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.

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 LX=π*(ΩX)⊕ℚ d’un CW complexe simplement connexe X.

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 LX=π*(ΩX)⊗ℚ of a simply connected space X. We deduce that the kernel of the map LX→LY induced by the attachment of a cell along an element in the center is contained in the center.

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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},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {1},
     year = {1996},
     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. https://www.numdam.org/articles/10.5802/aif.1513/

[1] H.J. Baues and J.-M. Lemaire, Minimal models in homotopy theory, Math. Ann., 225 (1977), 219-242. | MR | Zbl

[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. | MR | Zbl

[3] Y. Félix, S. Halperin, J.-M. Lemaire and J.-C. Thomas, Mod p loop space homology, Inventiones Math., 95 (1989), 247-262. | MR | Zbl

[4] Y. Félix, S. Halperin and J.-C. Thomas, Elliptic spaces II, Enseignement Mathématique, 39 (1993), 25-32. | MR | Zbl

[5] S. Halperin, Lectures on minimal models, Mémoire de la Société Mathématique de France 9/10 (1983). | Numdam | MR | Zbl

[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. | MR | Zbl

[7] K. Hess and J.-M. Lemaire, Nice and lazy cell attachments, Prépublication Nice, 1995. | Zbl

[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. | Numdam | MR | Zbl

[9] D. Quillen, Rational homotopy theory, Annals of Math., 90 (1969), 205-295. | MR | Zbl

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