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Measured Quantum Groupoids in Action
Michael Enock, Institut de Mathématiques de Jussieu, Paris, France
A publication of the Société Mathématique de France.
Mémoires de la Société Mathématique de France
2008; 150 pp; softcover
Number: 114
ISBN-10: 2-85629-265-8
ISBN-13: 978-2-85629-265-5
List Price: US$55
Individual Members: US$49.50
Order Code: SMFMEM/114
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In his thesis Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras, and a simplification of Lesieur's axioms is presented in an appendix of this publication.

In this book the author develops the notions of actions, crossed-product, and obtains a biduality theorem, following what had been done by Stefaan Vaes for locally compact quantum groups. Moreover, the author proves that the inclusion of the initial algebra into its crossed-product is depth 2, which gives a converse of a result proved by Jean-Michel Vallin and the author. More precisely, to any action of a measured quantum groupoid, the author associates another measured quantum groupoid. In particular, starting from an action of a locally compact quantum group, he obtains a measured quantum groupoid canonically associated to this action; when the action is outer, this measured quantum groupoid is the initial locally compact quantum group.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians interested in quantum groups.

Table of Contents

  • Introduction
  • Preliminaries
  • Measured quantum groupoids
  • Left invariance revisited
  • Corepresentations of measured quantum groupoids
  • Actions of measured quantum groupoids
  • Some technical properties of actions
  • The standard implementation of an action: The case of a \(\delta\)-invariant
  • Crossed-product and dual actions
  • An auxilliary weight on the crossed-product
  • Biduality
  • Characterization of crossed-products
  • Dual weight; bidual weight; depth 2 inclusion associated to an action
  • The measured quantum groupoid associated to an action
  • Appendix
  • Bibliography
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