Building on the work of the first part of this series of papers, we classify all ergodic actions of the quantum SU(2) groups (at some fixed parameter q in absolute value between 0 and 1) on operator algebras. The classification is in terms of graphs with a certain weight structure on them which may be interpreted as a random walk satisfying a particular reciprocality condition. In particular, the paper offers an easier and more conceptual proof of A. Wassermann's classification of all ergodic actions of SU(2) on operator algebras. The paper also calculates the K-theory of the C*-algebra underlying an ergodic action in terms of the associated graph (here the weight function does not enter).
It is well-known classically that if a compact Hausdorff group acts freely on a compact Hausdorff space, any pullback of a finite-dimensional representation of the group gives rise to a vector bundle on the quotient space. In this paper, we show that this property continues to hold in the quantum setting, namely if a compact quantum group acts freely on a (unital or non-unital) C*-algebra, by which is meant that a certain density condition, known as the Ellwood condition, is satisfied, then automatically any cotensor product with a finite-dimensional corepresentation of the quantum group gives a module (in fact a Hilbert C*-module) which is finitely generated projective over the algebra of coinvariants. The main tool is the connection of the above analytic condition of freeness with the notion of Galois map, well-known from the algebraic theory of Hopf algebras. In fact, we show that the above projectivity condition is equivalent with the Galois map admitting an adjoint with respect to suitable chosen Hilbert module structures.
We show that non-standard Podles spheres whose parameters differ by an integer are related to each other by an equivariant Morita equivalence. Our method uses the realization of Podles spheres as subquotients of the quantized enveloping algebra of SU(2) with a deformed *-operation. We investigate as well the situation for the quantum projective real plane.
We investigate the relation between compact quantum groups with a fixed corepresentation, and the associated quantum subgroups obtained by equating to zero the elements on the off-diagonals of the fixed corepresentation. Special emphasis is placed on those quantum groups for which all such quotients are commutative. Part of what triggered the investigations in this paper is the study of quantum symmetries of classical spaces, as studied before by T. Banica, D. Goswami and S. Wang.
We show how the function algebras of quantized SU(2) and quantized E(2) (double cover of the Euclidian transformation group of the plane) can be put into correspondence with each other (Morita equivalence), in a way which respects the comultiplication (comonoidal Morita equivalence). The result is obtained by integrating such a correspondence on the dual level of the infinitesimal QUE *-algebras.
In this elementary paper, we show how the QUE *-algebras of su(2), e(2) and su(1,1) can be put into correspondence with each other by means of a linking *-algebra. We also show that quotients of this linking *-algebra give rise to well-known homogeneous spaces such as Podles spheres.
We develop a theory of projective representations in the setting of Woronowicz' compact quantum groups. Basically, a projective representation of such a quantum group is just an action of it on B(H), for H a Hilbert space. We develop a Peter-Weyl-theory of such projective representations, and show that the obstruction of such a projective representation can be realized as a Galois object for its group von Neumann algebra. We give a characterisation of all projective representations of the duals of classical discrete groups. We also develop a theory of fusion between ordinary representations and projective representations, and illustrate it with a specific example of a projective representation for quantized SU(2).
We show that the function algebras of three quantum groups, namely quantized SU(2), quantized E(2) (double cover of Euclidian transformation group of the plane) and quantized extended SU(1,1), can be put into correspondence (Morita equivalence) in a way which carries over also the comultiplication (comonoidal Morita equivalence). The paper relies heavily on results from q-analysis (basic hypergeometric functions).
In this paper, we extend the notion of W*-Morita equivalence between von Neumann algebras to a notion of W*-Morita equivalence between von Neumann bialgebras, that is von Neumann algebras equipped with a comultiplication. It can be thought of as a combination of two processes: a `Morita deformation' of the underlying von Neumann algebra, and a deformation of the coproduct. We characterise the objects which provide W*-Morita equivalences (these play the role of `projective finitely generated generating modules' in this context). We show that W*-Morita equivalence preserves the property of having Haar weights, i.e. the Morita equivalence relation is internal to locally compact quantum groups. The particular case of twisting with 2-cocycles is also briefly discussed. This corresponds to the case where only the coproduct is deformed, while the underlying Morita equivalence of algebras becomes trivial.
