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Alfons Van Daele

Publications and source records attributed to Alfons Van Daele.

At least 37 records · Page 2Linked to original sources

arXiv2016

The Larson-Sweedler theorem for weak multiplier Hopf algebras

The Larson-Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra. The result has been generalized to finite-dimensional weak Hopf algebras by Vecsernyés. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by Böhm, Gómez-Torrecillas and López-Centella. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting. Our treatment of this material is motivated by the prospect of such a theory.

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arXiv2015

Weak Multiplier Hopf Algebras II. The source and target algebras

In this paper, we continue the study of weak multiplier Hopf algebras. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $Δ(1)$) belongs to the multiplier algebra $M(B\otimes C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a separability idempotent. We also consider special cases and examples in this paper. In particular, we see how for any weak multiplier Hopf algebra, it is possible to make $C\otimes B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\otimes C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

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arXiv2015

Separability Idempotents and Multiplier Algebras

Consider two non-degenerate algebras B and C over the complex numbers. We study a certain class of idempotent elements E in the multiplier algebra of the tensor product of B with C, called separability idempotents. The conditions include the existence of non-degenerate anti-homomorphisms from B to M(C) and from C to M(B), the multiplier algebras of C and B respectively. They are called the antipodal maps. There also exist what we call distinguished linear functionals. They are unique and faithful in the regular case. The notion is more restrictive than what is generally considered in the case of (finite-dimensional) algebras with identity. The separability idempotents we consider in this paper are of a Frobenius type. It seems to be quite natural to consider this type of separability idempotents in the case of non-degenerate algebras possibly without an identity. One example is coming from a discrete quantum group A. Here we take B=C=A and E is the image under the coproduct of h, where h is the normalized cointegral. The antipodal maps coincide with the original antipode and the distinguished linear functionals are the integrals. Another example is obtained from a weak multiplier Hopf algebra A. Now B and C are the images of the source and target maps. For E we take the canonical idempotent. Again the antipodal maps come from the antipode of the weak multiplier Hopf algebra. These two examples have motivated the study of separability idempotents as used in this paper. We use the separability idempotent to study modules over the base algebras B and C. In the regular case, we obtain a structure theorem for these algebras.

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arXiv2014

Locally Compact Quantum Groups. A von Neumann Algebra Approach

In this paper, we give an alternative approach to the theory of locally compact quantum groups, as developed by Kustermans and Vaes. We develop the theory completely within the von Neumann algebra framework. At various points, we also do things differently. We have a different treatment of the antipode. We obtain the uniqueness of the Haar weights in an early stage. We take advantage of this fact when deriving the other main results in the theory. We also give a slightly different approach to duality. Finally, we collect, in a systematic way, several important formulas. In an appendix, we indicate very briefly how the $C^*$-approach and the von Neumann algebra approach eventually yield the same objects.

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arXiv2014

Modified weak multiplier Hopf algebras

Let $(A,Δ)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A,Δ)$ and by $B$ the image of the source map. Recall that $B$ is a non-degenerate algebra, sitting nicely in the multiplier algebra $M(A)$ of $A$ so that also $M(B)$ can be viewed as a subalgebra of $M(A)$. Assume that $u,v$ are invertible elements in $M(B)$ so that $E(vu\otimes 1)E=E$. This last condition is obviously fulfilled if $u$ and $v$ are each other inverses, but there are also other cases. Now modify $Δ$ and define $Δ'(a)=(u\otimes 1)Δ(a)(v\otimes 1)$ for all $a \in A$. We show in this paper that $(A,Δ')$ is again a regular weak multiplier Hopf algebra and we obtain formulas for the various data of $(A,Δ')$ in terms of the data associated with the original pair $(A,Δ)$. In the case of a finite-dimensional weak Hopf algebra, the above deformation is a special case of the twists as studied by Nikshych and Vainerman. It is known that any regular weak multiplier Hopf algebra gives rise in a natural way to a regular multiplier Hopf algebroid. This result applies to both the original weak multiplier Hopf algebra $(A,Δ)$ and the modified version $(A,Δ')$. However, the same method can be used to associate another regular multiplier Hopf algebroid to the triple $(A,Δ,Δ')$. This turns out to give an example of a regular multiplier Hopf algebroid that does not arise from a regular weak multiplier Hopf algebra although the base algebra is separable Frobenius.

