arXiv subjects

A. Van Daele

Publications and source records attributed to A. Van Daele.

12 recordsLinked to original sources

arXiv2022

Pairing and duality of algebraic quantum groupoids

Algebraic quantum groupoids have been developed by two of the authors (AVD and SHW) of this note in a series of papers. Regular multiplier Hopf algebroids are obtained also by two authors (TT and AVD). Integral theory and duality for those have been studied by one author here (TT). Finally, again two authors of us (TT and AVD) have investigated the relation between weak multiplier Hopf algebras and multiplier Hopf algebroids. In the paper 'Weak multiplier Hopf algebras III. Integrals and duality' (by AVD and SHW), one of the main results is that the dual of an algebraic quantum groupoid, admits a dual of the same type. In the paper 'On duality of algebraic quantum groupoids' (by TT), a result of the same nature is obtained for regular multiplier Hopf algebroids with a single faithful integral. The duality of regular weak multiplier Hopf algebras with a single integral can be obtained from the duality of regular multiplier Hopf algebroids. That is however not the obvious way to obtain this result. It is more difficult and less natural than the direct way. We will discuss this statement further in the paper. Nevertheless, it is interesting to investigate the relation between the two approaches to duality in greater detail. This is what we do in this paper. We build further on the intimate relation between weak multiplier Hopf algebras and multiplier Hopf algebroids. We now add the presence of integrals. That seems to be done best in a framework of dual pairs. It is in fact more general than the duality of these objects coming with integrals.

math.RA↗

arXiv2010

Algebraic quantum groups 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 on A making it a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group. Now let H be a finite subgroup of G and consider the subalgebra 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 it so that it will leave the subalgebra invariant (in the sense that the image is in the multiplier algebra of the tensor product of this subalgebra with itself). However, the modified coproduct on the subalgebra 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. Group-like projections in a *-algebraic quantum group A 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. In this paper we push this result further. On the one hand, we no longer assume the *-structure 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.

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arXiv2007

Groups with compact open subgroups and multiplier Hopf $^*$-algebras

For a locally compact group $G$ we look at the group algebras $C_0(G)$ and $C_r^*(G)$, and we let $f\in C_0(G)$ act on $L^2(G)$ by the multiplication operator $M(f)$. We show among other things that the following properties are equivalent: 1. $G$ has a compact open subgroup. 2. One of the $C^*$-algebras has a dense multiplier Hopf $^*$-subalgebra (which turns out to be unique). 3. There are non-zero elements $a\in C_r^*(G)$ and $f\in C_0(G)$ such that $aM(f)$ has finite rank. 4. There are non-zero elements $a\in C_r^*(G)$ and $f\in C_0(G)$ such that $aM(f)=M(f)a$. If $G$ is abelian, these properties are equivalent to: 5. There is a non-zero continuous function with the property that both $f$ and $\hat f$ have compact support.

math.OA↗

arXiv2007

Locally compact quantum groups. Radford's $S^4$ formula

Let $A$ be a finite-dimensional Hopf algebra. The left and the right integrals on $A$ are related by means of a distinguished group-like element $δ$ of $A$. Similarly, there is this element $\hatδ$ in the dual Hopf algebra $\hat A$. Radford showed that $$S^4(a)=δ^{-1}(\hatδ\triangleright a \triangleleft \hatδ^{-1})δ$$ for all $a$ in $A$ where $S$ is the antipode of $A$ and where $\triangleright$ and $\triangleleft$ are used to denote the standard left and right actions of $\hat A$ on $A$. The formula still holds for multiplier Hopf algebras with integrals (algebraic quantum groups). In the theory of locally compact quantum groups, an analytical form of Radford's formula can be proven (in terms of bounded operators on a Hilbert space). In this talk, we do not have the intention to discuss Radford's formula as such, but rather to use it, together with related formulas, for illustrating various aspects of the road that takes us from the theory of Hopf algebras (including compact quantum groups) to multiplier Hopf algebras (including discrete quantum groups) and further to the more general theory of locally compact quantum groups.

math.QA↗

arXiv2007

Algebraic Quantum Hypergroups

An algebraic quantum group is a 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.

math.RA↗

arXiv2007

The Fourier transform in quantum group theory

The Fourier transform, known in classical analysis, and generalized in abstract harmonic analysis, can also be considered in the theory of locally compact quantum groups. In this note, I discuss some aspects of this more general Fourier transform. In order to avoid technical difficulties, typical for the analytical approach, I will restrict to the algebraic quantum groups. Roughly speaking, these are the locally compact quantum groups that can be treated with purely algebraic methods (in the framework of multiplier Hopf algebras). I will illustrate various notions and results using not only classical Fourier theory on the circle group $\Bbb T$, but also on the additive group $\Bbb Q_p$ of $p$-adic numbers. It should be observed however that these cases are still too simple to illustrate the full power of the more general theory.

