On duality of algebraic quantum groupoids

Timmermann, Thomas

doi:10.1016/j.aim.2017.01.009

Cited by 8 publications

(23 citation statements)

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“…This construction generalizes corresponding results of Van Daele for algebraic quantum groups [21] and of Enock and Lesieur for measured quantum groupoids [8,10], and will be studied in a separate article [14], see also [16].…”

Section: Introductionmentioning

confidence: 84%

“…Our approach yields an embedding of these four modules into the dual vector space of A and one subspace of the intersection where the four products coincide. In [14], we will show that this subspace can be equipped with the structure of a multiplier Hopf algebroid again; see also [16]. (4) Total integrals form the key to relate the algebraic approach to quantum groupoids to the operator-algebraic one, as we shall show in a forthcoming paper [13].…”

Section: Introductionmentioning

confidence: 99%

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Integration on algebraic quantum groupoids

Timmermann

2016

Int. J. Math.

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Abstract. In this article, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups beforefaithfulness, uniqueness up to scaling, existence of a modular element and existence of a modular automorphism -for algebraic quantum groupoids under reasonable assumptions. The approach to integration developed in this article forms the basis for the extension of Pontrjagin duality to algebraic quantum groupoids, and for the passage from algebraic quantum groupoids to operator-algebraic completions, which both will be studied in separate articles.

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Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Integration on algebraic quantum groupoids

Timmermann

2016

Int. J. Math.

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“…It also extends the duality of multiplier Hopf algebras with integrals, the so-called algebraic quantum groups. For this reason, we will sometimes call a regular weak multiplier Hopf algebra with enough integrals an algebraic quantum groupoid.We discuss the relation of our work with the work on duality for algebraic quantum groupoids by Timmermann [20].We also illustrate this duality with a particular example in a separate paper (see [30]). In this paper, we only mention the main definitions and results for this example.…”

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confidence: 93%

“…Relation with the work of Timmermann on duality of algebraic quantum groupoids [20] While preparing this manuscript, we came across recent work by Timmermann. He studies integrals and duality for regular multiplier Hopf algebroids in [19] and [20]. As an application, based on the notion of integrals on weak multiplier Hopf algebras as defined in [8], together with the characterization of regular multiplier Hopf algebroids in terms of weak multiplier Hopf algebras, studied in [22], he obtains also a duality theorem for regular weak multiplier Hopf algebras, similar as the one obtained in this paper.…”

Section: Introductionmentioning

confidence: 99%

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A Survey on Multiplier Hopf Algebras

Zhang¹

2019

Hopf Algebras and Quantum Groups

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Let (A, ∆) be a weak multiplier Hopf algebra as introduced in [32] (see also [31]). It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct ∆ on A, satisfying certain properties. The main difference with multiplier Hopf algebras is that now, the canonical maps T 1 and T 2 on A ⊗ A, defined byandare no longer assumed to be bijective. Also recall that a weak multiplier Hopf algebra is called regular if its antipode is a bijective map from A to itself.In this paper, we introduce and study the notion of integrals on such regular weak multiplier Hopf algebras. A left integral is a non-zero linear functional on A that is left invariant (in an appropriate sense). Similarly for a right integral.For a regular weak multiplier Hopf algebra (A, ∆) with (sufficiently many) integrals, we construct the dual ( A, ∆). It is again a regular weak multiplier Hopf algebra with (sufficiently many) integrals. This duality extends the known duality of finite-dimensional weak Hopf algebras to this more general case. It also extends the duality of multiplier Hopf algebras with integrals, the so-called algebraic quantum groups. For this reason, we will sometimes call a regular weak multiplier Hopf algebra with enough integrals an algebraic quantum groupoid.We discuss the relation of our work with the work on duality for algebraic quantum groupoids by Timmermann [20].We also illustrate this duality with a particular example in a separate paper (see [30]). In this paper, we only mention the main definitions and results for this example. However, we do consider the two natural weak multiplier Hopf algebras associated with a groupoid in detail and show that they are dual to each other in the sense of the above duality.a (1) S(a (2) ).We are using the Sweedler notation here (see further in this introduction under the item Conventions and notations). It has been carefully argued in [33] that these maps are well-defined with values in M (A).The images ε s (A) and ε t (A) are subalgebras of M (A). They are called the source and target algebras. They are commuting non-degenerate subalgebras of M (A). In the regular case, they embed in M (A) in such a way that their multiplier algebras M (ε s (A)) and M (ε t (A)) still embed in M (A). These multiplier algebras are denoted by A s and A t resp. They are still commuting subalgebras of M (A).The source and target algebras ε s (A) and ε t (A) can be identified resp. with the left and the right leg of E. In fact we have E ∈ M (ε s (A) ⊗ ε t (A)).Remark that in earlier papers on the subject, we called A s and A t the source and target algebras. However, in the second version of our second paper on the subject [33], we changed the terminology.For more details about the short review over regular multiplier Hopf algebra given above, we refer to earlier work, in particular [32] and [33]. The examples associated with a groupoidConsider a groupoid G. One can associate two regular weak multiplier Hopf algebras.

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Duality for weak multiplier Hopf algebras with sufficiently many integrals

Van Daele,

Wang

2024

Advances in Mathematics

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On duality of algebraic quantum groupoids

Cited by 8 publications

References 44 publications

Integration on algebraic quantum groupoids

Integration on algebraic quantum groupoids

A Survey on Multiplier Hopf Algebras

Duality for weak multiplier Hopf algebras with sufficiently many integrals

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