Algebraic Quantum Groups Research Articles

Algebraic quantum hypergroups of discrete type

In this paper we will study some structures of algebraic quantum hypergroups. First, we construct more examples of algebraic quantum hypergroups of discrete type. Next, we introduce the notion of a generalized quasi-Frobenius multiplier Hopf algebra and then show that generalized quasi-Frobenius multiplier Hopf algebras are a class of algebraic quantum hypergroups of discrete type. We also give some equivalent conditions for an algebraic quantum group to be of discrete type. Finally, we study sub-algebraic quantum hypergroups of discrete type and quotients of algebraic quantum hypergroups of discrete type.

ALGEBRAIC QUANTUM HYPERGROUPS II: CONSTRUCTIONS AND EXAMPLES

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 Δ on A making the pair (A,Δ) a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group as introduced and studied in [A. Van Daele, Adv. Math.140 (1998) 323]. Now let H be a finite subgroup of G and consider the subalgebra A1 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 Δ and define Δ1 as [Formula: see text] where f ∈ A1, p,q ∈ G and where n is the number of elements in the subgroup H. Then Δ1 will leave the subalgebra invariant (in the sense that the image is in the multiplier algebra M(A1 ⊗ A1) of the tensor product). However, it will no longer be an algebra map. So, in general we do not have an algebraic quantum group but a so-called algebraic quantum hypergroup as introduced and studied in [L. Delvaux and A. Van Daele, Adv. Math.226 (2011) 1134–1167]. Group-like projections in a *-algebraic quantum group A (as defined and studied in [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1]) give rise, in a natural way, to *-algebraic quantum hypergroups, very much like subgroups do as above for a *-algebraic quantum group associated to a group (again see [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1]). In this paper we push these results further. On the one hand, we no longer assume the *-structure as in [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1] while on the other hand, we allow the group-like projection to belong to the multiplier algebra M(A) of A and not only to A itself. Doing so, we not only get some well-known earlier examples of algebraic quantum hypergroups but also some interesting new ones.

Bounding Ext for modules for algebraic groups, finite groups and quantum groups

Given a finite root system Φ , we show that there is an integer c = c ( Φ ) such that dim Ext G 1 ( L , L ′ ) < c , for any reductive algebraic group G with root system Φ and any irreducible rational G -modules L , L ′ . There also is such a bound in the case of finite groups of Lie type, depending only on the root system and not on the underlying field. For quantum groups, a similar result holds for Ext n , for any integer n ⩾ 0 , using a constant depending only on n and the root system. When L is the trivial module, the same result is proved in the algebraic group case, thus giving similar bounded properties, independent of characteristic, for algebraic and generic cohomology. (A similar result holds for any choice of L = L ( λ ) , even allowing λ to vary, provided the p -adic expansion of lambda is limited to a fixed number of terms.) In particular, because of the interpretation of generic cohomology as a limit for underlying families of finite groups, the same boundedness properties hold asymptotically for finite groups of Lie type. The results both use, and have consequences for, Kazhdan–Lusztig polynomials. Appendix A proves a stable version, needed for small prime arguments, of Donkin's tilting module conjecture.

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.

Multiplier Hopf Algebras Imbedded in Locally Compact Quantum Groups

Let (A,∆) be a locally compact quantum group and (A0,∆0) a regular multiplier Hopf algebra. We show that if (A0,∆0) can be imbedded in (A,∆), then A0 will inherit some of the analytic structure of A. Under certain conditions on the imbedding, we will be able to conclude that (A0,∆0) is actually an algebraic quantum group with a full analytic structure. The techniques used to show this can also be applied to obtain the analytic structure of a ∗-algebraic quantum group in a purely algebraic fashion. Moreover, the reason that this analytic structure exists at all, is that the associated one-parameter groups, such as the modular group and the scaling group, are diagonizable. As an immediate corollary, we will show that the scaling constant μ of a ∗-algebraic quantum group equals 1. This solves an open problem posed in [13].

Constructing Quasitriangular Multiplier Hopf Algebras By Twisted Tensor Coproducts

Let A and B be multiplier Hopf algebras, and let R ∈ M(B ⊗ A) be an anti-copairing multiplier, i.e, the inverse of R is a skew-copairing multiplier in the sense of Delvaux [5]. Then one can construct a twisted tensor coproduct multiplier Hopf algebra A ⊗ R B. Using this, we establish the correspondence between the existence of quasitriangular structures in A ⊗ R B and the existence of such structures in the factors A and B. We illustrate our theory with a profusion of examples which cannot be obtained by using classical Hopf algebras. Also, we study the class of minimal quasitriangular multiplier Hopf algebras and show that every minimal quasitriangular Hopf algebra is a quotient of a Drinfel’d double for some algebraic quantum group.

