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.

scholarly journals A MONOIDAL STRUCTURE ON THE CATEGORY OF RELATIVE HOPF MODULES

2012 ◽  
Vol 11 (02) ◽  
pp. 1250026
Author(s):  
DANIEL BULACU ◽  
STEFAAN CAENEPEEL
Keyword(s):  
Tensor Product ◽  
Monoidal Category ◽  
Drinfeld Modules ◽  
Braided Monoidal Category ◽  
Monoidal Structure ◽  
Comodule Algebra ◽  
Hopf Modules

Let B be a bialgebra, and A be a left B-comodule algebra in a braided monoidal category [Formula: see text], and assume that A is also a coalgebra, with a not-necessarily associative or unital left B-action. Then we can define a right A-action on the tensor product of two relative Hopf modules, and this defines a monoidal structure on the category of relative Hopf modules if and only if A is a bialgebra in the category of left Yetter–Drinfeld modules over B. Some examples are given.

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

Symmetries and the u-condition in weak monoidal Hom–Yetter–Drinfeld categories

2020 ◽  
pp. 2150194
Author(s):  
Linlin Liu ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Drinfeld Modules ◽  
Braided Monoidal Category

The aim of this paper is to introduce and study Yetter–Drinfeld category over a weak monoidal Hom–Hopf algebra [Formula: see text]. We first show that the category [Formula: see text] of Yetter–Drinfeld modules over [Formula: see text] with a bijective antipode is a braided monoidal category. Secondly, we discuss some properties on the symmetries of the category [Formula: see text]. Finally, we prove that the representation category of triangular weak monoidal Hom–Hopf algebra is a symmetric braided monoidal subcategory of [Formula: see text]. Furthermore, a class of weak monoidal Hom–Yetter–Drinfeld modules are constructed by a quasitriangular weak monoidal Hom–Hopf algebra.

BRAIDED MIXED DATUMS AND THEIR APPLICATIONS ON HOM-QUANTUM GROUPS

2017 ◽  
Vol 60 (1) ◽  
pp. 231-251 ◽  
Author(s):  
XIAOHUI ZHANG ◽  
LIHONG DONG
Keyword(s):  
Quantum Groups ◽  
Monoidal Category ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drinfeld Modules ◽  
Distributive Law ◽  
Necessary And Sufficient

AbstractIn this paper, we mainly provide a categorical view on the braided structures appearing in the Hom-quantum groups. Let $\mathcal{C}$ be a monoidal category on which F is a bimonad, G is a bicomonad, and ϕ is a distributive law, we discuss the necessary and sufficient conditions for $\mathcal{C}^G_F(\varphi)$, the category of mixed bimodules to be monoidal and braided. As applications, we discuss the Hom-type (co)quasitriangular structures, the Hom–Yetter–Drinfeld modules, and the Hom–Long dimodules.

scholarly journals Twin TQFTs and Frobenius Algebras

2013 ◽  
Vol 2013 ◽  
pp. 1-25
Author(s):  
Carmen Caprau
Keyword(s):  
Monoidal Category ◽  
Topological Quantum Field Theory ◽  
Algebra Homomorphism ◽  
Frobenius Algebra ◽  
Quantum Field ◽  
Generators And Relations ◽  
Symmetric Monoidal Category ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Algebraic Description

We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on atwin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra(C,W,z,z∗)consists of a commutative Frobenius algebraC, a symmetric Frobenius algebraW, and an algebra homomorphismz:C→Wwith dualz∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.

Monad compositions II: Kleisli strength

2008 ◽  
Vol 18 (3) ◽  
pp. 613-643 ◽  
Author(s):  
ERNIE MANES ◽  
PHILIP MULRY
Keyword(s):  
Monoidal Category ◽  
Linear Equations ◽  
Large Collection ◽  
Data Types ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Symmetric Monoidal Category ◽  
Cartesian Closed

In this paper we introduce the concept of Kleisli strength for monads in an arbitrary symmetric monoidal category. This generalises the notion of commutative monad and gives us new examples, even in the cartesian-closed category of sets. We exploit the presence of Kleisli strength to derive methods for generating distributive laws. We also introduce linear equations to extend the results to certain quotient monads. Mechanisms are described for finding strengths that produce a large collection of new distributive laws, and consequently monad compositions, including the composition of monadic data types such as lists, trees, exceptions and state.

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.

scholarly journals The Monoidal Category of Yetter-Drinfeld Modules over a Weak Braided Hopf Algebra

2015 ◽  
Vol 22 (spec01) ◽  
pp. 871-902
Author(s):  
J.N. Alonso Álvarez ◽  
J.M. Fernández Vilaboa ◽  
R. González Rodríguez ◽  
C. Soneira Calvo
Keyword(s):  
Hopf Algebra ◽  
Monoidal Category ◽  
Adjoint Action ◽  
Drinfeld Modules ◽  
Braided Hopf Algebra

In this paper we introduce the notion of weak operators and the theory of Yetter-Drinfeld modules over a weak braided Hopf algebra with invertible antipode in a strict monoidal category. We prove that the class of such objects constitutes a non-strict monoidal category. It is also shown that this category is not trivial, that is to say, it admits objects generated by the adjoint action (coaction) associated to the weak braided Hopf algebra.

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