Lydia Delvaux

SEMI-DIRECT PRODUCTS OF MULTIPLIER HOPF ALGEBRAS: SMASH PRODUCTS

ABSTRACT The well-known concept of smash product, and the dual notion of smash coproduct were first formulated by Molnar for Hopf algebras. The object of this paper is to extend the theory of smash products to multiplier Hopf algebras. We prove that the result of Molnar is still true for regular multiplier Hopf algebras under an appropriate form. We consider integrals on the smash product and we obtain results in the -situation.

The size of the intrinsic group of a multiplier Hopf algebra

Abstract In the theory of Hopf algebras, grouplike elements are well studied. We argue that this notion is too restrictive when dealing with multiplier Hopf algebras. We introduce the “intrinsic” group, situated in the multiplier algebra of the multiplier Hopf algebra. When dealing with an algebraic quantum group A, the intrinsic group of the dual  characterizes a special class of automorphisms on A. For multiplier Hopf algebras which are paired in the sense of Drabant and Van Daele. (Drabant, B., Van Daele, A. Pairing and quantum double for multiplier Hopf algebras. (2001). Algebras and Representation Theory 4:109–132), we prove that the intrinsic group characterizes the semi-invariants of the action associated to the pairing.

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.

Traces on Multiplier Hopf Algebras

In this article we lay the algebraic foundations to establish the existence of trace functions on infinite-dimensional (multiplier) Hopf algebras. We solve the problem within the framework of multiplier Hopf algebra with integrals. By applying this theory to group-cograded multiplier Hopf algebras, we prove the existence of group-traces on group-cograded multiplier Hopf algebras with possibly infinite-dimensional components. We generalize the results as obtained by Virelizier in the case of finite-type Hopf group-coalgebras.

Semi-invariants of actions of multiplier hopf algebras

Let A be a multiplier Hopf algebra which acts on an algebra R. In this paper we study semi-invariants of this action. This idea has proved interesting in the case thatA is a Hopf algebra.