Non-semi-regular quantum groups coming from number theory

Saad Baaj (1) , Georges Skandalis (2) , Stefaan Vaes (2)

1 LMBP - Laboratoire de Mathématiques Blaise Pascal
2 IMJ - Institut de Mathématiques de Jussieu

In this paper, we study C*-algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi-regular: the crossed product of the quantum group acting on itself by translations does not contain any compact operator. We describe all corepresentations of these quantum groups and the associated universal C*-algebras. On the way, we provide several remarks on C*-algebraic properties of quantum groups and their actions.

Domaines: Algèbres d'opérateurs [math.OA]; Algèbres quantiques [math.QA]

Saad Baaj, Georges Skandalis, Stefaan Vaes. Non-semi-regular quantum groups coming from number theory. Communications in Mathematical Physics, 2003, 235 (1), pp.139-167. ⟨hal-00012250⟩