# The center of a graded connected Lie algebra is a nice ideal

Annales de l'institut Fourier (1996)

- Volume: 46, Issue: 1, page 263-278
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topFélix, Yves. "The center of a graded connected Lie algebra is a nice ideal." Annales de l'institut Fourier 46.1 (1996): 263-278. <http://eudml.org/doc/75173>.

@article{Félix1996,

abstract = {Let $(\{\Bbb L\}(V),d)$ be a free graded connected differential Lie algebra over the field $\{\Bbb Q\}$ of rational numbers. An ideal $I$ in the Lie algebra $H(\{\Bbb L\}(V),d)$ is called nice if, for every cycle $\alpha \in \{\Bbb L\}(V)$ such that $[\alpha ]$ belongs to $I$, the kernel of the map $H(\{\Bbb L\}(V),d) \rightarrow H(\{\Bbb L\}(V\oplus \{\Bbb Q\}x),d)$, $d(x) = \alpha $, is contained in $I$. We show that the center of $H(\{\Bbb L\}(V),d)$ is a nice ideal and we give in that case some informations on the structure of the Lie algebra $H(\{\Bbb L\}(V\oplus \{\Bbb Q\}x),d)$. We apply this computation for the determination of the rational homotopy Lie algebra $L_X = \pi _*(\Omega X) \otimes \{\Bbb Q\}$ of a simply connected space $X$. We deduce that the kernel of the map $L_X \rightarrow L_Y$ induced by the attachment of a cell along an element in the center is contained in the center.},

author = {Félix, Yves},

journal = {Annales de l'institut Fourier},

keywords = {differential graded Lie algebra; rational homotopy theory; inertia; graded differential Lie algebra; nice ideal; rational homotopy Lie algebra; simply connected space; attachment of a cell; center},

language = {eng},

number = {1},

pages = {263-278},

publisher = {Association des Annales de l'Institut Fourier},

title = {The center of a graded connected Lie algebra is a nice ideal},

url = {http://eudml.org/doc/75173},

volume = {46},

year = {1996},

}

TY - JOUR

AU - Félix, Yves

TI - The center of a graded connected Lie algebra is a nice ideal

JO - Annales de l'institut Fourier

PY - 1996

PB - Association des Annales de l'Institut Fourier

VL - 46

IS - 1

SP - 263

EP - 278

AB - Let $({\Bbb L}(V),d)$ be a free graded connected differential Lie algebra over the field ${\Bbb Q}$ of rational numbers. An ideal $I$ in the Lie algebra $H({\Bbb L}(V),d)$ is called nice if, for every cycle $\alpha \in {\Bbb L}(V)$ such that $[\alpha ]$ belongs to $I$, the kernel of the map $H({\Bbb L}(V),d) \rightarrow H({\Bbb L}(V\oplus {\Bbb Q}x),d)$, $d(x) = \alpha $, is contained in $I$. We show that the center of $H({\Bbb L}(V),d)$ is a nice ideal and we give in that case some informations on the structure of the Lie algebra $H({\Bbb L}(V\oplus {\Bbb Q}x),d)$. We apply this computation for the determination of the rational homotopy Lie algebra $L_X = \pi _*(\Omega X) \otimes {\Bbb Q}$ of a simply connected space $X$. We deduce that the kernel of the map $L_X \rightarrow L_Y$ induced by the attachment of a cell along an element in the center is contained in the center.

LA - eng

KW - differential graded Lie algebra; rational homotopy theory; inertia; graded differential Lie algebra; nice ideal; rational homotopy Lie algebra; simply connected space; attachment of a cell; center

UR - http://eudml.org/doc/75173

ER -

## References

top- [1] H.J. BAUES and J.-M. LEMAIRE, Minimal models in homotopy theory, Math. Ann., 225 (1977), 219-242. Zbl0322.55019MR55 #4174
- [2] Y. FÉLIX, S. HALPERIN, C. JACOBSSON, C. LÖFWALL and J.-C. THOMAS, The radical of the homotopy Lie algebra, Amer. Journal of Math., 110 (1988), 301-322. Zbl0654.55011MR89d:55029
- [3] Y. FÉLIX, S. HALPERIN, J.-M. LEMAIRE and J.-C. THOMAS, Mod p loop space homology, Inventiones Math., 95 (1989), 247-262. Zbl0667.55007MR89k:55010
- [4] Y. FÉLIX, S. HALPERIN and J.-C. THOMAS, Elliptic spaces II, Enseignement Mathématique, 39 (1993), 25-32. Zbl0786.55006MR94f:55008
- [5] S. HALPERIN, Lectures on minimal models, Mémoire de la Société Mathématique de France 9/10 (1983). Zbl0536.55003MR85i:55009
- [6] S. HALPERIN and J.-M. LEMAIRE, Suites inertes dans les algèbres de Lie graduées ("Autopsie d'un meurtre II"), Math. Scand., 61 (1987), 39-67. Zbl0655.55004MR89e:55022
- [7] K. HESS and J.-M. LEMAIRE, Nice and lazy cell attachments, Prépublication Nice, 1995. Zbl0859.55008
- [8] J.-M. LEMAIRE, "Autopsie d'un meurtre" dans l'homologie d'une algèbre de chaînes, Ann. Scient. Ecole Norm. Sup., 11 (1978), 93-100. Zbl0382.55011MR58 #18423
- [9] D. QUILLEN, Rational homotopy theory, Annals of Math., 90 (1969), 205-295. Zbl0191.53702MR41 #2678

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.