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

Let (𝕃(V),d) be a free graded connected differential Lie algebra over the field of rational numbers. An ideal I in the Lie algebra H(𝕃(V),d) is called nice if, for every cycle α𝕃(V) such that [α] belongs to I, the kernel of the map H(𝕃(V),d)H(𝕃(Vx),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(Vx),d). We apply this computation for the determination of the rational homotopy Lie algebra L X =π * (ΩX) 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 α𝕃(V) dont la classe appartient à I,I contient le noyau de l’application H(𝕃(V),d)H(𝕃(Vx),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(Vx),d). Ceci est ensuite appliqué à l’étude de la structure de l’algèbre de Lie d’homotopie rationnelle L X =π * (ΩX) 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},
     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://aif.centre-mersenne.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, Volume 46 (1996) no. 1, pp. 263-278. doi : 10.5802/aif.1513. https://aif.centre-mersenne.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: 55 #4174 | Zbl: 0322.55019

[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: 89d:55029 | Zbl: 0654.55011

[3] Y. Félix, S. Halperin, J.-M. Lemaire and J.-C. Thomas, Mod p loop space homology, Inventiones Math., 95 (1989), 247-262. | MR: 89k:55010 | Zbl: 0667.55007

[4] Y. Félix, S. Halperin and J.-C. Thomas, Elliptic spaces II, Enseignement Mathématique, 39 (1993), 25-32. | MR: 94f:55008 | Zbl: 0786.55006

[5] S. Halperin, Lectures on minimal models, Mémoire de la Société Mathématique de France 9/10 (1983). | Numdam | MR: 85i:55009 | Zbl: 0536.55003

[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: 89e:55022 | Zbl: 0655.55004

[7] K. Hess and J.-M. Lemaire, Nice and lazy cell attachments, Prépublication Nice, 1995. | Zbl: 0859.55008

[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. | Numdam | MR: 58 #18423 | Zbl: 0382.55011

[9] D. Quillen, Rational homotopy theory, Annals of Math., 90 (1969), 205-295. | MR: 41 #2678 | Zbl: 0191.53702

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