A generalized double crossproduct for monoidal Hom-Hopf algebras and the Drinfeld double

2017 ◽  
Vol 146 (2) ◽  
pp. 213-238
Author(s):  
Miman You ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebras ◽  
Drinfeld Double

scholarly journals Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

2019 ◽  
Vol 21 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Robert Laugwitz
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Crossed Product ◽  
Braided Monoidal Category ◽  
Drinfeld Double ◽  
Comodule Algebra ◽  
Category Of Modules ◽  
Cohomology Spaces ◽  
Quasitriangular Hopf Algebra

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

Drinfeld Double for Braided Infinitesimal Hopf Algebras

2014 ◽  
Vol 42 (5) ◽  
pp. 2195-2212 ◽  
Author(s):  
Shengxiang Wang ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebras ◽  
Drinfeld Double

TWISTED DRINFELD DOUBLES AND REPRESENTATIONS OF A HOPF ALGEBRA

2012 ◽  
Vol 11 (06) ◽  
pp. 1250118
Author(s):  
LI BAI ◽  
SHUANHONG WANG
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Direct Proof ◽  
Irreducible Representations ◽  
Quantum Double ◽  
Drinfeld Double ◽  
Braided Group ◽  
R Matrix ◽  
Coalgebra Structure

In this paper, we consider a class of non-commutative and non-cocommutative Hopf algebras Hp(α, q, m) and then show that these Hopf algebras can be realized as a quantum double of certain Hopf algebras with respect to Hopf skew pairings (Ap(q, m), Bp(q, m), τα). Furthermore, using the Hopf skew pairing with appropriate group homomorphisms: ϕ : π → Aut (Ap(q, m)) and ψ : π → Aut (Bp(q, m)), we construct a twisted Drinfeld double D(Ap(q, m), Bp(q, m), τ; ϕ, ψ) which is a Turaev [Formula: see text]-coalgebra, where the group [Formula: see text] is a twisted semi-direct square of a group π. Then we obtain its quasi-triangular Turaev [Formula: see text]-coalgebra structure. We also study irreducible representations of Hp(1, q, m) and construct a corresponding R-matrix. Finally, we introduce the notion of a left Yetter–Drinfeld category over a Turaev group coalgebra and show that such a category is a Turaev braided group category by a direct proof, without center construction. As an application, we consider the case of the quasi-triangular Turaev [Formula: see text]-coalgebra structure on our twisted Drinfeld double.

Left counital Hopf algebras on bi-decorated planar rooted forests and Rota-Baxter systems

2020 ◽  
Vol 27 (2) ◽  
pp. 219-243 ◽  
Author(s):  
Xiao-Song Peng ◽  
Yi Zhang ◽  
Xing Gao ◽  
Yan-Feng Luo
Keyword(s):  
Hopf Algebras

scholarly journals Monadic cointegrals and applications to quasi-Hopf algebras

2021 ◽  
Vol 225 (10) ◽  
pp. 106678
Author(s):  
Johannes Berger ◽  
Azat M. Gainutdinov ◽  
Ingo Runkel
Keyword(s):  
Hopf Algebras

Quotients of hopf algebras

1978 ◽  
Vol 6 (17) ◽  
pp. 1789-1800 ◽  
Author(s):  
Warren D. Nichols
Keyword(s):  
Hopf Algebras

scholarly journals Bialgebra cohomology, pointed Hopf algebras, and deformations

2009 ◽  
Vol 213 (7) ◽  
pp. 1399-1417 ◽  
Author(s):  
Mitja Mastnak ◽  
Sarah Witherspoon
Keyword(s):  
Hopf Algebras ◽  
Pointed Hopf Algebras
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