Frobenius cowreaths and Morita contexts

2020 ◽  
Vol 32 (5) ◽  
pp. 1143-1175
Author(s):  
Daniel Bulacu ◽  
Blas Torrecillas
Keyword(s):  
Hopf Algebras ◽  
Monoidal Category ◽  
Type Theorem ◽  
Morita Context ◽  
Total Integral

AbstractWe prove a uniqueness type theorem for (weak, total) integrals on a Frobenius cowreath in a monoidal category. When the cowreath is, moreover, pre-Galois, we construct a Morita context relating the subalgebra of coinvariants and a certain wreath algebra. Then we see that the strictness of the Morita context is related to the Galois property of the cowreath and the existence of a weak total integral on it. We apply our results to quasi-Hopf algebras.

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.

SMASH PRODUCTS, SEPARABLE EXTENSIONS AND A MORITA CONTEXT OVER HOPF ALGEBROIDS

2014 ◽  
Vol 13 (04) ◽  
pp. 1350124
Author(s):  
YONG WANG ◽  
GUANGQUAN GUO
Keyword(s):  
Type Theorem ◽  
Smash Product ◽  
Module Algebra ◽  
Morita Context ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Maschke Type Theorem ◽  
Module Algebras ◽  
Product Algebra ◽  
Separable Extensions

Let [Formula: see text] be a Hopf algebroid, and A a left [Formula: see text]-module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids, which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra100 (1986) 363–379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math.179 (2003) 75–121], we construct a Morita context connecting A#H and [Formula: see text] the invariant subalgebra of [Formula: see text] on A.

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.

scholarly journals Comodules over weak multiplier bialgebras

2014 ◽  
Vol 25 (05) ◽  
pp. 1450037 ◽  
Author(s):  
Gabriella Böhm
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Base Algebra ◽  
Finite Dimensional ◽  
Linear Maps ◽  
Multiplier Hopf Algebras ◽  
Hopf Modules ◽  
Total Algebra ◽  
Module Structures

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

scholarly journals A Few Remarks on the Tube Algebra of a Monoidal Category

2018 ◽  
Vol 61 (3) ◽  
pp. 735-758 ◽  
Author(s):  
Sergey Neshveyev ◽  
Makoto Yamashita
Keyword(s):  
Quantum Group ◽  
Monoidal Category ◽  
Compact Quantum Group ◽  
Morita Context ◽  
Tensor Categories ◽  
Drinfeld Double ◽  
Morita Equivalent

AbstractWe prove two results on the tube algebras of rigid C*-tensor categories. The first is that the tube algebra of the representation category of a compact quantum groupGis a full corner of the Drinfeld double ofG. As an application, we obtain some information on the structure of the tube algebras of the Temperley–Lieb categories 𝒯ℒ(d) ford> 2. The second result is that the tube algebras of weakly Morita equivalent C*-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the 2-category defining the Morita context.

Morita context of weak Hopf algebras

2011 ◽  
Vol 31 (3) ◽  
pp. 1133-1141
Author(s):  
Bo Hou ◽  
Zilong Zhang ◽  
Bingling Cai ◽  
Yan Li
Keyword(s):  
Hopf Algebras ◽  
Morita Context ◽  
Weak Hopf Algebras

scholarly journals The Maschke-Type Theorem and Morita Context for BiHom-Smash Products

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Bingliang Shen ◽  
Ling Liu
Keyword(s):  
Hopf Algebra ◽  
Type Theorem ◽  
Smash Product ◽  
Morita Context ◽  
Maschke Type Theorem

Let H , α H , β H , ω H , ψ H , S H be a BiHom-Hopf algebra and A , α A , β A be an H , α H , β H -module BiHom-algebra. Then, in this paper, we study some properties on the BiHom-smash product A # H . We construct the Maschke-type theorem for the BiHom-smash product A # H and form an associated Morita context A H , A H A A # H , A # H A A H , A # H .

scholarly journals The Tensor Category of Linear Maps and Leibniz Algebras

1998 ◽  
Vol 5 (3) ◽  
pp. 263-276
Author(s):  
J. L. Loday ◽  
T. Pirashvili
Keyword(s):  
Tensor Product ◽  
Hopf Algebras ◽  
Type Theorem ◽  
Tensor Category ◽  
Leibniz Algebra ◽  
Vector Spaces ◽  
Associative Algebras ◽  
Enveloping Algebra ◽  
Leibniz Algebras ◽  
Linear Maps

Abstract We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor–Moore type theorem for Leibniz algebras.

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