[Graduations sur les algèbres de Lie, croissance systolique, et propriétés cohopfiennes des groupes nilpotents]
We study the existence of various types of gradings on Lie algebras, such as Carnot gradings or gradings in positive integers, and prove that the existence of such gradings is invariant under extensions of scalars.
As an application, we prove that if is a finitely generated nilpotent group, its systolic growth is asymptotically equivalent to its word growth if and only if the Malcev completion of is Carnot.
We also characterize when is non-cohopfian, in terms of the existence of a non-trivial grading in non-negative integers, and deduce that this property only depends on its real (or even complex) Malcev completion.
On s'intéresse à l'existence de différentes sortes de graduations sur les algèbres de Lie, notamment les graduations Carnot ou en entiers positifs. On démontre notamment que l'existence de telles graduations est invariant par extension des scalaires.
On utilise ce résultat pour établir que si est un groupe nilpotent de type fini, sa croissance systolique est asymptotiquement équivalente à sa croissance des mots si et seulement si la complétion de Malcev de est Carnot.
On caractérise également quand est non-cohopfien, en termes de l'existence d'une graduation non triviale en entiers positifs, et on en déduit que cette propriété ne dépend que de sa complétion de Malcev réelle.
DOI : 10.24033/bsmf.2725
Keywords: Nilpotent Lie algebras, Lie algebra gradings, Carnot gradings, nilpotent groups, cohopian groups, systolic growth.
Mots-clés : Algèbres de Lie nilpotentes, graduations d'algèbres de Lie, graduations Carnot, groupes cohopfiens, croissance systolique
@article{BSMF_2016__144_4_693_0,
author = {Cornulier, Yves},
title = {Gradings on {Lie} algebras, systolic growth, and cohopfian properties of nilpotent groups},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {693--744},
year = {2016},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {144},
number = {4},
doi = {10.24033/bsmf.2725},
mrnumber = {3562610},
zbl = {1370.17015},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2725/}
}
TY - JOUR AU - Cornulier, Yves TI - Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups JO - Bulletin de la Société Mathématique de France PY - 2016 SP - 693 EP - 744 VL - 144 IS - 4 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2725/ DO - 10.24033/bsmf.2725 LA - en ID - BSMF_2016__144_4_693_0 ER -
%0 Journal Article %A Cornulier, Yves %T Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups %J Bulletin de la Société Mathématique de France %D 2016 %P 693-744 %V 144 %N 4 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2725/ %R 10.24033/bsmf.2725 %G en %F BSMF_2016__144_4_693_0
Cornulier, Yves. Gradings on Lie algebras, systolic growth, and cohopfian properties of nilpotent groups. Bulletin de la Société Mathématique de France, Tome 144 (2016) no. 4, pp. 693-744. doi: 10.24033/bsmf.2725
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