Locally compact quantum groups

In this paper we propose a simple definition of a locally compact quantum g roup in reduced form. By the word 'reduced' we mean that we suppose the Haa r weight to be faithful. So in fact we define and study an arbitrary locall y compact quantum group, represented on the L-2-space of its Haar weight. F or this locally compact quantum group we construct the antipode with polar decomposition. We construct the associated multiplicative unitary and prove that it is manageable in the sense of Woronowicz. We define the modular el ement and prove the uniqueness of the Haar weights. Following [15] we const ruct the reduced dual, which will again be a reduced locally compact quantu m group. Finally we prove that the second dual is canonically isomorphic to the original reduced locally compact quantum group, extending the Pontryag in duality theorem. (C) 2000 Editions scientifiques et medicales Elsevier S AS.