Measured quantum groupoids associated to proper dynamical quantum groups

@article{Timmermann2015MeasuredQG,
  title={Measured quantum groupoids associated to proper dynamical quantum groups},
  author={Thomas Timmermann},
  journal={Journal of Noncommutative Geometry},
  year={2015},
  volume={9},
  pages={35-82}
}
  • T. Timmermann
  • Published 28 June 2012
  • Mathematics
  • Journal of Noncommutative Geometry
Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang-Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II_1 factors. In this article, we associate to suitable dynamical quantum groups, which are a purely algebraic objects, Hopf C*-bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical… 

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References

SHOWING 1-10 OF 37 REFERENCES

Measured quantum groupoids

In this article, we give a definition for measured quantum groupoids. We want to get objects with duality extending both quantum groups and groupoids. We base ourselves on J. Kustermans and S. Vaes'

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation

Inclusions of Von Neumann Algebras and Quantum Groupoı̈ds II

Abstract From a depth 2 inclusion of von Neumann algebras M 0 ⊂M1 , with an operator-valued weight verifying a regularity condition, we construct a pseudo-multiplicative unitary, which leads to two

Free dynamical quantum groups and the dynamical quantum group SU_q(2)

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on

Weak Hopf algebras and quantum groupoids

We give a detailed comparison between the notion of a weak Hopf algebra (also called a quantum groupoid by Nikshych and Vainerman), and that of a $\times_R$-bialgebra due to Takeuchi (and also called

Solutions of the Quantum Dynamical Yang–Baxter Equation and Dynamical Quantum Groups

Abstract:The quantum dynamical Yang–Baxter (QDYB) equation is a useful generalization of the quantum Yang–Baxter (QYB) equation. This generalization was introduced by Gervais, Neveu, and Felder.

Exchange Dynamical Quantum Groups

Abstract:For any simple Lie algebra ? and any complex number q which is not zero or a nontrivial root of unity, %but may be equal to 1 we construct a dynamical quantum group (Hopf algebroid), whose

On the dynamical Yang-Baxter equation

This talk is inspired by two previous ICM talks, by V.Drinfeld (1986) and G.Felder (1994). Namely, one of the main ideas of Drinfeld's talk is that the quantum Yang-Baxter equation (QYBE), which is

C*-pseudo-multiplicative unitaries, Hopf C*-bimodules and their Fourier algebras

  • T. Timmermann
  • Mathematics
    Journal of the Institute of Mathematics of Jussieu
  • 2011
Abstract We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules

The Dynamical Yang-Baxter Equation, Representation Theory, and Quantum Integrable Systems

0. Preface 1. Introduction 2. Background material 3. Intertwiners, fusion and exchange operators for Lie algebras 4. Quantum groups 5. Intertwiners, fusion and exchange operators for U_q (g) 6.