no. 1
The center of a graded connected Lie algebra is a nice ideal
Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 263-278

Let (𝕃(V),d) be a free graded connected differential Lie algebra over the field ℚ of rational numbers. An ideal I in the Lie algebra H(𝕃(V),d) is called nice if, for every cycle α∈𝕃(V) such that [α] belongs to I, the kernel of the map H(𝕃(V),d)→H(𝕃(V⊕ℚx),d), d(x)=α, is contained in I. We show that the center of H(𝕃(V),d) is a nice ideal and we give in that case some informations on the structure of the Lie algebra H(𝕃(V⊕ℚx),d). We apply this computation for the determination of the rational homotopy Lie algebra L X =π * (ΩX)⊗ℚ of a simply connected space X. We deduce that the kernel of the map L X →L Y induced by the attachment of a cell along an element in the center is contained in the center.

Soit (𝕃(V),d) une algèbre de Lie différentielle graduée définie sur le corps des nombres rationnels. Un idéal I dans l’algèbre de Lie H(𝕃(V),d) est dit gentil si pour tout cycle α∈𝕃(V) dont la classe appartient à I,I contient le noyau de l’application H(𝕃(V),d)→H(𝕃(V⊕ℚx),d),d(x)=α. Nous montrons que le centre de H(𝕃(V),d) est un gentil idéal et nous donnons dans ce cas des informations sur la structure de H(𝕃(V⊕ℚx),d). Ceci est ensuite appliqué à l’étude de la structure de l’algèbre de Lie d’homotopie rationnelle L X =π * (ΩX)⊕ℚ d’un CW complexe simplement connexe X.

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     author = {F\'elix, Yves},
     title = {The center of a graded connected {Lie} algebra is a nice ideal},
     journal = {Annales de l'Institut Fourier},
     pages = {263--278},
     year = {1996},
     publisher = {Association des Annales de l'Institut Fourier},
     volume = {46},
     number = {1},
     doi = {10.5802/aif.1513},
     mrnumber = {97d:55021},
     zbl = {0836.55005},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1513/}
}
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Félix, Yves. The center of a graded connected Lie algebra is a nice ideal. Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 263-278. doi: 10.5802/aif.1513

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