Smash coproduct and braided groups in braided monoidal category

1997 ◽  
Vol 40 (11) ◽  
pp. 1121-1128
Author(s):  
Jinqi Li ◽  
Yonghua Xu
Keyword(s):  
Monoidal Category ◽  
Braided Monoidal Category

scholarly journals The Hopf algebra structure of a crossed product in a braided monoidal category

2002 ◽  
Vol 26 (2) ◽  
pp. 299-311 ◽  
Author(s):  
J. N. Alonso Alvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodriguez
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Crossed Product ◽  
Algebra Structure ◽  
Braided Monoidal Category ◽  
Hopf Algebra Structure

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.

Hopf Quasimodules and Yetter-Drinfeld Modules over Hopf Quasigroups

2021 ◽  
Vol 28 (02) ◽  
pp. 213-242
Author(s):  
Tao Zhang ◽  
Yue Gu ◽  
Shuanhong Wang
Keyword(s):  
Monoidal Category ◽  
Drinfeld Modules ◽  
Braided Monoidal Category ◽  
Symmetric Monoidal Category ◽  
Category Equivalence ◽  
Hopf Modules ◽  
Hopf Quasigroup

We introduce the notions of a four-angle Hopf quasimodule and an adjoint quasiaction over a Hopf quasigroup [Formula: see text] in a symmetric monoidal category [Formula: see text]. If [Formula: see text] possesses an adjoint quasiaction, we show that symmetric Yetter-Drinfeld categories are trivial, and hence we obtain a braided monoidal category equivalence between the category of right Yetter-Drinfeld modules over [Formula: see text] and the category of four-angle Hopf modules over [Formula: see text] under some suitable conditions.

Transmutation theory and rank for quantum braided groups

1993 ◽  
Vol 113 (1) ◽  
pp. 45-70 ◽  
Author(s):  
Shahn Majid
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Braided Monoidal Category ◽  
Quasitriangular Hopf Algebra ◽  
Cocommutative Hopf Algebra

AbstractLet f: H1 → H2be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B(H1, f, H2) in the braided monoidal category of H1-modules. It consists in the same algebra as H2 with a modified comultiplication and has a quasitriangular structure given by the ratio of those of H1 and H2. This transmutation procedure trades a non-cocommutative Hopf algebra in the category of k-modules for a more cocommutative object in a more non-commutative category. As an application, every Hopf algebra containing the group algebra of ℤ2 becomes transmuted to a super-Hopf algebra.

Weak Galois and Weak Cocleft Coextensions

2007 ◽  
Vol 14 (02) ◽  
pp. 229-244
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez ◽  
A. B. Rodríguez Raposo
Keyword(s):  
Monoidal Category ◽  
Normal Basis ◽  
Braided Monoidal Category

For a weak entwining structure (A, C, ψ) living in a braided monoidal category with equalizers and coequalizers, we formulate the notion of weak A-Galois coextension with normal basis and we show that these Galois coextensions are equivalent to the weak A-cocleft coextensions introduced by the authors.

Relative Hopf modules in the braided monoidal category _LM

2014 ◽  
Vol 8 ◽  
pp. 733-738
Author(s):  
Wenqiang Li ◽  
Xineng Hu ◽  
Jinqi Li
Keyword(s):  
Monoidal Category ◽  
Braided Monoidal Category ◽  
Hopf Modules

scholarly journals The Brauer Group of a Braided Monoidal Category

1998 ◽  
Vol 202 (1) ◽  
pp. 96-128 ◽  
Author(s):  
Fred Van Oystaeyen ◽  
Yinhuo Zhang
Keyword(s):  
Monoidal Category ◽  
Brauer Group ◽  
Braided Monoidal Category

The Braided Monoidal Structure on the Category of Comodules of Bimonads

2019 ◽  
Vol 26 (04) ◽  
pp. 565-578
Author(s):  
Bingliang Shen ◽  
Xiaoguang Zou
Keyword(s):  
Monoidal Category ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Braided Monoidal Category ◽  
Compatibility Relation ◽  
Monoidal Structure ◽  
Necessary And Sufficient

We investigate how the category of comodules of bimonads can be made into a monoidal category. It suffices that the monad and comonad in question are bimonads, with some extra compatibility relation. On a monoidal category of comodules of bimonads, we construct a braiding and get the necessary and sufficient conditions making it a braided monoidal category. As an application, we consider the category of comodules of corings and the category of entwined modules.

ON THE BRAIDED MONOIDAL CATEGORY OF W-SMASH COPRODUCT COMODULE

2010 ◽  
Vol 03 (02) ◽  
pp. 295-305
Author(s):  
Lihong Dong ◽  
Zhengming Jiao
Keyword(s):  
Monoidal Category ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Braided Monoidal Category ◽  
Necessary And Sufficient

In this paper, we discuss the W-smash coproduct comodule category [Formula: see text] and give the necessary and sufficient conditions for the W-smash coproduct comodule category [Formula: see text] to be a braided monoidal category.

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