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Induced representations of locally compact quantum groups in the bornological setting

Abstract : In the first part of this thesis, we continue the development of bornological quantum groups, introduced by Voigt. We add the hypothesis of *-involution to its original definition and then we show in a second part that such a bornological quantum group gives rise to a locally compact quantum group in the sense of Kustermans & Vaes. For this we use a method similar to the one used by Van Daele and Kustermans in the case of algebraic quantum groups. Then we develop a general theory of induction of unitary representations for boundological quantum groups, generalizing the original work of Rieffel on locally compact groups. We then show that our induction functor coincides, in the bornological framework, with the (more general) one developed by Vaes. Finally, we apply our induction method to the special case of complex semisimple quantum groups. We first show that the parabolically induced representations given by our method coincide with the ad hoc definition, due to Arano. Then we construct a homogeneous space, similar to the one found in the works of Clare, Crisp & Higson, in order to present the theory of parabolically induced representations of a semisimple quantum group in a geometric way, leading to a reformulation of results of Monk & Voigt.
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Submitted on : Friday, November 18, 2022 - 4:22:11 PM
Last modification on : Saturday, November 19, 2022 - 3:21:40 AM


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  • HAL Id : tel-03860533, version 1


Damien Rivet. Induced representations of locally compact quantum groups in the bornological setting. General Mathematics [math.GM]. Université Clermont Auvergne, 2021. English. ⟨NNT : 2021UCFAC106⟩. ⟨tel-03860533⟩



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