[Sur la structure d’algèbre de Hopf des algèbres de puissances divisées quantiques de Lusztig]
Nous étudions la structure d’algèbre de Hopf des groupes quantiques de Lusztig.Tout d’abord, nous montrons que la partie zéro est le produit tensoriel de l’algèbre de groupe d’un groupe abélien fini avec l’algèbre enveloppante d’une algèbre de Lie abélienne. Ensuite, nous les construisons à partir des parties plus, moins et zéro au moyen d’actions et de coactions appropriées par le formalisme de Sommerhäuser pour décrire des décompositions triangulaires.
We study the Hopf algebra structure of Lusztig’s quantum groups. First we show that the zero part is the tensor product of the group algebra of a finite abelian group with the enveloping algebra of an abelian Lie algebra. Second we build them from the plus, minus and zero parts by means of suitable actions and coactions within the formalism presented by Sommerhäuser to describe triangular decompositions.
Keywords: Quantum groups, Lusztig quantum divided power algebras, Nichols algebras
Mot clés : Groupes quantiques, algèbres de puissance divisée quantique de Lusztig, algèbres de Nichols
@article{AMBP_2020__27_2_131_0,
author = {Andruskiewitsch, Nicol\'as and Angiono, Iv\'an and Vay, Cristian},
title = {On the {Hopf} algebra structure of the {Lusztig} quantum divided power algebras},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {131--157},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {27},
number = {2},
year = {2020},
doi = {10.5802/ambp.393},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/ambp.393/}
}
TY - JOUR AU - Andruskiewitsch, Nicolás AU - Angiono, Iván AU - Vay, Cristian TI - On the Hopf algebra structure of the Lusztig quantum divided power algebras JO - Annales mathématiques Blaise Pascal PY - 2020 SP - 131 EP - 157 VL - 27 IS - 2 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - http://archive.numdam.org/articles/10.5802/ambp.393/ DO - 10.5802/ambp.393 LA - en ID - AMBP_2020__27_2_131_0 ER -
%0 Journal Article %A Andruskiewitsch, Nicolás %A Angiono, Iván %A Vay, Cristian %T On the Hopf algebra structure of the Lusztig quantum divided power algebras %J Annales mathématiques Blaise Pascal %D 2020 %P 131-157 %V 27 %N 2 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U http://archive.numdam.org/articles/10.5802/ambp.393/ %R 10.5802/ambp.393 %G en %F AMBP_2020__27_2_131_0
Andruskiewitsch, Nicolás; Angiono, Iván; Vay, Cristian. On the Hopf algebra structure of the Lusztig quantum divided power algebras. Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 2, pp. 131-157. doi : 10.5802/ambp.393. http://archive.numdam.org/articles/10.5802/ambp.393/
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