A. Van Daele

Multiplier hopf-algebras

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.

C*-Algebraic Quantum Groups Arising from Algebraic Quantum Groups

In this paper, we construct a universal C*-algebraic quantum group out of an algebraic one. We show that this universal C*-algebraic quantum group has the same rich structure as its reduced companion. This universal C*-algebraic quantum group also satifies an upcoming definition of Masuda, Nakagami & Woronowicz except for the possible non-faithfulness of the left Haar weight.

Galois Theory for Multiplier Hopf Algebras with Integrals

In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.

Unitary equivalence of Fock representations on the Weyl algebra

A necessary and sufficient condition for unitary equivalence of pure quasifree states over the Weyl algebra is proved. Some partial results on states over the Weyl algebra are formulated in Theorem 1, and Lemmas 1, 4, 5 and 6.

Multiplier Hopf algebras and duality

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

Multiplier Hopf Algebras Imbedded in Locally Compact Quantum Groups

Let (A,∆) be a locally compact quantum group and (A0,∆0) a regular multiplier Hopf algebra. We show that if (A0,∆0) can be imbedded in (A,∆), then A0 will inherit some of the analytic structure of A. Under certain conditions on the imbedding, we will be able to conclude that (A0,∆0) is actually an algebraic quantum group with a full analytic structure. The techniques used to show this can also be applied to obtain the analytic structure of a ∗-algebraic quantum group in a purely algebraic fashion. Moreover, the reason that this analytic structure exists at all, is that the associated one-parameter groups, such as the modular group and the scaling group, are diagonizable. As an immediate corollary, we will show that the scaling constant μ of a ∗-algebraic quantum group equals 1. This solves an open problem posed in [13].

A Note on the Antipode for Algebraic Quantum Groups

Recently, Beattie, Bulacu ,and Torrecillas proved Radford’s formula for the fourth power of the antipode for a co-Frobenius Hopf algebra. In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finitedimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier 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, in particular 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. Department of Mathematics, University of Hasselt, Agoralaan, B-3590 Diepenbeek, Belgium e-mail: Lydia.Delvaux@uhasselt.be Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium e-mail: Alfons.VanDaele@wis.kuleuven.be Department of Mathematics, Southeast University, Nanjing 210096, China e-mail: shuanhwang@seu.edu.cn Received by the editors March 13, 2009. Published electronically April 25, 2011. This work was partly supported by the Research Council of the K.U. Leuven (through a fellowship for Shuanhong Wang) AMS subject classification: 16W30, 46L65.

ALGEBRAIC QUANTUM HYPERGROUPS 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 Δ on A making the pair (A,Δ) a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic quantum group as introduced and studied in [A. Van Daele, Adv. Math.140 (1998) 323]. Now let H be a finite subgroup of G and consider the subalgebra A1 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 Δ and define Δ1 as where f ∈ A1, p,q ∈ G and where n is the number of elements in the subgroup H. Then Δ1 will leave the subalgebra invariant (in the sense that the image is in the multiplier algebra M(A1 ⊗ A1) of the tensor product). However, it 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 as introduced and studied in [L. Delvaux and A. Van Daele, Adv. Math.226 (2011) 1134–1167]. Group-like projections in a *-algebraic quantum group A (as defined and studied in [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1]) 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 (again see [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1]). In this paper we push these results further. On the one hand, we no longer assume the *-structure as in [M. B. Landstad and A. Van Daele, arXiv:math.OA/0702458v1] 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.

BICROSSPRODUCTS OF ALGEBRAIC QUANTUM GROUPS

Let A and B be two algebraic quantum groups. 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 Δ# on the smash product A#B making the pair (A#B, Δ#) into an algebraic quantum group. In this paper we study the various data of the bicrossproduct A#B, such as the modular automorphisms, the modular elements, … and we 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

Constructing Quasitriangular Multiplier Hopf Algebras By Twisted Tensor Coproducts

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