A new characterisation of idempotent states on finite and compact quantum groups

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. To cite this article: U. Franz, A. Skalski, C R. Acad. Sci. Paris, Ser. 1347 (2009). (C) 2009 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.