Duality functors for quantum groupoids

Journal of Algebra, 2006

In this paper we realize the dynamical categories introduced in our previous paper as categories of modules over bialgebroids; we study the bialgebroids arising in this way. We define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to the dynamical categories. We show that dynamical twists over an arbitrary base give rise to bialgebroid twists.

2003

In this work we prove, in a self contained way, that any finite dimensional C *-quantum groupoid can be deformed in order that the square of the antipode is the identity on the base. We also prove that for any C *-quantum groupoid with non abelian base, there is uncountably many C *-quantum groupoids with the same underlying algebra structure but which are not isomorphic to it. In fact, the C *quantum groupoids are closed in an analog of the procedure presented by D.Nikshych ([N] 3.7) in a more general situation.

We give a survey of the theory of finite quantum groupoids (weak Hopf algebras), including foundations of the theory and applications to finite depth subfactors, dynamical deformations of quantum groups, and invariants of knots and 3-manifolds.

Demonstratio Mathematica, 2004

We show that the Hopf algebra on a transformation groupoid T = E x G, where G is a finite group acting on the total space of a principal fibre boundle over M = E/G, is the cross product of the algebras C'(E) and CG. We study duality properties of this algebra, and consider quantization on orbit spaces program in this context.

Nuclear Physics B - Proceedings Supplements, 1990

Mathematical Physics in Mathematics and Physics, 2001

Journal of Noncommutative Geometry, 2016

In this article, when G is a locally compact quantum group, we associate, to a braided-commutative G-Yetter-Drinfel'd algebra (N, a, a) equipped with a normal faithful semi-finite weight verifying some appropriate condition (in particular if it is invariant with respect to a, or to a), a structure of a measured quantum groupoid. The dual structure is then given by (N, a, a). Examples are given, especially the situation of a quotient type co-ideal of a compact quantum group. This construction generalizes the standard construction of a transformation groupoid. Most of the results were announced by the second author in 2011, at a conference in Warsaw. Contents 1. Introduction 2. Preliminaries 3. Invariant weights on Yetter-Drinfel'd algebras 4. The Hopf bimodule associated to a braided-commutative Yetter-Drinfel'd algebra 5. Measured quantum groupoid structure associated to a braided-commutative Yetter-Drinfel'd algebra equipped with an appropriate weight 6. Duality 7. Examples 8. Quotient type co-ideals and Morita equivalence References Date: october 15.

2007

For a complex or real algebraic group G, with g := Lie(G) , quantizations of global type are suitable Hopf algebras F q [G] or U q (g) over C q, q −1. Any such quantization yields a structure of Poisson group on G, and one of Lie bialgebra on g : correspondingly, one has dual Poisson groups G * and a dual Lie bialgebra g *. In this context, we introduce suitable notions of quantum subgroup and, correspondingly, of quantum homogeneous space, in three versions: weak, proper and strict (also called flat in the literature). The last two notions only apply to those subgroups which are coisotropic, and those homogeneous spaces which are Poisson quotients; the first one instead has no restrictions whatsoever. We end the paper with some explicit examples of application of our recipes.

2000

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 measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and toetale

International Journal of Mathematics, 2003

We develop a general framework to deal with the unitary representations of quantum groups using the language of C * -algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.