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arXiv2014

Multiplier Hopf algebroids arising from weak multiplier Hopf algebras

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra.

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arXiv2012

Weak Multiplier Hopf Algebras. The main theory

A weak multiplier Hopf algebra is a pair (A,Δ) of a non-degenerate idempotent algebra A and a coproduct $Δ$ on A. The coproduct is a coassociative homomorphism from A to the multiplier algebra M(A\otimes A) with some natural extra properties (like the existence of a counit). Further we impose extra but natural conditions on the ranges and the kernels of the canonical maps T_1 and T_2 defined from A\otimes A to M(A\otimes A) by T_1(a\otimes b)=Δ(a)(1\otimes b) and T_2(a\ot b)=(a\otimes 1)Δ(b). The first condition is about the ranges of these maps. It is assumed that there exists an idempotent element E\in M(A\otimes A) such that Δ(A)(1\ot A)=E(A\ot A) and (A\otimes 1)Δ(A)=(A\otimes A)E. The second condition determines the behavior of the coproduct on the legs of E. We require (Δ\otimes ι)(E)=(ι\otimesΔ)(E)=(1\otimes E)(E\ot 1)=(E\otimes 1)(1\otimes E) where $ι$ is the identity map and where $Δ\otimes ι$ and $ι\otimesΔ$ are extensions to the multipier algebra M(A\otimes A). Finally, the last condition determines the kernels of the canonical maps T_1 and T_2 in terms of this idempotent E by a very specific relation. From these conditions we develop the theory. In particular, we construct a unique antipode satisfying the expected properties and various other data. Special attention is given to the regular case (that is when the antipode is bijective) and the case of a *-algebra (where regularity is automatic). Weak Hopf algebras are special cases of such weak multiplier Hopf algebras. Conversely, if the underlying algebra of a (regular) weak multiplier Hopf algebra has an identity, it is a weak Hopf algebra. Also any groupoid, finite or not, yields two weak multiplier Hopf algebras in duality.

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arXiv2012

Weak Multiplier Hopf Algebras. Preliminaries, motivation and basic examples

Let $G$ be a {\it finite group}. Consider the algebra $A$ of all complex functions on G (with pointwise product). Define a coproduct $Δ$ on A by $Δ(f)(p,q)=f(pq)$ where $f\in A$ and $p,q\in G$. Then $(A,Δ)$ is a Hopf algebra. If $G$ is only a {\it groupoid}, so that the product of two elements is not always defined, one still can consider $A$ and define $Δ(f)(p,q)$ as above when $pq$ is defined. If we let $Δ(f)(p,q)=0$ otherwise, we still get a coproduct on $A$, but $Δ(1)$ will no longer be the identity in $A\ot A$. The pair $(A,Δ)$ is not a Hopf algebra but a weak Hopf algebra. If $G$ is a {\it group}, but {\it no longer finite}, one takes for $A$ the algebra of functions with finite support. Then $A$ has no identity and $(A,Δ)$ is not a Hopf algebra but a multiplier Hopf algebra. Finally, if $G$ is a {\it groupoid}, but {\it not necessarily finite}, the standard construction above, will give, what we call in this paper, a weak multiplier Hopf algebra. Indeed, this paper is devoted to the development of this 'missing link': {\it weak multiplier Hopf algebras}. We spend a great part of this paper to the motivation of our notion and to explain where the various assumptions come from. The goal is to obtain a good definition of a weak multiplier Hopf algebra. Throughout the paper, we consider the basic examples and use them, as far as this is possible, to illustrate what we do. In particular, we think of the finite-dimensional weak Hopf algebras. On the other hand however, we are also inspired by the far more complicated existing analytical theory. In forthcoming papers on the subject, we develop the theory further.

