# Article

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Keywords:
differential graded Lie algebra; free Lie algebra on a differential graded module; universal enveloping algebra
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.
References:
[1] Anick D.J.: An $R$-local Milnor-Moore theorem. Adv. Math. 77 (1989), 116-136. MR 1014074 | Zbl 0684.55010
[2] Cohen F.R., Moore J.C., Niesendorfer J.A.: Torsion in homotopy groups. Ann. of Math. 109 (1979), 121-168. MR 0519355
[3] Eilenberg S., Moore J.C.: Homology and fibrations I, Coalgebras, cotensor product and its derived functors. Comment. Math. Helv. 40 (1966), 199-236. MR 0203730 | Zbl 0148.43203
[4] Halperin S.: Universal enveloping algebras and loop space homology. J. Pure Appl. Algebra 83 (1992), 237-282. MR 1194839 | Zbl 0769.57025
[5] Hilton P., Wu Y.-C.: A Course in Modern Algebra (especially Chapter 5, Section 4). John Wiley and Sons, Inc., New York, London, Sydney, Toronto, 1974. MR 0345744
[6] Jacobson N.: Basic Algebra I (especially Chapter 3). second edition, W.H. Freeman and Co., New York, 1985. MR 0780184
[7] Milnor J.W., Moore J.C.: On the structure of Hopf algebras. Ann. of Math. 81 (1965), 211-264. MR 0174052 | Zbl 0163.28202
[8] Popescu C.: Non-Isomorphic UHL and HUL. Rapport no. 257, Séminaire Mathématique, Institut de Mathématique, Université Catholique de Louvain, Belgique, February, 1996.
[9] Popescu C.: On UHL and HUL. Rapport no. 267, Séminaire Mathématique, Institut de Mathématique, Université Catholique de Louvain, Belgique, December, 1996; to appear in the Bull. Belgian Math. Soc. Simon Stevin. MR 1705144 | Zbl 1073.17502
[10] Quillen D.G.: Rational homotopy theory. Ann. of Math. 90 (1969), 205-295. MR 0258031 | Zbl 0191.53702
[11] Scheerer H., Tanré D.: The Milnor-Moore theorem in tame homotopy theory. Manuscripta Math. 70 (1991), 227-246. MR 1089059
[12] Tanré D.: Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan. LNM 1025, Springer-Verlag, Berlin, Heidelberg, New York, 1982. MR 0764769

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