On the homology of free Lie algebras

Calin Popescu

Commentationes Mathematicae Universitatis Carolinae (1998)

  • Volume: 39, Issue: 4, page 661-669
  • ISSN: 0010-2628

Abstract

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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 𝕃 F H ( V ) → F H 𝕃 ( V ) is an isomorphism of graded Lie algebras over R , and deduce thereby that the natural arrow U F H 𝕃 ( V ) → F H U 𝕃 ( V ) is an isomorphism of graded cocommutative Hopf algebras over R ; as usual, F stands for free part, H for homology, 𝕃 for free Lie algebra, and U for universal enveloping algebra. Related facts and examples are also considered.

How to cite

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Popescu, Calin. "On the homology of free Lie algebras." Commentationes Mathematicae Universitatis Carolinae 39.4 (1998): 661-669. <http://eudml.org/doc/248286>.

@article{Popescu1998,
abstract = {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 $ \mathbb \{L\} \{F\hspace\{-0.8pt\}H\}(V) \rightarrow \{F\hspace\{-0.8pt\}H\} \mathbb \{L\} (V)$ is an isomorphism of graded Lie algebras over $R$, and deduce thereby that the natural arrow $\{U\hspace\{-1.0pt\}F\hspace\{-0.8pt\}H\}\mathbb \{L\} (V) \rightarrow \{F\hspace\{-0.8pt\}H\hspace\{-0.4pt\}U\} \mathbb \{L\} (V)$ is an isomorphism of graded cocommutative Hopf algebras over $R$; as usual, $F$ stands for free part, $H$ for homology, $\mathbb \{L\}$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also considered.},
author = {Popescu, Calin},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {differential graded Lie algebra; free Lie algebra on a differential graded module; universal enveloping algebra; differential graded Lie algebra; free Lie algebra; universal enveloping algebra; Quillen's model},
language = {eng},
number = {4},
pages = {661-669},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the homology of free Lie algebras},
url = {http://eudml.org/doc/248286},
volume = {39},
year = {1998},
}

TY - JOUR
AU - Popescu, Calin
TI - On the homology of free Lie algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 4
SP - 661
EP - 669
AB - 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 $ \mathbb {L} {F\hspace{-0.8pt}H}(V) \rightarrow {F\hspace{-0.8pt}H} \mathbb {L} (V)$ is an isomorphism of graded Lie algebras over $R$, and deduce thereby that the natural arrow ${U\hspace{-1.0pt}F\hspace{-0.8pt}H}\mathbb {L} (V) \rightarrow {F\hspace{-0.8pt}H\hspace{-0.4pt}U} \mathbb {L} (V)$ is an isomorphism of graded cocommutative Hopf algebras over $R$; as usual, $F$ stands for free part, $H$ for homology, $\mathbb {L}$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also considered.
LA - eng
KW - differential graded Lie algebra; free Lie algebra on a differential graded module; universal enveloping algebra; differential graded Lie algebra; free Lie algebra; universal enveloping algebra; Quillen's model
UR - http://eudml.org/doc/248286
ER -

References

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  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. Zbl1073.17502MR1705144
  10. Quillen D.G., Rational homotopy theory, Ann. of Math. 90 (1969), 205-295. (1969) Zbl0191.53702MR0258031
  11. Scheerer H., Tanré D., The Milnor-Moore theorem in tame homotopy theory, Manuscripta Math. 70 (1991), 227-246. (1991) MR1089059
  12. Tanré D., Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan, LNM 1025, Springer-Verlag, Berlin, Heidelberg, New York, 1982. MR0764769

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