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In the paper Algebraic quantum groups and duality I, we consider a pairing $(a,b)\mapsto\langle a,b\rangle$ of regular multiplier Hopf algebras $A$ and $B$. When $A$ has integrals and when $B$ is the dual of $A$, we can describe the duality with an element $V$ in the multiplier algebra $M(B\otimes A)$ satisfying and defined by $\langle V, a\otimes b\rangle=\langle a,b\rangle$ for all $a,b$. Properties of the dual pair are formulated in terms of this multiplier $V$. It acts, in a natural way, on $A\otimes A$ as the canonical map $T$, given by $T(a\otimes a')=\Delta(a)(1\otimes a')$. In this second paper on the subject, we assume that the pairing is coming from a multiplier Hopf $^*$-algebra with positive integrals. In this case, the positive right integral on $A$ can be used to construct a Hilbert space $\mathcal H$. The duality $V$ now acts as a unitary operator on the Hilbert space tensor product $\mathcal H\otimes\mathcal H$. This eventually makes it possible to complete the algebraic quantum group to a locally compact quantum group. The procedure to pass from the algebraic quantum group to the operator algebraic completion has been treated in the literature but the construction is rather involved because of the necessary use of left Hilbert algebras. In this paper, we give a comprehensive, yet concise and somewhat simpler approach. It should be considered as a springboard to the more complicated theory of locally compact quantum groups.

Let $(A,\Delta)$ be a finite-dimensional Hopf algebra. The linear dual $B$ of $A$ is again a finite-dimensional Hopf algebra. The duality is given by an element $V\in B\otimes A$, defined by $\langle V,a\otimes b\rangle=\langle a,b\rangle$ where $a\in A$ and $b\in B$. We use $\langle\,\cdot\, , \,\cdot\,\rangle$ for the pairings. In the introduction of this paper, we recall the various properties of this element $V$ as sitting in the algebra $B\otimes A$. More generally, we can consider an algebraic quantum group $(A,\Delta)$. We use the term here for a regular multiplier Hopf algebra with integrals. For $B$ we now take the dual $\widehat A$ of $A$. It is again an algebraic quantum group. In this case, the duality gives rise to an element $V$ in the multiplier algebra $M(B\otimes A)$. Still, most of the properties of $V$ in the finite-dimensional case are true in this more general setting. The focus in this paper lies on various aspects of the duality between $A$ and its dual $\widehat A$. Among other things we include a number of formulas relating the objects associated with an algebraic quantum group and its dual. This note is meant to give a comprehensive, yet concise (and sometimes simpler) account of these known results. This is part I of a series of three papers on this subject. The case of a multiplier Hopf $^*$-algebra with positive integrals is treated in detail in part II and part III.

This is the last part of a series of three papers on the subject. In the first part we have considered the duality of algebraic quantum groups. In that paper, we use the term algebraic quantum group for a regular multiplier Hopf algebra with integrals. We treat the duality as studied in that theory. In the second part we have considered the duality for multiplier Hopf $^*$-algebras with positive integrals. The main purpose of that paper is to explain how it gives rise to a locally compact quantum group in the sense of Kustermans and Vaes. In a preliminary section of the second part we have mentioned the analytic structure of such a $^*$-algebraic quantum group. We have not gone into the details, except for that part needed to obtain that the scaling constant is trivial and for proving that the composition of a positive left integral with the antipode gives a right integral that is again positive. Also the construction of the square root. of the modular element is discussed and used to obtain the two Haar weights on the associated locally compact quantum group. In this paper we treat the analytic structure of the $^*$-algebraic quantum group in detail. The analytic structure is intimately related with the modular structure. We obtain a collection of interesting formulas, relating the two structures. Further we compare these results with formulas obtained in the framework of general locally compact quantum groups. As for the first two papers in this series, also here no new results are obtained. On the other hand the approach is different, more direct and instructive. It might help the reader for understanding the more general theory of locally compact quantum groups.

A finite quantum hypergroup is a finite-dimensional unital algebra $A$ over the field of complex numbers. There is a coproduct on $A$, a coassociative map from $A$ to $A\otimes A$ assumed to be unital, but it is not required to be an algebra homomorphism. There is a counit that is supposed to be a homomorphism. Finally, the main extra requirement is the existence of a faithful left integral with the right properties. For such a finite quantum hypergroup, the dual can be constructed. It is again a finite quantum hypergroup. The more general concept of an algebraic quantum hypergroup is studied in \cite{De-VD1, De-VD2}. If the underlying algebra of an algebraic quantum hypergroup is finite-dimensional, it is a finite quantum hypergroup in the sense of this paper. Here we treat finite quantum hypergroups independently with an emphasis on the development of the notion. It is meant to clarify the various steps taken in \cite{La-VD3a} and \cite{La-VD3b}. In \cite{La-VD3b} we introduce and study a still more general concept of quantum hypergroups. It not only contains the algebraic quantum hypergroups from \cite{De-VD1}, but also some topological cases. We naturally encounter these quantum hypergroups in our work on bicrossproducts, see \cite{La-VD4, La-VD5, La-VD7}. We include some examples to illustrate the theory. One kind is coming from a finite group and a subgroup. The other examples are taken from the bicrossproduct theory.

Let $B$ and $C$ be non-degenerate idempotent algebras and assume that $E$ is a regular separability idempotent in $M(B\otimes C)$. Define $A=C\otimes B$ and $\Delta:A\to M(A\otimes A)$ by $\Delta(c\otimes b)=c\otimes E\otimes b$. The pair $(A,\Delta)$ is a weak multiplier Hopf algebra. Because we assume that $E$ is regular, it is a regular weak multiplier Hopf algebra. There is a faithful left integral on $(A,\Delta)$ that is also right invariant. Therefore, we call $(A,\Delta)$ a unimodular algebraic quantum groupoid. By the general theory, the dual $(\widehat A,\widehat \Delta)$ can be constructed and it is again an algebraic quantum groupoid. In this paper, we treat this algebraic quantum groupoid and its dual in great detail. The main purpose is to illustrate various aspects of the general theory. For this reason, we will also recall the basic notions and results of separability idempotents and weak multiplier Hopf algebras with integrals. The paper is to be considered as an expository note., Comment: arXiv admin note: text overlap with arXiv:1701.04951

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.

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.

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.

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.

We have written down a set of notes on compact quantum groups from which all the different aspects can be learned in an easy way and such that a lot of insight can be obtained without too much effort. Compact quantum groups have been studied by several authors, from different points of view. The difference lies mainly in the choice of the axioms and consequently, in the way the main results are proven. These results however are essentially the same in all these cases. In these notes, we mainly follow the approach of Woronowicz and we extensively motivate this choice. We give a complete and rather detailed treatment, starting from a simple set of axioms and obtaining the main results. We also discuss the most common examples and show how they fit into the framework. During this process, we compare with the existing other treatments., Comment: 43 pages, latex2e