Algebraic group actions on noncommutative spectra

Let G be an affine algebraic group and let R be an associative algebra with a rational action of G by algebra automorphisms. We study the induced G-action on the set Spec R of all prime ideals of R, viewed as a topological space with the Jacobson–Zariski topology, and on the subspace Rat R ⊆ Spec R consisting of all rational ideals of R. Here, a prime ideal P of R is said to be rational if the extended centroid $$ \mathcal{C}\left( {R/P} \right) $$ is equal to the base field. Our results generalize the work of Mœglin and Rentschler and of Vonessen to arbitrary associative algebras while also simplifying some of the earlier proofs. The map P ↦ ⋂g ∈ G g.P gives a surjection from Spec R onto the set G-Spec R of all G-prime ideals of R. The fibers of this map yield the so-called G-stratification of Spec R which has played a central role in the recent investigation of algebraic quantum groups, in particular, in the work of Goodearl and Letzter. We describe the G-strata of Spec R in terms of certain commutative spectra. Furthermore, we show that if a rational ideal P is locally closed in Spec R then the orbit G.P is locally closed in Rat R. This generalizes a standard result on G-varieties. Finally, we discuss the situation where G-Spec R is a finite set.

On topological centre problems and SIN quantum groups

Let A be a Banach algebra with a faithful multiplication and 〈 A ∗ A 〉 ∗ be the quotient Banach algebra of A ∗ ∗ with the left Arens product. We introduce a natural Banach algebra, which is a closed subspace of 〈 A ∗ A 〉 ∗ but equipped with a distinct multiplication. With the help of this Banach algebra, new characterizations of the topological centre Z t ( 〈 A ∗ A 〉 ∗ ) of 〈 A ∗ A 〉 ∗ are obtained, and a characterization of Z t ( 〈 A ∗ A 〉 ∗ ) by Lau and Ülger for A having a bounded approximate identity is extended to all Banach algebras. The study of this Banach algebra motivates us to introduce the notion of SIN locally compact quantum groups and the concept of quotient strong Arens irregularity. We give characterizations of co-amenable SIN quantum groups, which are even new for locally compact groups. Our study shows that the SIN property is intrinsically related to topological centre problems. We also give characterizations of quotient strong Arens irregularity for all quantum group algebras. Within the class of Banach algebras introduced recently by the authors, we characterize the unital ones, generalizing the corresponding result of Lau and Ülger. We study the interrelationships between strong Arens irregularity and quotient strong Arens irregularity, revealing the complex nature of topological centre problems. Some open questions by Lau and Ülger on Z t ( 〈 A ∗ A 〉 ∗ ) are also answered.

Galois objects for algebraic quantum groups

The basic elements of Galois theory for algebraic quantum groups were given in the paper ‘Galois Theory for Multiplier Hopf Algebras with Integrals’ by Van Daele and Zhang. In this paper, we supplement their results in the special case of Galois objects: algebras equipped with a Galois coaction by an algebraic quantum group, such that only the scalars are coinvariants. We show how the structure of these objects is as rich as the one of the quantum groups themselves: there are two distinguished weak K.M.S. functionals, related by a modular element, and there is an analogue of the antipode squared. We show how to reflect the quantum group across a Galois object to obtain a (possibly) new algebraic quantum group. We end by considering an example.

Bornological quantum groups

We introduce and study the concept of a bornological quantum group. This generalizes the theory of algebraic quantum groups in the sense of van Daele from the algebraic setting to the framework of bornological vector spaces. Working with bornological vector spaces, the scope of the latter theory can be extended considerably. In particular, the bornological theory covers smooth convolution algebras of arbitrary locally compact groups and their duals. Moreover Schwartz algebras of nilpotent Lie groups are bornological quantum groups in a natural way, and similarly one may consider algebras of functions on finitely generated discrete groups defined by various decay conditions. Another source of examples arises from deformation quantization in the sense of Rieffel. Apart from describing these examples we obtain some general results on bornological quantum groups. In particular, we construct the dual of a bornological quantum group and prove the Pontrjagin duality theorem.

From multiplicative unitaries to quantum groups II

It is shown that all important features of a C ∗ -algebraic quantum group ( A , Δ ) defined by a modular multiplicative W depend only on the pair ( A , Δ ) rather than the multiplicative unitary operator W. The proof is based on thorough study of representations of quantum groups. As an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary—without assuming existence of Haar weights.

