We show that idempotent states on finite quantum groups correspond to pre-subgroups in the sense of Baaj, Blanchard, and Skandalis. It follows that the lattices formed by the idempotent states on a finite quantum group and by its coidalgebras are isomorphic. We show, furthermore, that these lattices are also isomorphic for compact quantum groups, if one restricts to expected coidalgebras.
Nous donnons une caractérisation des états idempotents sur un groupe quantique fini en termes des pré-sous-groupes introduits par Baaj, Blanchard, et Skandalis, et en déduisons un isomorphisme entre le réseau des états idempotents et le réseau des sous-algèbres coïdéales d'un groupe quantique fini. Cet isomorphisme s'étend aux groupes quantiques compacts, si on le restreind au sous-algèbres coïdéales expectées.
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@article{CRMATH_2009__347_17-18_991_0,
author = {Franz, Uwe and Skalski, Adam},
title = {A new characterisation of idempotent states on finite and compact quantum groups},
journal = {Comptes Rendus. Math\'ematique},
pages = {991--996},
publisher = {Elsevier},
volume = {347},
number = {17-18},
year = {2009},
doi = {10.1016/j.crma.2009.06.015},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.06.015/}
}
TY - JOUR AU - Franz, Uwe AU - Skalski, Adam TI - A new characterisation of idempotent states on finite and compact quantum groups JO - Comptes Rendus. Mathématique PY - 2009 SP - 991 EP - 996 VL - 347 IS - 17-18 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.06.015/ DO - 10.1016/j.crma.2009.06.015 LA - en ID - CRMATH_2009__347_17-18_991_0 ER -
%0 Journal Article %A Franz, Uwe %A Skalski, Adam %T A new characterisation of idempotent states on finite and compact quantum groups %J Comptes Rendus. Mathématique %D 2009 %P 991-996 %V 347 %N 17-18 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.06.015/ %R 10.1016/j.crma.2009.06.015 %G en %F CRMATH_2009__347_17-18_991_0
Franz, Uwe; Skalski, Adam. A new characterisation of idempotent states on finite and compact quantum groups. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 991-996. doi : 10.1016/j.crma.2009.06.015. http://www.numdam.org/articles/10.1016/j.crma.2009.06.015/
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