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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.

Consider a C*-algebra $A$ with a comultiplication $\Delta$. 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 $\Delta(a)(1\otimes h)=a \otimes h$ 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.

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.

We define and investigate pairings of multiplier Hopf algebras. It is shown that two dually paired regular multiplier Hopf ($*$-)algebras $A$ and $B$ yield a quantum double multiplier Hopf ($*$-)algebra which is again regular. Integrals on $A$ and $B$ induce an integral on the quantum double. The results generalize pairing and quantum double construction from ordinary Hopf algebras to multiplier Hopf algebras., AMS-TeX, 21 pages