A simple definition for locally compact quantum groups |
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| Affiliation: | 1. Department of Mathematics, University College Cork, Western Road, Cork, Ireland;2. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium;1. Institut Montpellierain Alexander Grothendieck, UMR 5149 du CNRS, France;2. Department of Mathematics, University of Adelaide, Adelaide 5005, Australia;1. Department of Mathematics, Shanghai Jiaotong University, Shanghai, 200240, China;2. Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA;1. Guangdong Provincial Key Lab of Agro-Animal Genomics and Molecular Breeding, College of Animal Science, South China Agricultural University, Guangzhou 510642, P.R. China;2. College of Biological and Food Engineering, Guangdong University of Education, Guangzhou 510303, P.R. China |
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| Abstract: | In this Note we propose a simple definition of a locally compact quantum group in reduced form. By the word “reduced” we mean that we suppose the Haar weight to be faithful, and hence we define in fact arbitrary locally compact quantum groups represented on the L2-space of the Haar weight. We construct the multiplicative unitary associated with our quantum group. We construct the antipode with its polar decomposition, and the modular element. We prove the unicity of the Haar weights, define the dual and prove a Pontryagin duality theorem. |
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