Progress of Research on Weak Hopf Algebra

In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

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On Braided Lie Structures of Algebras in the Categories of Weak Hopf Bimodules

Algebra Colloquium ◽

10.1142/s1005386710000659 ◽

2010 ◽

Vol 17 (04) ◽

pp. 685-698 ◽

Cited By ~ 1

Author(s):

Shuan-hong Wang ◽

Hai-xing Zhu

Keyword(s):

Hopf Algebra ◽

Monoidal Category ◽

Weak Hopf Algebra ◽

Commutator Ideal ◽

Commutative Subalgebras

Let H be a weak Hopf algebra. In this paper, it is proved that the monoidal category [Formula: see text] of weak Hopf bimodules studied in Wang [19] is equivalent to the monoidal category [Formula: see text] of weak Yetter–Drinfel'd modules introduced in Böhm [2]. When H has a bijective antipode, a braiding in the category [Formula: see text] is constructed by the braiding on [Formula: see text], generalizing the main result in Schauenburg [14]. Finally, the braided Lie structures of an algebra A in the category [Formula: see text] are investigated, by showing that if A is a sum of two braided commutative subalgebras, then the braided commutator ideal of A is nilpotent.

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Comparison of two notions of weak crossed product

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500596 ◽

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].

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Automorphism group of representation ring of the weak Hopf algebra $\widetilde{H_8}$

10.21136/cmj.2018.0131-17 ◽

2018 ◽

Vol 68 (4) ◽

pp. 1131-1148 ◽

Cited By ~ 2

Author(s):

Dong Su ◽

Shilin Yang

Keyword(s):

Automorphism Group ◽

Hopf Algebra ◽

Weak Hopf Algebra ◽

Representation Ring

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Weak Hopf Algebra

Concise Encyclopedia of Supersymmetry ◽

10.1007/1-4020-4522-0_685 ◽

2004 ◽

pp. 503-503

Author(s):

François Gieres ◽

Fang Li ◽

Peter Trotter

Keyword(s):

Hopf Algebra ◽

Weak Hopf Algebra

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The Duality Theorem for Weak Hopf Algebra (Co) Actions

Communications in Algebra ◽

10.1080/00927870903443303 ◽

2010 ◽

Vol 38 (12) ◽

pp. 4613-4632 ◽

Cited By ~ 7

Author(s):

Xuan Zhou ◽

Shuanhong Wang

Keyword(s):

Hopf Algebra ◽

Duality Theorem ◽

Weak Hopf Algebra

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Quantum Groupoids Acting on Semiprime Algebras

Advances in Mathematical Physics ◽

10.1155/2011/546058 ◽

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.

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