A simple definition for locally compact quantum groups

In this Note we propose a simple definition of a locally compact quantum group in reduced form. By the word “reduced” we mean that we suppose the Haar weight to be faithful, and hence we define in fact arbitrary locally compact quantum groups represented on the L²-space of the Haar weight. We construct the multiplicative unitary associated with our quantum group. We construct the antipode with its polar decomposition, and the modular element. We prove the unicity of the Haar weights, define the dual and prove a Pontryagin duality theorem.     

Pontryagin Duality for a Class of Locally Compact Quantum Groups

A class of locally compact quantum semigroups in the sense of WORONOWICZ , containing the quantized Heisenberg group, is studied. The underlying quantum space is described in terms of a crossed product. We show the existence of a universal corepresentation and a Pontryagin dual, which turns out to be a twisted crossed product. The Pontryagin duality for these quantum semigroups is investigated.     

Aspects of compact quantum group theory

We show that if a compact quantum semigroup satisfies certain weak cancellation laws, then it admits a Haar measure, and using this we show that it is a compact quantum group. Thus, we obtain a new characterization of a compact quantum group. We also give a necessary and sufficient algebraic condition for the Haar measure of a compact quantum group to be faithful, in the case that its coordinate -algebra is exact. A representation is given for the linear dual of the Hopf -algebra of a compact quantum group, and a functional calculus for unbounded linear functionals is derived.

     

Structural properties of compact groups with measure-theoretic applications

Every compact group is Baire isomorphic to a product of compact metric spaces; the isomorphism takes the Haar measure on the group to a direct product measure. This topological connection between compact groups and products of compact metric spaces provides a unified treatment for (Baire) measures on compact groups and for measures on topological products of metric spaces.     

Uniform structures and square roots in topological groups

Topological groups which are free from small subgroups and topological groups with locally uniformly continuous group multiplication are considered. Results concerning square roots, one-parameter subgroups and extensions of local groups are obtained as well as some generalisations of theorems for locally compact groups.     

紧矩阵量子群G的余表示与其对偶量子群的关系

设 G=（A,△）为紧矩阵量子群,G为A的所有有限维光滑的、不可约余表示等价类的集合.本文通过（A,△）的一个余表示Vo构造了两个相互配对的集合,利用Hilbert C＊-模的理论证明它们分别为A和Baaj与Skandalis构造的量子群A,并且证明了对任意的α∈G,在A中都对应一个有限维投影算子Pα,满足 dim（α）=dim（pα）.     

Representations of Algebraic Quantum Groups and Reconstruction Theorems for Tensor Categories

We give a pedagogical survey of those aspects of the abstract representation theory of quantum groups which are related to the Tannaka–Krein reconstruction problem. We show that every concrete semisimple tensor *-category with conjugates is equivalent to the category of finite-dimensional nondegenerate *-representations of a discrete algebraic quantum group. Working in the self-dual framework of algebraic quantum groups, we then relate this to earlier results of S. L. Woronowicz and S. Yamagami. We establish the relation between braidings and R-matrices in this context. Our approach emphasizes the role of the natural transformations of the embedding functor. Thanks to the semisimplicity of our categories and the emphasis on representations rather than corepresentations, our proof is more direct and conceptual than previous reconstructions. As a special case, we reprove the classical Tannaka–Krein result for compact groups. It is only here that analytic aspects enter, otherwise we proceed in a purely algebraic way. In particular, the existence of a Haar functional is reduced to a well-known general result concerning discrete multiplier Hopf *-algebras.     

A Remark on Twists and the Notion of Torsion-free Discrete Quantum Groups

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki–Takai duality. The twisted Takesaki–Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld–Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.     

On projective representations for compact quantum groups

We study actions of compact quantum groups on type I-factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz?s results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).     

The Haar measure on finite quantum groups

By a finite quantum group, we will mean in this paper a finite-dimensional Hopf algebra. A left Haar measure on such a quantum group is a linear functional satisfying a certain invariance property. In the theory of Hopf algebras, this is usually called an integral. It is well-known that, for a finite quantum group, there always exists a unique left Haar measure. This result can be found in standard works on Hopf algebras. In this paper we give a direct proof of the existence and uniqueness of the left Haar measure on a finite quantum group. We introduce the notion of a faithful functional and we show that the Haar measure is faithful. We consider the special case where the underlying algebra is a -algebra with a faithful positive linear functional. Then the left and right Haar measures coincide. Finally, we treat an example of a root of unity algebra. It is an example of a finite quantum group where the left and right Haar measures are different. This note does not contain many new results but the treatment of the finite-dimensional case is very concise and instructive.

