A Hopf algebra is a pair (A, 2) where A is an associative algebra with identity and 2 a homomorphism form A to A A satisfying certain conditions. If we drop the assumption that A has an identity and if we allow 2 to have values in the socalled multiplier algebra M(A A), we get a natural extension of the notion of a Hopf algebra. We call this a multiplier Hopf algebra. The motivating example is the algebra of complex functions with finite support on a group with the comultiplication defined as dual to the product in the group. Also for these multiplier Hopf algebras, there is a natural notion of left and right invariance for linear functionals (called integrals in Hopf algebra theory). We show that, if such invariant functionals exist, they are unique (up to a scalar) and faithful. For a regular multiplier Hopf algebra (A, 2) (i.e., with invertible antipode) with invariant functionals, we construct, in a canonical way, the dual (A , 2 ). It is again a regular multiplier Hopf algebra with invariant functionals. It is also shown that the dual of (A , 2 ) is canonically isomorphic with the original multiplier Hopf algebra (A, 2). It is possible to generalize many aspects of abstract harmonic analysis here. One can define the Fourier transform; one can prove Plancherel's formula. Because any finite-dimensional Hopf algebra is a regular multiplier Hopf algebra and has invariant functionals, our duality theorem applies to all finite-dimensional Hopf algebras. Then it coincides with the usual duality for such Hopf algebras. But our category of multiplier Hopf algebras also includes, in a certain way, the discrete (quantum) groups and the compact (quantum) groups. Our duality includes the duality between discrete quantum groups and compact quantum groups. In particular, it includes the duality between compact abelian groups and discrete abelian groups. One of the nice features of our theory is that we have an extension of this duality to the non-abelian case, but within one category. This is shown in the last section of our paper where we introduce the algebras of compact type and the algebras of discrete type. We prove that also these are dual to each other. We treat an example that is sufficiently general to illustrate most of the different features of our theory. It is also possible to construct the quantum double of Drinfel'd within this category. This provides a still wider class of examples. So, we obtain many more than just the compact and discrete quantum within this setting.1998 Academic Press

Abstract. In this paper we generalize the notion of Hopf algebra. We consider an algebra A , with or without identity, and a homomorphism A from A to the multiplier algebra M(A ® A) of A ® A . We impose certain conditions on A (such as coassociativity). Then we call the pair {A, A) a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where (Af)(s, t) = f(st) with s, t £ G and f € A . We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a *-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a *-algebra.

For each invertible m×m matrix Q a compact matrix quantum group Au(Q) is constructed. These quantum groups are shown to be universal in the sense that any compact matrix quantum group is a quantum subgroup of some of them. Their orthogonal version Ao(Q) is also constructed. Finally, we discuss related constructions in the literature.

A weak multiplier Hopf algebra is a pair .A; / of a non-degenerate idempotent algebra A and a coproduct on A. The coproduct is a coassociative homomorphism from A to the multiplier algebra M.A˝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˝A to M.A˝A/ by T 1 .a˝b/ D .a/.1˝b/ and T 2 .a˝b/ D .a˝1/.b/. The first condition is about the ranges of these maps. It is assumed that there exists an idempotent element E 2 M.A˝A/ such that .A/.1˝A/ D E.A˝A/ and .A˝1/.A/ D .A˝A/E. This element is unique if it exists. Then it is possible to extend the coproduct in a unique way to a homomorphism e W M.A/ ! M.A˝A/ such that e .1/ D E. In the case of a multiplier Hopf algebra we have E D 1˝1 but this is no longer assumed for weak multiplier Hopf algebras. The second condition determines the behavior of the coproduct on the legs of E. We require .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. We will give more and interesting examples of weak multiplier Hopf algebras in parts II and III of this series of papers.

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