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Keywords:

multiplicative unitaries, products, algebras, Science & Technology, Physical Sciences, Physics, Mathematical, Physics, MULTIPLICATIVE UNITARIES, PRODUCTS, math.OA, math.QA, 0101 Pure Mathematics, 0105 Mathematical Physics, 0206 Quantum Physics, Mathematical Physics, 4902 Mathematical physics, 4904 Pure mathematics, 5107 Particle and high energy physics

Abstract:

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.