| 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 |
| . |