On Amenability and Co-Amenability of Algebraic Quantum Groups and Their Corepresentations

AbstractWe introduce and study several notions of amenability for unitary corepresentations and *-representations of algebraic quantum groups, which may be used to characterize amenability and co-amenability for such quantumgroups. As a background for this study, we investigate the associated tensor C*-categories.

From double Lie groups to quantum groups

It is shown, using geometric methods, that there is a $C^*$-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a di

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.

The Drinfel'd double versus the Heisenberg double for an algebraic quantum group

Let A be a regular multiplier Hopf algebra with integrals. The dual of A, denoted by Â, is a multiplier Hopf algebra so that 〈 A ̂ ,A〉 is a pairing of multiplier Hopf algebras. We consider the Drinfel'd double, D= A ̂ ⋈A cop , associated to this pair. We prove that D is a quasitriangular multiplier Hopf algebra. More precisely, we show that the pair 〈 A ̂ ,A〉 has a “canonical multiplier” W∈M( A ̂ ⊗A) . The image of W in M( D⊗ D) is a generalized R-matrix for D. We use this image of W to deform the product of the dual multiplier Hopf algebra D̂ via the right action of D on D̂ which defines the pair 〈 D ̂ ,D〉 . As expected from the finite-dimensional case, we find that the deformation of the product in D̂ is related to the Heisenberg double A# A ̂ .

Twisted tensor coproduct of multiplier Hopf algebras

We put a non-trivial comultiplication on the natural tensor product algebra of two multiplier Hopf algebras by means of a “cotwisting map.” As a special case we characterize the dual of the Drinfel'd double of an algebraic quantum group. Because any finite-dimensional Hopf algebra is an algebraic quantum group, our characterization applies to the dual of the Drinfel'd double of a finite-dimensional Hopf algebra. Then it coincides with a result of Lu.

Amenability and co-amenability of algebraic quantum groups II

We continue our study of the concepts of amenability and co-amenability for algebraic quantum groups in the sense of A. Van Daele and our investigation of their relationship with nuclearity and injectivity. One major tool for our analysis is that every non-degenerate ∗ -representation of the universal C ∗ -algebra associated to an algebraic quantum group has a unitary generator which may be described in a concrete way.

Locally compact quantum groups in the von Neumann algebraic setting

In this paper we complete in several aspects the picture of locally compact quantum groups. First of all we give a definition of a locally compact quantum group in the von Neumann algebraic setting and show how to deduce from it a $C^*$-algebraic quantum group. Further we prove several results about locally compact quantum groups which are important for applications, but were not yet settled in our paper [8]. We prove a serious strengthening of the left invariance of the Haar weight, and we give several formulas connecting the locally compact quantum group with its dual. Loosely speaking we show how the antipode of the locally compact quantum group determines the modular group and modular conjugation of the dual locally compact quantum group.

The Size of the Intrinsic Group of a Multiplier Hopf Algebra

In the theory of Hopf algebras, grouplike elements are well studied. We argue that this notion is too restrictive when dealing with multiplier Hopf algebras. We introduce the “intrinsic” group, situated in the multiplier algebra of the multiplier Hopf algebra. When dealing with an algebraic quantum group A, the intrinsic group of the dual  characterizes a special class of automorphisms on A. For multiplier Hopf algebras which are paired in the sense of Drabant and Van Daele. (Drabant, B., Van Daele, A. Pairing and quantum double for multiplier Hopf algebras. (2001). Algebras and Representation Theory 4:109–132), we prove that the intrinsic group characterizes the semi-invariants of the action associated to the pairing.

The analytic structure of algebraic quantum groups

In [Adv. Math. 140 (1998) 323–366], Van Daele introduced the notion of an algebraic quantum group. We proved in [Internat. J. Math. 8 (8) (1997) 1067–1139] that such algebraic quantum groups give rise to reduced C∗-algebraic quantum groups in the sense of [J. Kustermans, S. Vaes, Ann. Sci. École Norm. Sup. (4) 33 (2000) 837–934]. After introducing the universal C∗-algebraic quantum group associated to an algebraic one, we will pull down the analytic structure of this C∗-algebraic quantum group to the algebraic quantum group. The multiplier algebra of the dual quantum group will be realized as a space of linear functionals on the original algebra. We also identify the analytic structure of the dual quantum group in terms of the analytic structure of the original algebraic quantum group.