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 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 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].

An extended form of Majid’s double biproduct

2017 ◽  
Vol 16 (04) ◽  
pp. 1750061 ◽  
Author(s):  
Tianshui Ma ◽  
Huihui Zheng
Keyword(s):  
Hopf Algebra ◽  
Sufficient Conditions ◽  
Smash Product ◽  
Algebra Structure ◽  
Module Algebra ◽  
Extended Form ◽  
Linear Map ◽  
Product Algebra ◽  
Necessary And Sufficient ◽  
Crossed Coproducts

Let [Formula: see text] be a bialgebra. Let [Formula: see text] be a linear map, where [Formula: see text] is a left [Formula: see text]-module algebra, and a coalgebra with a left [Formula: see text]-weak coaction. Let [Formula: see text] be a linear map, where [Formula: see text] is a right [Formula: see text]-module algebra, and a coalgebra with a right [Formula: see text]-weak coaction. In this paper, we extend the construction of two-sided smash coproduct to two-sided crossed coproduct [Formula: see text]. Then we derive the necessary and sufficient conditions for two-sided smash product algebra [Formula: see text] and [Formula: see text] to be a bialgebra, which generalizes the Majid’s double biproduct in [Double-bosonization of braided groups and the construction of [Formula: see text], Math. Proc. Camb. Philos. Soc. 125(1) (1999) 151–192] and the Wang–Wang–Yao’s crossed coproduct in [Hopf algebra structure over crossed coproducts, Southeast Asian Bull. Math. 24(1) (2000) 105–113].

scholarly journals Quantum Groupoids Acting on Semiprime Algebras

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Inês Borges ◽  
Christian Lomp
Keyword(s):  
Hopf Algebra ◽  
Polynomial Identity ◽  
Weak Hopf Algebra ◽  
Smash Product ◽  
Module Algebra

Following Linchenko and Montgomery's arguments we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime.

Relations of A*H and AH

2010 ◽  
Vol 143-144 ◽  
pp. 828-831
Author(s):  
Yan Yan ◽  
Cui Lan Mi ◽  
Xin Chun Wang
Keyword(s):  
Hopf Algebra ◽  
Weak Hopf Algebra ◽  
Smash Product ◽  
Trace Function ◽  
Module Algebra ◽  
Finiteness Conditions ◽  
Module Algebras ◽  
The Right ◽  
Product Algebra

In this paper, we study the concept of the right twisted smash product algebra A*H over weak Hopf algebra. Let H be a weak Hopf algebra and A an H-module algebra, using the properties of the trace function we describe the finiteness conditions for H-module algebras.

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 Hopf algebras arising from partial (co)actions

2020 ◽  
pp. 2140006
Author(s):  
Danielle Azevedo ◽  
Grasiela Martini ◽  
Antonio Paques ◽  
Leonardo Silva
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Product Formula ◽  
Matched Pair ◽  
Matched Pairs ◽  
Partial Action

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra 9 (1981) 841–882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra 130(1) (1990) 17–64.], we introduce the notion of partial matched pair [Formula: see text] involving the concepts of partial action and partial coaction between two bialgebras [Formula: see text] and [Formula: see text]. Furthermore, we present sufficient conditions for the corresponding bismash product [Formula: see text] to generate a new Hopf algebra and, as illustration, a family of examples is provided. Moreover, under some hypotheses such sufficient conditions are also necessary conditions.

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.

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).

Sign in / Sign up

Export Citation Format

Share Document