On the homology of free Lie algebras

summary: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 $ \Bbb{L} {F\kern -0.8pt H}(V) \rightarrow {F\kern -0.8pt H} \Bbb{L} (V)$ is an isomorphism of graded Lie algebras over $R$, and deduce thereby that the natural arrow ${U\kern -1pt F\kern -0.8pt H}\Bbb{L} (V) \rightarrow {F\kern -0.8pt H\kern -0.4pt U} \Bbb{L} (V)$ is an isomorphism of graded cocommutative Hopf algebras over $R$; as usual, $F$ stands for free part, $H$ for homology, $\Bbb{L}$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also considered