The notion of a Galois object was introduced by P. Schauenburg in the setting of Hopf algebras. In this paper, we give a similar theory in the setting of locally compact quantum groups, i.e. `the operator algebraic approach to quantum groups'. The results are very similar to the ones obtained by P. Schauenburg, although the techniques are very different (and much more technical). They heavily depend on Tomita-Takesaki theory, and more generally the theory of non-commutative integration. The main result is that a Galois object for a locally compact quantum group can always be completed to a bi-Galois object, that is, there exists a second quantum group which also acts on the Galois object, and such that the two quantum groups are then fully symmetric with respect to each other. Thus, from a different viewpoint, one may also see the theory of Galois objects as a construction method for new quantum groups.
We study inclusions of multiplier Hopf algebras (algebraic objects) into von Neumann or C*-algebraic quantum group (analytic objects), and examine how the existence of such an inclusion puts constraints on both structures. The main importance of this paper lies in the techniques we use: we obtain some simplifications to earlier results by J. Kustermans, which allow us to push the structure theory for *-algebraic quantum groups a lot further (e.g., we prove triviality of the so-called scaling constant in this case).
This paper constructs a special example of a 2-cocycle twist of a quantum group. The peculiarity is that the original quantum group is compact, while the twisted quantum group is not (although it still has (unbounded) Haar weights). One may interpret this result in the following way: in the setting of C*-algebras, one can have non-unital C*-algebras which are C*-Morita equivalent with unital ones (e.g. the complex numbers are C*-Morita equivalent with the C*-algebra of compact operators on a separable Hilbert space). The above example is an instance of this, but also takes into account a deformation of the coproduct at the same time. Another peculiarity of this example is that while the Morita equivalence is trivial on the von Neumann algebra level, the Morita equivalence is no longer trivial on the C*-algebra level. The example itself is constructed by means of infinite tensor products of compact quantum groups.
In this short note, we show that the universal unitary quantum groups associated to a 2-by-2 matrix are either free Araki-Woods factors or the group factor of the free group on 2 generators. This extends a result of T. Banica.
The notion of a Galois object was introduced by P. Schauenburg in the setting of Hopf algebras. In this paper, we develop this theory in the setting of Van Daeles algebraic quantum groups. This article is in a sense a preparation for the article 'Galois coactions and cocycle twisting for locally compact quantum groups', giving much insight into how to proceed in the technically more complicated setting of locally compact quantum groups.
We deform quantized enveloping algebras of a semi-simple Lie algebra by certain characters on the positive part of the root lattice. More precisely, the relation linking the positive and negative Borel part is modified. The resulting family of algebras are not Hopf algebras, but rather the family as a whole has a joined coproduct. In particular, a subclass of these algebras admits right coactions by the unmodified quantized enveloping algebra. In general, we study the Verma module theory of these algebras, paying special attention to their associated real structure and the unitarity question for Verma modules. As a result, we obtain a new way of producing quantum homogeneous spaces for the quantizations of certain compact semi-simple Lie groups.
The Haagerup property for and the weak amenability of the duals of the compact Kac quantum groups A_o(n) were shown to hold by respectively M. Brannan and A. Freslon. In this note, we show by different techniques that these properties hold for all A_o(F), with F an arbitrary matrix. The proof hinges on three separate techniques: first, one reduces the question to SU_q(2) by monoidal equivalence (one considers first the case where A_o(F) does not split as a free product). Then, on SU_q(2), one tries to find a sufficient supply of appropriate approximation functionals which are central. To find these, we make the connection with the representation theory of the quantum double of SU_q(2) (the 'quantum Lorentz group'). Finally, using a technique due to Pytlik and Szwarc involving holomorphicity of a family of completely bounded maps, one proves the weak amenability and Haagerup property in one sweep.
Given a compact quantum group and a quantum homogeneous space for it, one can consider the category of equivariant `quantum vector bundles' over the quantum homogeneous space. There is a natural action of the representation category of the compact quantum group on this new category. The aim of this paper is to give an abstract characterization of such module C*-categories, and to show that one can completely recover the quantum homogeneous space from this data (establishing thus a Tannaka-Krein reconstruction for quantum homogeneous spaces).
We explain in detail how the notion of freeness of a compact quantum group action can be equivalently defined in an analytic and an algebraic fashion. Here the analytic definition is a priori less demanding, and thus easier to check. By means of examples, we show how the analytic characterization is indeed sometimes more useful than the algebraic one in proving freeness of certain given actions.
We show how one can construct algebraic quantum hypergroups from algebraic quantum groups by passing to the set of fixed points for the square of the antipode. We show how this is related with the bicrossed product construction, performed with respect to the scaling group. We give some concrete computations in the case of SUq(2).