scholarly journals A Class of Weak Hopf Algebras

2010 ◽  
Vol 2010 ◽  
pp. 1-8 ◽  
Author(s):  
Dongming Cheng
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Direct Sum ◽  
Weak Hopf Algebras

We introduce a class of noncommutative and noncocommutative weak Hopf algebras with infinite Ext quivers and study their structure. We decompose them into a direct sum of two algebras. The coalgebra structures of these weak Hopf algebras are described by their Ext quiver. The weak Hopf extension of Hopf algebraHnhas a quotient Hopf algebra and a sub-Hopf algebra which are isomorphic toHn.

A braided T-category over weak monoidal Hom-Hopf algebras

2019 ◽  
Vol 19 (08) ◽  
pp. 2050159
Author(s):  
Guohua Liu ◽  
Wei Wang ◽  
Shuanhong Wang ◽  
Xiaohui Zhang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Weak Hopf Algebras ◽  
T Category

In this paper, we define and study weak monoidal Hom-Hopf algebras, which generalize both weak Hopf algebras and monoidal Hom-Hopf algebras. Let [Formula: see text] be a weak monoidal Hom-Hopf algebra with bijective antipode and let [Formula: see text] be the set of all automorphisms of [Formula: see text], we introduce a category [Formula: see text] with [Formula: see text] and construct a braided [Formula: see text]-category [Formula: see text] having all the categories [Formula: see text] as components.

scholarly journals Automorphism groups of representation rings of the weak Sweedler Hopf algebras

2021 ◽  
Vol 7 (2) ◽  
pp. 2318-2330
Author(s):  
Dong Su ◽  
◽  
Shilin Yang ◽  
Keyword(s):  
Hopf Algebra ◽  
Solvable Group ◽  
Hopf Algebras ◽  
Automorphism Groups ◽  
Representation Rings ◽  
Weak Hopf Algebras

<abstract><p>Let $ \mathfrak{w}^{s}_{2, 2}(s = 0, 1) $ be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and $ r(\mathfrak{w}^{s}_{2, 2}) $ be the representation rings of $ \mathfrak{w}^{s}_{2, 2} $. In this paper, we investigate the automorphism groups $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $ of $ r(\mathfrak{w}^{s}_{2, 2}) $, and discuss some properties of $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $. We obtain that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2})) $ is isomorphic to $ K_4 $, where $ K_4 $ is the Klein four-group. It is shown that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is a non-commutative infinite solvable group, but it is not nilpotent. In addition, $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is isomorphic to $ (\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2} $, and its centre is isomorphic to $ \mathbb{Z}_{2} $.</p></abstract>

scholarly journals Comparison of two notions of weak crossed product

2020 ◽  
pp. 2150059
Author(s):  
Jorge A. Guccione ◽  
Juan J. Guccione
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Crossed Product ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Weak Crossed Product ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

scholarly journals A duality theorem for weak multiplier Hopf algebra actions

2017 ◽  
Vol 28 (05) ◽  
pp. 1750032 ◽  
Author(s):  
Nan Zhou ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Duality Theorem ◽  
Smash Product ◽  
Module Algebra ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Weak Hopf Algebras

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by Van Daele and Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids.

scholarly journals Rota–Baxter Operators on Cocommutative Weak Hopf Algebras

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 95
Author(s):  
Zhongwei Wang ◽  
Zhen Guan ◽  
Yi Zhang ◽  
Liangyun Zhang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Matrix Algebra ◽  
Weak Hopf Algebra ◽  
Weak Hopf Algebras ◽  
Baxter Operator

In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a new multiplication “∗” and an antipode SB from a Rota–Baxter operator B on H such that HB=(H,∗,η,Δ,ε,SB) becomes a new weak Hopf algebra. Finally, all Rota–Baxter operators on a weak Hopf algebra of a matrix algebra are given.

