On Coquasitriangular Pointed Majid Algebras

Huang, Haowen; Liu, Gongxiang

doi:10.1080/00927872.2011.582059

Cited by 4 publications

(6 citation statements)

References 18 publications

(52 reference statements)

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“…A Hopf quiver representation is defined in [15] and some structures are given in this regard. We generalize this notion as a weak Hopf quiver representation and discuss its structures.…”

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

“…Definition 7 (see [15]). If R and S be two representations of the weak Hopf quiver Γ(S, r), then Φ: R ⟶ S as a representation morphism is a collection of k-linear maps Suppose φ i λ,μ : V i,λ ⟶ W i,μ is invertible for each i ∈ Γ 0 and all μ ≥ λ; λ, μ ∈ Y, we have the morphism Φ: R ⟶ S, which is called isomorphism from R to S.…”

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

“…A representation R of a weak Hopf quiver Γ is decomposable if there exist two nonzero representations S and T, such that R � S ⊕ T, and a nonzero representation is indecomposable if it is not decomposable [15].…”

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

“…We introduce the notion of canonical representation of Γ and observe that it is also a simple one. Definition 8 (see [15]). A canonical representation R � R λ,μ : μ ≥ λ; λ, μ ∈ Y 􏽮 􏽯 for weak Hopf quiver Γ(S, r) is a collection of representations R λ,μ , such that…”

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

“…By [14], the categories of Hopf algebra are discussed for the representation has tensor structures induced from the graded Hopf structures of kΓ. By [15], the path coalgebra kΓ of a quiver Γ admits a coquasitriangular Majid algebra structure if and only if Γ is a Hopf quiver of the form Γ(G, R) with G abelian. Here, the authors gave a classification of the set of graded coquasitriangular Majid structures on connected Hopf quiver.…”

Section: Introductionmentioning

confidence: 99%

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Weak Hopf Algebra and Its Quiver Representation

Khan

Munir

Arshad

et al. 2021

Mathematical Problems in Engineering

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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.

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“…A Hopf quiver representation is defined in [15] and some structures are given in this regard. We generalize this notion as a weak Hopf quiver representation and discuss its structures.…”

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

Section: Representation Of Weak Hopf Quivermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Weak Hopf Algebra and Its Quiver Representation

Khan

Munir

Arshad

et al. 2021

Mathematical Problems in Engineering

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Quasi-quantum planes and quasi-quantum groups of dimension $p^3$ and $p^4$

Huang

Yang

2015

Proc. Amer. Math. Soc.

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Abstract. The aim of this paper is to contribute more examples and classification results of finite pointed quasi-quantum groups within the quiver framework initiated in [5,6]. The focus is put on finite dimensional graded Majid algebras generated by group-like elements and two skew-primitive elements which are mutually skew-commutative. Such quasi-quantum groups are associated to quasi-quantum planes in the sense of nonassociative geomertry [14,15]. As an application, we obtain an explicit classification of graded pointed Majid algebras with abelian coradical of dimension p 3 and p 4 for any prime number p. IntroductionThe classification problem of finite pointed tensor categories and the underlying quasi-quantum groups in accordance to the Tannaka-Krein duality has been an active research theme for quite some time. In [5,6], the quiver framework was proposed by the first author to tackle this problem and some interesting results have been obtained in this direction, see for example [7,8,12].The aim of this paper is to contribute more examples and classification results of finite pointed quasiquantum groups within the quiver framework. We focus on the class of finite dimensional graded pointed Majid algebras which are generated by group-like elements and two skew-primitive elements which are mutually skew-commutative. The reason is twofold. On the one hand, such Majid algebras are relatively easy and a complete classification may be attainable. On the other hand, this class of Majid algebras are interesting in nonassociative geometry, namely, they may be viewed as the "coordinate algebra" of nonassociative planes, see [14,15].The notion of Majid algebras adopted here stands for coquasi-Hopf algebras, or dual quasi-Hopf algebras used by some authors. We remark that the study of Majid algebras and their comodule categories is the main task of the classification problem of tensor categories and quasi-quantum groups, see for instance [5] for an explanation.Throughout we work over the field which is algebraically closed with characteristic 0. We aim to classify finite dimensional graded pointed Majid algebras M over satisfying (R1): M is generated by an abelian group G and two skew-primitive elements {X, Y }; and (R2): the skew-primitive elements are skew-commutative, i.e., XY = qY X for some q ∈ * .Here, by graded we mean M is coradically graded, that is, M admits a decomposition M = n≥0 M (n) and M n := 0≤i≤n M (i) is the n-th term of its coradical filtration. By the assumption, the coradical of M is M 0 = ( G, Φ), which is a Majid subalgebra of M and the associator Φ is in fact a normalized 3-cocycle on G. By Key words and phrases. quasi-quantum plane, quasi-quantum group, Hopf quiver.

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On the classification of finite quasi-quantum groups

2017

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We give an overview of the classification results obtained so far for finite quasi-quantum groups over an algebraically closed field of characteristic zero. The main classification results on basic quasi-Hopf algebras are obtained by Etingof, Gelaki, Nikshych, and Ostrik, and on dual quasi-Hopf algebras by Huang, Liu and Ye. The objective of this survey is to help in understanding the tools and methods used for the classification.

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On Coquasitriangular Pointed Majid Algebras

Cited by 4 publications

References 18 publications

Weak Hopf Algebra and Its Quiver Representation

Weak Hopf Algebra and Its Quiver Representation

Quasi-quantum planes and quasi-quantum groups of dimension $p^3$ and $p^4$

On the classification of finite quasi-quantum groups

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