# Article

 Title: On the homology of free Lie algebras (English) Author: Popescu, Calin Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 39 Issue: 4 Year: 1998 Pages: 661-669 . Category: math . 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. (English) Keyword: differential graded Lie algebra Keyword: free Lie algebra on a differential graded module Keyword: universal enveloping algebra MSC: 17B01 MSC: 17B35 MSC: 17B55 MSC: 17B70 idZBL: Zbl 1059.17503 idMR: MR1715456 . Date available: 2009-01-08T18:47:18Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/119042 . Reference: [1] Anick D.J.: An $R$-local Milnor-Moore theorem.Adv. Math. 77 (1989), 116-136. Zbl 0684.55010, MR 1014074 Reference: [2] Cohen F.R., Moore J.C., Niesendorfer J.A.: Torsion in homotopy groups.Ann. of Math. 109 (1979), 121-168. MR 0519355 Reference: [3] Eilenberg S., Moore J.C.: Homology and fibrations I, Coalgebras, cotensor product and its derived functors.Comment. Math. Helv. 40 (1966), 199-236. Zbl 0148.43203, MR 0203730 Reference: [4] Halperin S.: Universal enveloping algebras and loop space homology.J. Pure Appl. Algebra 83 (1992), 237-282. Zbl 0769.57025, MR 1194839 Reference: [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 Reference: [6] Jacobson N.: Basic Algebra I (especially Chapter 3).second edition, W.H. Freeman and Co., New York, 1985. MR 0780184 Reference: [7] Milnor J.W., Moore J.C.: On the structure of Hopf algebras.Ann. of Math. 81 (1965), 211-264. Zbl 0163.28202, MR 0174052 Reference: [8] Popescu C.: Non-Isomorphic UHL and HUL.Rapport no. 257, Séminaire Mathématique, Institut de Mathématique, Université Catholique de Louvain, Belgique, February, 1996. Reference: [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. Zbl 1073.17502, MR 1705144 Reference: [10] Quillen D.G.: Rational homotopy theory.Ann. of Math. 90 (1969), 205-295. Zbl 0191.53702, MR 0258031 Reference: [11] Scheerer H., Tanré D.: The Milnor-Moore theorem in tame homotopy theory.Manuscripta Math. 70 (1991), 227-246. MR 1089059 Reference: [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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