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arXiv2012

Bicrossproducts of algebraic quantum groups

Let $A$ and $B$ be two algebraic quantum groups (i.e. multiplier Hopf algebras with integrals). Assume that $B$ is a right $A$-module algebra and that $A$ is a left $B$-comodule coalgebra. If the action and coaction are matched, it is possible to define a coproduct $Δ_#$ on the smash product $A # B$ making the pair $(A # B,Δ_#)$ into an algebraic quantum group. In this paper, we continue the study of these objects. First, we study the various data of the bicrossproduct $A # B$, such as the modular automorphisms, the modular elements, ... and obtain formulas in terms of the data of the components $A$ and $B$. Secondly, we look at the dual of $A # B$ (in the sense of algebraic quantum groups) and we show it is itself a bicrossproduct (of the second type) of the duals $\hatA$ and $\hatB$. The result is immediate for finite-dimensional Hopf algebras and therefore it is expected also for algebraic quantum groups. However, it turns out that some aspects involve a careful argument, mainly due to the fact that coproducts and coactions have ranges in the multiplier algebras of the tensor products and not in the tensor product itself. Finally, we also treat some examples in this paper. We have included some of the examples that are known for finite-dimensional Hopf algebras and show how they can also be obtained for more general algebraic quantum groups. We also give some examples that are more typical for algebraic quantum groups. In particular, we focus on the extra structure, provided by the integrals and associated objects. It should be mentioned that with examples of bicrossproducts of algebraic quantum groups, we do get examples that are essentially different from those commonly known in Hopf algebra theory.

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arXiv2009

Bicrossproducts of multiplier Hopf algebras

In this paper, we generalize Majid's bicrossproduct construction. We start with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A # B. The left coaction of B on A gives a possible coproduct on A # B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for this to be a proper coproduct on A # B. The result is again a regular multiplier Hopf algebra. Majid's construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair (C,D) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We will show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D will yield a duality between A # B and the smash product C # D. We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The *-algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature. Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G=KH and the intersection of H and K is trivial will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention.

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arXiv2008

Tools for working with multiplier Hopf algebras

Let $(A,Δ)$ be a multiplier Hopf algebra. In general, the underlying algebra $A$ need not have an identity and the coproduct $Δ$ does not map $A$ into $A\otimes A$ but rather into its multiplier algebra $M(A\otimes A)$. In this paper, we study {\it some tools} that are frequently used when dealing with such multiplier Hopf algebras and that are typical for working with algebras without identity in this context. The {\it basic ingredient} is a unital left $A$-module $X$. And the basic construction is that of extending the module by looking at linear maps $ρ:A\to X$ satisfying $ρ(aa')=aρ(a')$ where $a,a'\in A$. We write the module action as multiplication. Of course, when $x\in X$, and when $ρ(a)=ax$, we get such a linear map. And if $A$ has an identity, all linear maps $ρ$ have this form for $x=ρ(1)$. However, the point is that in the case of a non-unital algebra, the space of such maps is in general strictly bigger than $X$ itself. We get an {\it extended module}, denoted by $\bar X$ (for reasons that will be explained in the paper). We study all sorts of more complicated situations where such extended modules occur and we illustrate all of this with {\it several examples}, from very simple ones to more complex ones where iterated extensions come into play. We refer to cases that appear in the literature. We use this basic idea of extending modules to explain, in a more rigorous way, the so-called {\it covering technique}, which is needed when using {\it Sweedler notations} for coproducts and coactions. Again, we give many examples and refer to the existing literature where this technique is applied.

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arXiv2004

The Larson-Sweedler theorem for multiplier Hopf algebras

Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of Hopf algebras and work with multiplier Hopf algebras. Moreover, whereas in the finite-dimensional case, there is a complete symmetry between the bialgebra and its dual, this is no longer the case in infinite dimensions. Therefore we consider a direct version (with integrals) and a dual version (with cointegrals) of the Larson-Sweedler theorem. We also add some results about the antipode. Furthermore, in the process of this paper, we obtain a new approach to multiplier Hopf algebras with integrals.

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arXiv2004

Quasitriangular (G-cograded) multiplier Hopf algebras

We put the known results on the antipode of a usual quasitriangular Hopf algebra into the framework of multiplier Hopf algebras. We illustrate with examples which can not be obtained using classical Hopf Algebras. The focus of the present paper lies on the class of the so-called group cograded multiplier Hopf algebras. By doing so, we bring the results of quasitriangular Hopf group-coalgebras (as introduced by Turaev) to the more general framework of multiplier Hopf algebras.

