C^*-algebraic quantum groups arising from algebraic quantum groups

It is shown that there is a $C^*$-algebraic quantum group related to any double Lie group. An algebra underlying this quantum group is an algebra of a differential groupoid naturally associated with a double Lie group

We give some sufficient conditions for the injectivity of actions of compact quantum groups on $C^{\ast}$-algebra. As an application, we prove that any faithful smooth action by a compact quantum group on a compact smooth (not necessarily connected) manifold is injective. A similar result is proved for actions on $C^{\ast}$- algebras obtained by Rieffel-deformation of compact, smooth manifolds.

It is known that any covering space of a topological group has the natural structure of a topological group. This article discusses a noncommutative generalization of this fact. A noncommutative generalization of the topological group is a quantum group. Also there is a noncommutative generalization of a covering. The combination of these algebraic constructions yields a motive to research the generalization of coverings of topological groups. In contrary to a topological gro...

A tutorial introduction is given to general Hopf algebras and to general compact quantum groups. In the definition and further treatment of compact quantum groups C*-algebras are avoided. Contact with Woronowicz's compact matrix quantum groups is made at a later stage.

Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct on A making it a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group. Now let H be a finite subgroup of G and consider the subalgebra of functions in A that are constant on double cosets of H. The coproduct in general will not leave this algebra invariant but we can modify it so that it will leave ...

We have written down a set of notes on compact quantum groups from which all the different aspects can be learned in an easy way and such that a lot of insight can be obtained without too much effort. Compact quantum groups have been studied by several authors, from different points of view. The difference lies mainly in the choice of the axioms and consequently, in the way the main results are proven. These results however are essentially the same in all these cases. In ...

In this article, we establish the duality between the generalised Drinfeld double and generalised quantum codouble within the framework of modular or manageable (not necessarily regular) multiplicative unitaries, and discuss several properties.

In this paper, we quantize universal gauge groups such as SU(\infty), in the sigma-C*-algebra setting. More precisely, we propose a concise definition of sigma-C*-quantum groups and explain the concept here. At the same time, we put this definition in the mathematical context of countably compactly generated groups as well as C*-compact quantum groups.

We survey some of our old results given in [CE95] and [CE10] and present some new ones in the last three sections.We survey some of our old results given in [CE95] and [CE10] and present some new ones in the last three sections.

In this paper we construct and study the representation theory of a Hopf C^*-algebra with approximate unit, which constitutes quantum analogue of a compact group C^*-algebra. The construction is done by first introducing a convolution-product on an arbitrary Hopf algebra H with integral, and then constructing the L_2 and C^*-envelopes of H (with the new convolution-product) when H is a compact Hopf *-algebra.