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arXiv2007

Compact and discrete subgroups of algebraic quantum groups I

Let $G$ be a locally compact group. Consider the C$^*$-algebra $C_0(G)$ of continuous complex functions on $G$, tending to 0 at infinity. The product in $G$ gives rise to a coproduct $Δ_G$ on the C$^*$-algebra $C_0(G)$. A locally compact {\it quantum} group is a pair $(A,Δ)$ of a C$^*$-algebra $A$ with a coproduct $Δ$ on $A$, satisfying certain conditions. The definition guarantees that the pair $(C_0(G),Δ_G)$ is a locally compact quantum group and that conversely, every locally compact quantum group $(A,Δ)$ is of this form when the underlying C$^*$-algebra $A$ is abelian. Assume now that $G$ is a locally compact group with a compact open subgroup $K$. The algebra of complex functions on $G$ of {\it polynomial type} is a dense multiplier Hopf $^*$-algebra with positive integrals (i.e. an algebraic quantum group}. The characteristic function of $K$ is a group-like projection in this algebraic quantum group. In this paper, we study group-like projections in an arbitrary algebraic quantum group. We find several associated objects that generalize the classical objects associated to a compact open subgroup of a locally compact group.

math.OA↗

arXiv2006

Multiplier Hopf algebras imbedded in C$^*$-algebraic quantum groups

Let $(A,Δ)$ be a locally compact quantum group and $(A_0,Δ_0)$ a regular multiplier Hopf algebra. We show that if $(A_0,Δ_0)$ can in some sense be imbedded in $(A,Δ)$, then $A_0$ will inherit some of the analytic structure of $A$. Under certain conditions on the imbedding, we will be able to conclude that $(A_0,Δ_0)$ is actually an algebraic quantum group with a full analytic structure. The techniques used to show this, can be applied to obtain the analytic structure of a $^*$-algebraic quantum group {\it in a purely algebraic fashion}. Moreover, the {\it reason} that this analytic structure exists at all, is that the one-parameter groups, such as the modular group and the scaling group, are diagonizable. In particular, we will show that necessarily the scaling constant $μ$ of a $^*$-algebraic quantum group equals 1. This solves an open problem.

math.OA↗

arXiv2006

A note on Radford's $S^4$ formula

In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a finite-dimensional Hopf algebra but also the case of any Hopf algebra with integrals (co-Frobenius Hopf algebras). The proof follows in a few lines from well-known formulas in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

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arXiv2004

Group-cograded multiplier Hopf (*-)algebras

Let $G$ be a group and assume that $(A_p)_{p\in G}$ is a family of algebras with identity. We have a {\it Hopf $G$-coalgebra} (in the sense of Turaev) if, for each pair $p,q\in G$, there is given a unital homomorphism $\co_{p,q}:A_{pq}\to A_p \ot A_q$ satisfying certain properties. Consider now the direct sum $A$ of these algebras. It is an algebra, without identity, except when $G$ is a finite group, but the product is non-degenerate. The maps $\co_{p,q}$ can be used to define a coproduct $\co$ on $A$ and the conditions imposed on these maps give that $(A,\co)$ is a multiplier Hopf algebra. It is $G$-cograded as explained in this paper. We study these so-called {\it group-cograded multiplier Hopf algebras}. They are, as explained above, more general than the Hopf group-coalgebras as introduced by Turaev. Moreover, our point of view makes it possible to use results and techniques from the theory of multiplier Hopf algebras in the study of Hopf group-coalgebras (and generalizations). In a separate paper, we treat the quantum double in this context and we recover, in a simple and natural way (and generalize) results obtained by Zunino. In this paper, we study integrals, in general and in the case where the components are finite-dimensional. Using these ideas, we obtain most of the results of Virelizier on this subject and consider them in the framework of multiplier Hopf algebras.

math.QA↗

arXiv2004

The Drinfel'd double for group-cograded multiplier Hopf algebras

Let $G$ be any group and let $K(G)$ denote the multiplier Hopf algebra of complex functions with finite support in $G$. The product in $K(G)$ is pointwise. The comultiplication on $K(G)$ is defined with values in the multiplier algebra $M(K(G) \otimes K(G))$ by the formula $(Δ(f)) (p,q) = f(pq)$ for all $f \in K(G)$ and $p, q \in G$. In this paper we consider multiplier Hopf algebras $B$ (over $\Bbb C$) such that there is an embedding $I: K(G) \to M(B)$. This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps $K(G)$ into the center of $M(B)$. These multiplier Hopf algebras are called {\it $G$-cograded multiplier Hopf algebras.} They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an {\it admissible} action $π$ of the group $G$ on a $G$-cograded multiplier Hopf algebra $B$. When $B$ is paired with a multiplier Hopf algebra $A$, we construct the Drinfel'd double $D^π$ where the coproduct and the product depend on the action $π$. We also treat the $^*$-algebra case. If $π$ is the trivial action, we recover the usual Drinfel'd double associated with the pair $ $. On the other hand, also the Drinfel'd double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a $G$-cograded multiplier Hopf algebra.

math.QA↗

arXiv1998

Actions of Multiplier Hopf Algebras

For an action $α$ of a group $G$ on an algebra $R$ (over $\Bbb C$), the crossed product $R\times_αG$ is the vector space of $R$-valued functions with finite support in $G$, together with the twisted convolution product given by $$(ξη)(p) = \sum_{q \in G} ξ(q) α_q (η(q^{-1}p))$$ where $p\in G$. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra $A$ on an algebra $R$, it is possible to make the tensor product $R\ot A$ into an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by $R# A$. In the group case, the action $α$ of $G$ on $R$ yields an action of the group algebra $\Bbb C G$ as a Hopf algebra on $R$ and the crossed $R\times_αG$ coincides with the smash product $R# \Bbb C G$. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras. The main result in the paper is a duality theorem for such actions. We consider dual pairs of multiplier Hopf algebras to formulate this duality theorem. We prove a result in the case of an algebraic quantum group and its dual. The more general case is only stated and will be proven in a separate paper on coactions. These duality theorems for actions are substantial generalizations of the corresponding theorem for Hopf algebras. Also the techniques that are used here to prove this result are slightly different and simpler.

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