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arXiv2004

Quasi-Discrete Locally Compact Quantum Groups

Consider a C*-algebra $A$ with a comultiplication $Δ$. This pair is usually thought of as locally compact quantum semi-group. When these notes were written, in 1993, it was not at all clear what the extra assumptions on the comultiplication should be for this pair to be a 'locally compact quantum group'. This only became clear in 1999 thanks to the work of Kustermans and Vaes. In the compact case however, rather natural conditions are formulated by Woronowicz and a good notion of a compact quantum group was available in 1993. In these notes, we consider another class of locally compact quantum groups. We assume the existence of a non-zero element $h$ in $A$ satisfying $Δ(a)(1\otimes h)=a \otimes h$ for all $a$ in $A$. We add some natural conditions and we speak of a 'quasi-discrete locally compact quantum group'. We also discuss the discrete case. We prove the existence of the Haar measure, the regular representation, the fundamental unitary and we obtain the reduced dual. The quasi-discrete case is, in principle, more general than the discrete case. Shortly after these notes were written, Kustermans showed that in fact, the two classes coincided. For this reason, these notes have not been published. Nevertheless, some of the results and techniques seem to be usful and in recent work, we came across similar settings. Therefore, we have decided to make these notes available on the net. We have added some comments at the end of the introduction and we have updated the reference list. But apart from these minor changes, the notes are still as they were written and distributed in 1993.

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arXiv2002

The multiplicative unitary as a basis for duality

The classical duality theory associates to an abelian group a dual companion. Passing to a non-abelian group, a dual object can still be defined, but it is no longer a group. The search for a broader category which should include both the groups and their duals, points towards the concept of quantization. Classically, the regular representation of a group contains the complete information about the structure of this group and its dual. In this article, we follow Baaj and Skandalis and study duality starting from an abstract version of such a representation: the multiplicative unitary. We suggest extra conditions which will replace the regularity and irreducibility of the multiplicative unitary. From the proposed structure of a "quantum group frame", we obtain two objects in duality. We equip these objects with certain group-like properties, which make them into candidate quantum groups. We consider the concrete example of the quantum az+b-group, and discuss how it fits into this framework. Finally, we construct the crossed product of a quantum group frame with a locally compact group.

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arXiv2002

Multiplier Hopf *-algebras with positive integrals: A laboratory for locally compact quantum groups

Any multiplier Hopf *-algebra} with positive integrals gives rise to a locally compact quantum group (in the sense of Kustermans and Vaes). As a special case of such a situation, we have the compact quantum groups (in the sense of Woronowicz) and the discrete quantum groups (as introduced by Effros and Ruan). In fact, the class of locally compact quantum groups arising from such multiplier Hopf *-algebras} is self-dual. The most important features of these objects are (1) that they are of a purely algebraic nature and (2) that they have already a great complexity, very similar to the general locally compact quantum groups. This means that they can serve as a good model for the general objects, at least from the purely algebraic point of view. They can therefore be used to study various aspects of the general case, without going into the more difficult technical aspects, due to the complicated analytic structure of a general locally compact quantum group. In this paper, we will first recall the notion of a multiplier Hopf *-algebra} with positive integrals. Then we will illustrate how these {\it algebraic quantum groups} can be used to gain a deeper understanding of the general theory. An important tool will be the Fourier transform. We will also concentrate on certain actions and how they behave with respect to this Fourier transform. On the one hand, we will study this in a purely algebraic context while on the other hand, we will also pass to the Hilbert space framework.

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arXiv2001

The Haar measure on some locally compact quantum groups

The Haar measure on some locally compact quantum groups is constructed. The main example we treat is the az+b-group of Woronowicz. We also briefly consider some other examples (like the ax+b-group). We get the first examples of a locally compact quantum group where the Haar measure is not invariant with respect to the scaling group.

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arXiv1999

Hopf C*-algebras

In this paper we introduce the notion of a Hopf C*-algebra and construct the counit and antipode. A Hopf C*-algebra is a C*-algebra with comultiplication satisfying some extra condition which makes possible the construction of the counit and antipode. The leading example is of course the C*-algebra of continuous, vanishing at infinity functions on a locally compact group. Also locally compact quantum groups will be examples. We include several formulas for the counit and antipode which are familiar from Hopf algebra theory.

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