational homotopy theory. The original motivation for introducing the envelope we present was for extending the classical Milnor-Moore theorem [24] to L∞ algebras in the rational setting. We uncover an interesting relationship between the higher order rational Whitehead products on π∗ (Ω X ) ⊗ Q and the higher order Pontryagin-Massey products of H∗ (Ω X ; Q) of connected spaces: the former are antisymmetrizations of the latter, whenever these are defined
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