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Zbl 0836.55005
Félix, Yves
The center of a graded connected Lie algebra is a nice ideal. (English)
[J] Ann. Inst. Fourier 46, No.1, 261-276 (1996). ISSN 0373-0956; ISSN 1777-5310/e

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

MSC 2000:: *55P62 Rational homotopy theory
17B70 Graded Lie algebras

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

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