A Class of Multiplier Hopf Algebras

Daele, Alfons Van; Wang, Shuanhong

doi:10.1007/s10468-007-9045-6

Cited by 3 publications

(5 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These equations (47) and (48) "force" the equality (44) to be in ι B⊗A⊗A (B ⊗ A ⊗ A), hence obtaining an equality in B ⊗ A ⊗ A (by the injectivity of ι B⊗A⊗A ). It is actually this equality that expresses the coassociativity in [14]. Furthermore, (46) states that the coaction ρ : B → M(B ⊗ A) is counital in the sense of [14].…”

Section: 2mentioning

confidence: 99%

“…It is actually this equality that expresses the coassociativity in [14]. Furthermore, (46) states that the coaction ρ : B → M(B ⊗ A) is counital in the sense of [14].…”

Section: 2mentioning

confidence: 99%

“…In [14] comodule algebras and module algebras were defined over a multiplier Hopf algebra. Just like multiplier Hopf algebras are in particular multiplier bialgebras, (co)module algebras over a multiplier Hopf algebra are particular instances of (co)module algebras over the underlying multiplier bialgebra.…”

Section: 2mentioning

confidence: 99%

“…Note that (44) is an equality in M(B ⊗ A ⊗ A). In [14], a so-called coassociative coaction of A on B is an algebra map ρ : B → M(B ⊗ A) such that a similar coassociativity condition as (44) holds, but under the extra assumption that…”

Section: 2mentioning

confidence: 99%

“…We show that a multiplier Hopf algebra is always a multiplier bialgebra and we give an interpretation of the antipode as a type of convolution inverse. We conclude the paper by providing a categorical way to introduce the notions of module algebra and comodule algebra over a multiplier bialgebra, which in the multiplier Hopf algebra case were studied in [14].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Multiplier Bi- And Hopf Algebras

Janssen

Vercruysse

2010

J. Algebra Appl.

View full text Add to dashboard Cite

Abstract. We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele in [11] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly non-unital, idempotent, non-degenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

show abstract

Section: 2mentioning

confidence: 99%

“…It is actually this equality that expresses the coassociativity in [14]. Furthermore, (46) states that the coaction ρ : B → M(B ⊗ A) is counital in the sense of [14].…”

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Multiplier Bi- And Hopf Algebras

Janssen

Vercruysse

2010

J. Algebra Appl.

View full text Add to dashboard Cite

show abstract

Bicrossproducts of multiplier Hopf algebras

2011

View full text Add to dashboard Cite

In this paper, we generalize Majid's bicrossproduct construction. We start with a pair (A, B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A#B. The left coaction of B on A gives a possible coproduct ∆ # on A#B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for ∆ # to be a proper coproduct on A#B. The result is again a regular multiplier Hopf algebra. Majid's construction is obtained when we have Hopf algebras.We also look at the dual case, constructed from a pair (C, D) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We will show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D will yield a duality between A#B and the smash product C#D.We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The * -algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature.Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G = KH and H ∩ K = {e} (where e is the identity of G) will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention.

show abstract

Bicrossproducts of Algebraic Quantum Groups

2013

View full text Add to dashboard Cite

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 ∆ # on the smash product A#B making the pair (A#B, ∆ # ) into an algebraic quantum group. This result is proven in 'Bicrossproducts of multiplier Hopf algebras' (reference [De-VD-W]) where the precise matching conditions are explained in detail, as well as the construction of this bicrossproduct. 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 A and B. 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. In [L-VD] related examples are considered, involving totally disconnected locally compact groups, but as these examples require some analysis, they fall outside the scope of this paper. It should be mentioned however 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.February 2012 (Version 1.0) Actions and coactions, integrals and cointegralsConsider two regular multiplier Hopf algebras A and B. Let ⊳ be a right action of A on B making B into a right A-module algebra. Let Γ be a left coaction of B on A making A a left B-comodule coalgebra. Assume that the action and coaction make of (A, B) a matched pair as discussed already in the introduction.In this section, we will consider integrals and cointegrals for the components A and B and we will investigate the relation with the action and the coaction. The case where B has cointegralsWe will not be able to deduce very much however, but fortunately, this case is not so important. We mainly include it for motivational purposes. In particular, we will use this case to explain what kind of problems we encounter.Recall that an element h ∈ B is called a left cointegral if it is non-zero and if bh = ε(b)h for all b ∈ B. Similarly, a non-zero element k ∈ B satisfying kb = ε(b)k for all ...

show abstract

A Class of Multiplier Hopf Algebras

Cited by 3 publications

References 12 publications

Multiplier Bi- And Hopf Algebras

Multiplier Bi- And Hopf Algebras

Bicrossproducts of multiplier Hopf algebras

Bicrossproducts of Algebraic Quantum Groups

Contact Info

Product

Resources

About