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

Félix, 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 -