Geodesic
On the homology of free Lie algebras
Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 4, pp. 661-669

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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 $ \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.
Classification : 17B01, 17B35, 17B55, 17B70
Keywords: differential graded Lie algebra; free Lie algebra on a differential graded module; universal enveloping algebra 
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     author = {Popescu, Calin},
     title = {On the homology of free {Lie} algebras},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {661--669},
     publisher = {mathdoc},
     volume = {39},
     number = {4},
     year = {1998},
     mrnumber = {1715456},
     zbl = {1059.17503},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1998__39_4_a2/}
}
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Popescu, Calin. On the homology of free Lie algebras. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) no. 4, pp. 661-669. http://geodesic.mathdoc.fr/item/CMUC_1998__39_4_a2/