Pseudotriangular weak Hopf algebras

2016 ◽  
Vol 16 (07) ◽  
pp. 1750137
Author(s):  
Xiaofan Zhao ◽  
Guohua Liu ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Special Class ◽  
Weak Hopf Algebra ◽  
Extra Condition ◽  
Weak Hopf Algebras ◽  
Algebraic Counterpart

In this paper, in looking for the weak Hopf algebraic counterpart of pseudosymmetric braidings, we introduce the concept of a pseudotriangular weak Hopf algebra, which is a quasitriangular weak Hopf algebra satisfying an extra condition. Then, we investigate the question, when a quasitriangular weak Hopf algebra is pseudotriangular. As an application, we study a special class of pseudotriangular weak Hopf algebras, under the name almost-triangular weak Hopf algebras and list some nontrivial examples. Finally, in order to construct more examples of pseudotriangular weak Hopf algebras, we show that the pseudosymmetry of the Yetter–Drinfeld category [Formula: see text] is determined by the commutativity and cocommutativity of [Formula: see text], where [Formula: see text] is a weak Hopf algebra with a bijective antipode.

scholarly journals Weak Multiplier Hopf Algebras II: The Source and Target Algebras

2020 ◽  
Author(s):  
Shuanhong Wang ◽  
Alfons Van Daele
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Weak Hopf Algebra ◽  
Multiplier Algebra ◽  
Special Cases ◽  
Multiplier Hopf Algebras ◽  
Group A ◽  
Multiplier Hopf Algebra ◽  
Weak Hopf Algebras ◽  
Algebra Part

Let $(A,\Delta)$ be a {\it weak multiplier Hopf algebra} as introduced in [VD-W3] (see also [VD-W2]). It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. If the algebra has an identity and the coproduct is unital, then we have a Hopf algebra. If the algebra has no identity, but if the coproduct is non-degenerate (which is the equivalent of being unital if the algebra has an identity), then the pair would be a multiplier Hopf algebra. If the algebra has an identity, but the coproduct is not unital, we have a weak Hopf algebra. In the general case, we neither assume $A$ to have an identity nor do we assume $\Delta$ to be non-degenerate and so we work with a {\it genuine} weak multiplier Hopf algebra. It is called {\it regular} if its antipode is a bijective map from $A$ to itself. \snl In this paper, we {\it continue the study of weak multiplier Hopf algebras}. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\ot C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a {\it separability idempotent} (as studied in [VD4.v2]). If the weak multiplier Hopf algebra is regular, then also $E$ is a {\it regular} separability idempotent. \snl We also consider {\it special cases and examples} in this paper. In particular, we see how for any weak multiplier Hopf algebra $(A,\Delta)$, it is possible to make $C\ot B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\ot C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

scholarly journals Weak Multiplier Hopf Algebras II: Source and Target Algebras

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1975
Author(s):  
Alfons Van Daele ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Hopf Algebras ◽  
Symmetric Pair ◽  
Multiplier Algebra ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Weak Hopf Algebras ◽  
Algebra Part

Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps εs and εt are studied, and their symmetric pair of images, the source algebra and the target algebra εs(A) and εt(A), are also investigated. We show that the canonical idempotent E (which is eventually Δ(1)) belongs to the multiplier algebra M(B⊗C), where (B=εs(A), C=εt(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Δ), it is possible to make C⊗B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the ’Hopf algebra part’ of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E∈M(B⊗C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced).

scholarly journals Inner actions of weak Hopf algebras

2017 ◽  
Vol 16 (06) ◽  
pp. 1750118
Author(s):  
Dirceu Bagio ◽  
Daiana Flôres ◽  
Alveri Sant’ana
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Product Formula ◽  
Weak Hopf Algebra ◽  
Smash Product ◽  
Module Algebra ◽  
Weak Hopf Algebras

Let [Formula: see text] be an associative ring and [Formula: see text] idempotent elements of [Formula: see text]. In this paper we introduce the notion of [Formula: see text]-invertibility for an element of [Formula: see text] and use it to define inner actions of weak Hopf algebras. Given a weak Hopf algebra [Formula: see text] and an algebra [Formula: see text], we present sufficient conditions for [Formula: see text] to admit an inner action of [Formula: see text]. We also prove that if [Formula: see text] is a left [Formula: see text]-module algebra then [Formula: see text] acts innerly on the smash product [Formula: see text] if and only if [Formula: see text] is a quantum commutative weak Hopf algebra.

scholarly journals Partial actions of weak Hopf algebras on coalgebras

2020 ◽  
pp. 2250012
Author(s):  
Eneilson Fontes ◽  
Grasiela Martini ◽  
Graziela Fonseca
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Weak Hopf Algebra ◽  
Partial Action ◽  
Dual Relation ◽  
Weak Hopf Algebras

In this work, the notions of a partial action of a weak Hopf algebra on a coalgebra and of a partial action of a groupoid on a coalgebra will be introduced, together with some important properties. An equivalence between these notions will be presented. Finally, a dual relation between the structures of a partial action on a coalgebra and of a partial action on an algebra will be established, as well as a globalization theorem for partial module coalgebras will be presented.

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