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Given a principal ideal domain $R$ of characteristic zero, containing $1/2$, and a connected differential non-negatively graded free finite type $R$-module $V$, we prove that the natural arrow $𝕃FH\left(V\right)\to FH𝕃\left(V\right)$ is an isomorphism of graded Lie algebras over $R$, and deduce thereby that the natural arrow $UFH𝕃\left(V\right)\to FHU𝕃\left(V\right)$ is an isomorphism of graded cocommutative Hopf algebras over $R$; as usual, $F$ stands for free part, $H$ for homology, $𝕃$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also considered....

Given a principal ideal domain $R$ of characteristic zero, containing 1/2, and a two-cone $X$ of appropriate connectedness and dimension, we present a sufficient algebraic condition, in terms of Adams-Hilton models, for the Hopf algebra $FH(\Omega X;R)$ to be isomorphic with the universal enveloping algebra of some $R$-free graded Lie algebra; as usual, $F$ stands for free part, $H$ for homology, and $\Omega$ for the Moore loop space functor.