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Let $A$ be a non-degenerate algebra over the complex numbers and $\Delta$ a homomorphism from $A$ to the multiplier algebra $M(A\otimes A)$. Consider the linear maps $T_1$ and $T_2$ from $A\otimes A$ to $M(A\otimes A)$ defined by \begin{equation*} T_1(a\otimes b)=\Delta(a)(1\otimes b) \qquad\text{and}\qquad T_2(c\otimes a)=(c\otimes 1)\Delta(a). \end{equation*} The pair $(A,\Delta)$ is a multiplier Hopf algebra if these two maps have range in $A\otimes A$ and are bijections from $A\otimes A$ to itself. In our recent paper on the Larson-Sweedler theorem, single sided multiplier Hopf algebras emerge in a natural way. For this case, instead of requiring the above for the maps $T_1$ and $T_2$, we now have this property for the maps $T_1$ and $T_4$ or for $T_2$ and $T_3$ where \begin{equation*} T_3(a\otimes b)=(1\otimes b)\Delta(a) \qquad\text{and}\qquad T_4(c\otimes a)=\Delta(a)(c\otimes 1). \end{equation*} As it turns out, also for these single sided multiplier Hopf algebras, the existence of a unique counit and antipode can be proven. In fact, rather surprisingly, using the properties of the antipode, one can actually show that for a single sided multiplier Hopf algebra all four canonical maps are bijections from $A\otimes A$ to itself. In other words, $(A,\Delta)$ is automatically a regular multiplier Hopf algebra. We take the advantage of this approach to reconsider some of the known results for a regular multiplier Hopf algebra.

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.

Let $(A,\Delta)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A,\Delta)$ 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 $\Delta$ and define $\Delta'(a)=(u\otimes 1)\Delta(a)(v\otimes 1)$ for all $a \in A$. We show in this paper that $(A,\Delta')$ is again a regular weak multiplier Hopf algebra and we obtain formulas for the various data of $(A,\Delta')$ in terms of the data associated with the original pair $(A,\Delta)$. 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 $(A,\Delta)$ and the modified version $(A,\Delta')$. However, the same method can be used to associate another regular multiplier Hopf algebroid to the triple $(A,\Delta,\Delta')$. 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.

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.

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discuss invertibility of the antipode, and present some examples and special cases., Comment: now covers the non-regular case; to appear in Comm. in Algebra

A weak multiplier Hopf algebra is a pair (A,\Delta) of a non-degenerate idempotent algebra A and a coproduct $\Delta$ 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)=\Delta(a)(1\otimes b) and T_2(a\ot b)=(a\otimes 1)\Delta(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 \Delta(A)(1\ot A)=E(A\ot A) and (A\otimes 1)\Delta(A)=(A\otimes A)E. The second condition determines the behavior of the coproduct on the legs of E. We require (\Delta\otimes \iota)(E)=(\iota\otimes\Delta)(E)=(1\otimes E)(E\ot 1)=(E\otimes 1)(1\otimes E) where $\iota$ is the identity map and where $\Delta\otimes \iota$ and $\iota\otimes\Delta$ 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.

Let $G$ be a {\it finite group}. Consider the algebra $A$ of all complex functions on G (with pointwise product). Define a coproduct $\Delta$ on A by $\Delta(f)(p,q)=f(pq)$ where $f\in A$ and $p,q\in G$. Then $(A,\Delta)$ 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 $\Delta(f)(p,q)$ as above when $pq$ is defined. If we let $\Delta(f)(p,q)=0$ otherwise, we still get a coproduct on $A$, but $\Delta(1)$ will no longer be the identity in $A\ot A$. The pair $(A,\Delta)$ 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 $(A,\Delta)$ 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.

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 $\Delta_#$ on the smash product $A # B$ making the pair $(A # B,\Delta_#)$ 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.