Related papers: Measured quantum groupoids associated to proper dy…

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' works about locally compact quantum…

In this article, part of the author's thesis, we propose a definition for measured quantum groupoid. The aim is the construction of objects with duality including both quantum groups and groupoids. We base ourselves on J. Kustermans and S.…

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 the purely algebraic level and on the…

Given a dynamical twist for a finite dimensional Hopf algebra we construct two weak Hopf algebras, using methods of Xu and Etingof-Varchenko, and show that they are dual to each other. We generalize the theory of dynamical quantum groups to…

In the setting of von Neumann algebras, measurable quantum groupoids have successfully been axiomatized and studied by Enock, Vallin, and Lesieur, whereas in the setting of $C^{*}$-algebras, a similar theory of locally compact quantum…

Mimicking the von Neumann version of Kustermans and Vaes' locally compact quantum groups, Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras. In this article, we suppose that the…

A fundamental feature of quantum groups is that many come in pairs of mutually dual objects, like finite-dimensional Hopf algebras and their duals, or quantisations of function algebras and of universal enveloping algebras of Poisson-Lie…

The theory of measured quantum groupoids, as defined by Lesieur and myself, was made to generalize the theory of quantum groups made by Kustarmans and Vaes, but was only defined in a von Neumann algebra setting; Th. Timmermann constructed…

We introduce a notion of partial algebraic quantum group. This is an important special case of a weak multiplier Hopf algebra with integrals, as introduced in the work of Van Daele and Wang. At the same time, it generalizes the notion of…

Franck Lesieur had introduced in his thesis (now published in an expended and revised version in the {\it M\'emoires de la SMF} (2007)) a notion of measured quantum groupoid, in the setting of von Neumann algebras and a simplification of…

In this paper, we introduce C*-algebraic partial compact quantum groups, which are quantizations of topological groupoids with discrete object set and compact morphism spaces. These C*-algebraic partial compact quantum groups are…

We establish the equivalence of three versions of a finite dimensional quantum groupoid: a generalized Kac algebra introduced by T. Yamanouchi, a weak $C^*$-Hopf algebra introduced by G. Bohm, F. Nill and K. Szlachanyi (with an involutive…

The first example of a quantum group was introduced by P.~Kulish and N.~Reshetikhin. In their paper "Quantum linear problem for the sine-Gordon equation and higher representations" published in Zap. Nauchn. Sem. LOMI, 1981, Volume 101…

We propose a definition of compact quantum groupoids in the setting of C*-algebras, associate to such a quantum groupoid a regular C*-pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C*-bimodule and to pass to a…

We present a formal algebraic language to deal with quantum deformations of Lie-Rinehart algebras - or Lie algebroids, in a geometrical setting. In particular, extending the ice-breaking ideas introduced by Xu in [Ping Xu, "Quantum…

In his thesis ([L1]), which is published in an expended and revised version ([L2]), Franck Lesieur had introduced a notion of measured quantum groupoid, in the setting of von Neumann algebras, using intensively the notion of…

In this article, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups before -…

We define a natural concept of duality for the h-Hopf algebroids introduced by Etingof and Varchenko. We prove that the special case of the trigonometric SL(2) dynamical quantum group is self-dual, and may therefore be viewed as a…

Finite quantum groupoids can be described in many equivalent ways: In terms of the weak Hopf C*-algebras of B\"ohm, Nill, and Szlach\'anyi or the finite-dimensional Hopf-von Neumann bimodules of Vallin, and in terms of finite-dimensional…

An algebraic quantum group is a multiplier Hopf algebra with integrals. In this paper we will develop a theory of algebraic quantum hypergroups. It is very similar to the theory of algebraic quantum groups, except that the comultiplication…