Weak Multiplier Hopf Algebras II: Source and Target Algebras

Daele, Alfons Van; Wang, Shuanhong

doi:10.3390/sym12121975

Cited by 5 publications

(3 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Naseer Khan et al [19] worked on the WHA and its quiver representation. Li [20] solved the quantum Yang-Baxter equation, Montgomery [21] showed the action of HA on rings, Radford [22] described the projection of structures of HA, Daele and Wang [23] defined the multipliers of HA, Swedler [24], Yang and Zhang [25], and Smith [26] also extended the theory of Hopf Algebra.…”

Section: Introductionmentioning

confidence: 99%

Double Weak Hopf Quiver and Its Path Coalgebra

Khan

Munir

Arshad

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

The main input of this research is the introduction of the concept of double weak Hopf quiver (DWHQ). In addition, the structures of weak Hopf algebra (WHA) are obtained through path coalgebra of the proposed quivers. Furthermore, the module and comodule structures on the said WHA are discussed. Moreover, the classification of the semilattice-graded WHA structures of the path coalgebra obtained from the so called DWHQ was presented. This contributes a step further in the development of the module and comodule structures of more general algebras. These structures are important in the physics for dynamic systems.

show abstract

Section: Introductionmentioning

confidence: 99%

Double Weak Hopf Quiver and Its Path Coalgebra

Khan

Munir

Arshad

et al. 2022

Journal of Function Spaces

View full text Add to dashboard Cite

show abstract

“…For the theory of multiplier Hopf algebras, the main (original) reference is [6] and for the theory of multiplier Hopf algebras with integrals, sometimes called algebraic quantum groups, the main reference is [7]. Weak multiplier Hopf algebras are studied in a number of papers, see [16] and [17] (and also [15]), but we will say very little about them in this paper.…”

Section: Basic Referencesmentioning

confidence: 99%

A Survey on Multiplier Hopf Algebras

Zhang¹

2019

Hopf Algebras and Quantum Groups

View full text Add to dashboard Cite

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.

show abstract

“…With this map ρ, H becomes a Hopf algebra. Montgomery [2] described the action of Hopf algebra on rings, Me [3] wrote a series of mathematics lecture notes, Redford [4] deliberated the structure of Hopf algebras with a projection, Daele and Wang [5] discussed the source and target algebras for weak multiplier Hopf algebras, Yang and Zhang [6] proposed the ore extensions for Sweedler's Hopf algebra, Smith [7] formulated the quantum Yang-Baxter equation and quantum quasigroups, Nichita [8] introduced the Yang-Baxter equation with open problems, and Cibils and Rosso [9] introduced the Hopf quiver. According to them, a Hopf quiver is just a Cayley graph of a group.…”

Section: Introductionmentioning

confidence: 99%

Weak Hopf Algebra and Its Quiver Representation

Khan

Munir

Arshad

et al. 2021

Mathematical Problems in Engineering

View full text Add to dashboard Cite

This study induced a weak Hopf algebra from the path coalgebra of a weak Hopf quiver. Moreover, it gave a quiver representation of the said algebra which gives rise to the various structures of the so-called weak Hopf algebra through the quiver. Furthermore, it also showed the canonical representation for each weak Hopf quiver. It was further observed that a Cayley digraph of a Clifford monoid can be embedded in its corresponding weak Hopf quiver of a Clifford monoid. This lead to the development of the foundation structures of weak Hopf algebra. Such quiver representation is useful for the classification of its path coalgebra. Additionally, some structures of module theory of algebra were also given. Such algebras can also be applied for obtaining the solutions of “quantum Yang–Baxter equation” that has many applications in the dynamical systems for finding interesting results.

show abstract

Weak Multiplier Hopf Algebras II: Source and Target Algebras

Cited by 5 publications

References 29 publications

Double Weak Hopf Quiver and Its Path Coalgebra

Double Weak Hopf Quiver and Its Path Coalgebra

A Survey on Multiplier Hopf Algebras

Weak Hopf Algebra and Its Quiver Representation

Contact Info

Product

Resources

About