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.     

Algebraic quantum hypergroups

An algebraic quantum group is a regular 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 is no longer assumed to be a homomorphism. We still require the existence of a left and of a right integral. There is also an antipode but it is characterized in terms of these integrals. We construct the dual, just as in the case of algebraic quantum groups and we show that the dual of the dual is the original quantum hypergroup. We define algebraic quantum hypergroups of compact type and discrete type and we show that these types are dual to each other. The algebraic quantum hypergroups of compact type are essentially the algebraic ingredients of the compact quantum hypergroups as introduced and studied (in an operator algebraic context) by Chapovsky and Vainerman.We will give some basic examples in order to illustrate different aspects of the theory. In a separate note, we will consider more special cases and more complicated examples. In particular, in that note, we will give a general construction procedure and show how known examples of these algebraic quantum hypergroups fit into this framework.     

A Natural Extension of a Left Invariant Lower Semi–Continuous Weight

In this paper, we describe a natural method to extend left invariant weights on C*–algebraic quantum groups. This method is then used to improve the invariance property of a left invariant weight. We also prove some kind of uniqueness result for left Haar weights on C*–algebraic quantum groups arising from algebraic ones.     

Examples of non-compact quantum group actions

We present two examples of actions of non-regular locally compact quantum groups on their homogeneous spaces. The homogeneous spaces are defined in a way specific to these examples, but the definitions we use have the advantage of being expressed in purely C^∗-algebraic language. We also discuss continuity of the obtained actions. Finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups.     

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.     

Twisted Partial Actions: A Classification of Regular C*-Algebraic Bundles

We introduce the notion of continuous twisted partial actionsof a locally compact group on a C*-algebra. With such, we constructan associated C*-algebraic bundle called the semidirect productbundle. Our main theorem shows that, given any C*-algebraicbundle which is second countable and which satisfies a certainregularity condition (automatically satisfied if the unit fibrealgebra is stable), there is a continuous twisted partial actionof the base group on the unit fibre algebra, whose associatedsemidirect product bundle is isomorphic to the given one. 1991Mathematics Subject Classification: 46L05, 46L40, 46L55.     

Quantum families of quantum group homomorphisms

The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper, we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.     

Embeddable quantum homogeneous spaces

We discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups. On the von Neumann algebra level we recall an interesting duality for such objects studied earlier by M. Izumi, R. Longo, S. Popa for compact Kac algebras and by M. Enock in the general case of locally compact quantum groups. A definition of a quantum homogeneous space is proposed along the lines of the pioneering work of Vaes on induction and imprimitivity for locally compact quantum groups. The concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces. Among various examples we single out the quantum analog of the quotient of the Cartesian product of a quantum group with itself by the diagonal subgroup, analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles. The former turns out to be an interesting application of the duality mentioned above.     

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.     

拟信息基,ω-代数Cpo,和SFP domain

G.Sam b in引入了(代数)信息基的概念,并证明了代数Scott D om a in范畴和信息基范畴是等价的.B.R.C.Bedrega l给出了ω-代数cpo和SFP dom a in的刻划.而G.Q.Zhang通过序结构给出了SFP dom a in的刻划.本文将引入了拟信息基的概念并给出了ω-代数cpo和SFP dom a in的刻划.     

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.

     

Uncertainty principles for locally compact quantum groups

In this paper, we prove the Donoho–Stark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the Hirschman–Beckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.     

Gluing Compact Matrix Quantum Groups

We study glued tensor and free products of compact matrix quantum groups with cyclic groups – so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing.

     

Connected components of compact matrix quantum groups and finiteness conditions

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of constructing the identity component by introducing canonical approximating transfinite sequences of subgroups. These sequences have lengths ≤1 in the classical case but can be countably infinite for duals of discrete groups. We give examples of free product quantum groups where the identity component is not normal and the associated sequence has length 1.     

A Quantum Modification of Relative Chen–Ruan Cohomology

In this paper, by using the de Rham model of Chen–Ruan cohomology, we define the relative Chen–Ruan cohomology ring for a pair of almost complex orbifold(G, H) with H being an almost sub-orbifold of G. Then we use the Gromov–Witten invariants ofG, the blow-up of G along H,to give a quantum modification of the relative Chen–Ruan cohomology ring H*CR(G, H) when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.     

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).     

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）.     

AF代数中子代数间等距代数同构的扩张

本文证明了具有二次交换子性质的ＡＦ代数中的子代数间的等距代数同构可以扩张成其包络Ｃ＊代数的＊同构．由此肯定地回答了［４］中２．１２提出的问题．     

A new form and a ⋆-algebraic structure of quantum stochastic integrals in Fock space

An algebraic definition of the basic quantum process for the noncommutative stochastic calculus is given in terms of the Fock representation of a Lie ⋆-algebra of matrices in a pseudo-Euclidean space. An operator definition of the quantum stochastic integral is given and its continuity is proved in a projective limit uniform operator topology. A new form of quantum stochastic equations, revealing the ⋆-algebraic structure of quantum Ito's formula, is given. (Conferenza tenuta il 21 settembre 1988)