     

A compact quantum semigroup generated by an isometry

In this paper we construct a compact quantum semigroup structure on a Toeplitz algebra. We prove the existence of a subalgebra in the dual algebra isomorphic to the algebra of regular Borel measures on a circle with the convolution product. We also prove the existence of Haar functionals in the dual algebra and in the mentioned subalgebra. We show that this compact quantum semigroup contains a dense subalgebra with the structure of a weak Hopf algebra.     

Quantum Mechanics and Representation Theory: The New Synthesis

Quantum mechanics and representation theory, in the sense of unitary representations of groups on Hilbert spaces, were practically born together between 1925–1927, and have continued to enrich each other till the present day. Following a brief historical introduction, we focus on a relatively new aspect of the interaction between quantum mechanics and representation theory, based on the use of K-theory of C ^*-algebras. In particular, the study of the K-theory of the reduced C ^*-algebra of a locally compact group (which for a compact group is just its representation ring) has culminated in two fundamental conjectures, which are closely related to quantum theory and index theory, namely the Baum–Connes conjecture and the Guillemin–Sternberg conjecture. Although these conjectures were both formulated in 1982, and turn out to be closely related, so far there has been no interplay between them whatsoever, either mathematically or sociologically. This is presumably because the Baum–Connes conjecture is nontrivial only for noncompact groups, with current emphasis entirely on discrete groups, whereas the Guillemin–Sternberg conjecture has so far only been stated for compact Lie groups. As an elementary introduction to both conjectures in one go, indicating how the latter can be generalized to the noncompact case, this paper is a modest attempt to change this state of affairs.     

On sets of Haar measure zero in abelian polish groups

It is shown that the concept of zero set for the Haar measure can be generalized to abelian Polish groups which are not necessarily locally compact. It turns out that these groups, in many respects, behave like locally compact groups. Suitably modified, many theorems from harmonic analysis carry over to this case. A few applications are given and some open problems are mentioned.     

Compact quantum groups and group duality

In this paper, we consider the *-representations of compact quantum groups and group duality. The main results in the paper are: (1) there is a one-to-one correspondence between the *-representations of compact quantum groups and *-representations of the dual Banach *-algebra; (2) the category of commutative compact quantum groups (semigroups) is a dual category to the category of compact groups (semigroups); (3) the dual category of the category of locally compact groups (semigroups) is the category of commutative Hopf C^*-algebras with a particular property. Our group duality has the flavor of a Gelfand-Naimark type theorem for compact quantum groups, and for Hopf C^*-algebras.     

Haar system on a product of zero-dimensional compact groups

In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d=2, we describe all Haar functions.     

A wavelet theory for local fields and related groups

Let G be a locally compact abelian group with compact open subgroup H. The best known example of such a group is G = ℚ_p, the field of padic rational numbers (as a group under addition), which has compact open subgroup H = ℤ_p, the ring of padic integers. Classical wavelet theories, which require a non trivial discrete subgroup for translations, do not apply to G, which may not have such a subgroup. A wavelet theory is developed on G using coset representatives of the discrete quotient Ĝ/H^⊥ to circumvent this limitation. Wavelet bases are constructed by means of an iterative method giving rise to socalled wavelet sets in the dual group Ĝ. Although the Haar and Shannon wavelets are naturally antipodal in the Euclidean setting, it is observed that their analogues for G are equivalent.     

Quantum double constructions for compact quantum group

For a non-degenerate pair of compact quantum groups, we first construct the quantum double as an algebraic compact quantum group in an algebraic framework. Then by adopting some completion procedure, we give the universal and reduced quantum double constructions in the correspondence C＊-algebraic settings, which generalize Drinfeld＇s quantum double construction and yield new C＊-algebraic compact quantum groups.     

Quantum actions on discrete quantum spaces and a generalization of Clifford’s theory of representations

To any action of a compact quantum group on a von Neumann algebra which is a direct sum of factors we associate an equivalence relation corresponding to the partition of a space into orbits of the action. We show that in case all factors are finite-dimensional (i.e., when the action is on a discrete quantum space) the relation has finite orbits. We then apply this to generalize the classical theory of Clifford, concerning the restrictions of representations to normal subgroups, to the framework of quantum subgroups of discrete quantum groups, itself extending the context of closed normal quantum subgroups of compact quantum groups. Finally, a link is made between our equivalence relation in question and another equivalence relation defined by R. Vergnioux.