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 .

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.

On the Hopf algebra dual of an enveloping algebra

1982 ◽  
Vol 91 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Stephen Donkin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Groups ◽  
Field Extension ◽  
Smash Product ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Polycyclic Groups ◽  
Algebra Over A Field

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

Hom-L-R-smash biproduct and the category of Hom–Yetter–Drinfel’d–Long bimodules

2018 ◽  
Vol 17 (07) ◽  
pp. 1850133 ◽  
Author(s):  
Daowei Lu ◽  
Xiaohui Zhang
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Monoidal Category ◽  
Admissible Pair ◽  
Smash Product ◽  
Monoidal Categories ◽  
Finite Dimensional

Let [Formula: see text] be a Hom-bialgebra. In this paper, we firstly introduce the notion of Hom-L-R smash coproduct [Formula: see text], where [Formula: see text] is a Hom-coalgebra. Then for a Hom-algebra and Hom-coalgebra [Formula: see text], we introduce the notion of Hom-L-R-admissible pair [Formula: see text]. We prove that [Formula: see text] becomes a Hom-bialgebra under Hom-L-R smash product and Hom-L-R smash coproduct. Next, we will introduce a prebraided monoidal category [Formula: see text] of Hom–Yetter–Drinfel’d–Long bimodules and show that Hom-L-R-admissible pair [Formula: see text] actually corresponds to a bialgebra in the category [Formula: see text], when [Formula: see text] and [Formula: see text] are involutions. Finally, we prove that when [Formula: see text] is finite dimensional Hom-Hopf algebra, [Formula: see text] is isomorphic to the Yetter–Drinfel’d category [Formula: see text] as braid monoidal categories where [Formula: see text] is the tensor product Hom–Hopf algebra.

SEPARABLE FUNCTORS FOR THE CATEGORY OF PARTIAL ENTWINED MODULES

2012 ◽  
Vol 11 (05) ◽  
pp. 1250101
Author(s):  
SHUANG-JIAN GUO ◽  
SHUAN-HONG WANG
Keyword(s):  
Type Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Maschke Type Theorem ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for the functor F from the category of partial entwined modules to the category of right A-modules to be separable. This leads to a generalized notion of integrals of partial entwining structures. As an application, we prove a Maschke-type theorem for partial entwined modules.

scholarly journals A duality theorem for weak multiplier Hopf algebra actions

2017 ◽  
Vol 28 (05) ◽  
pp. 1750032 ◽  
Author(s):  
Nan Zhou ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Duality Theorem ◽  
Smash Product ◽  
Module Algebra ◽  
Multiplier Hopf Algebras ◽  
Multiplier Hopf Algebra ◽  
Weak Hopf Algebras

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by Van Daele and Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids.

An extended form of Majid’s double biproduct

2017 ◽  
Vol 16 (04) ◽  
pp. 1750061 ◽  
Author(s):  
Tianshui Ma ◽  
Huihui Zheng
Keyword(s):  
Hopf Algebra ◽  
Sufficient Conditions ◽  
Smash Product ◽  
Algebra Structure ◽  
Module Algebra ◽  
Extended Form ◽  
Linear Map ◽  
Product Algebra ◽  
Necessary And Sufficient ◽  
Crossed Coproducts

Let [Formula: see text] be a bialgebra. Let [Formula: see text] be a linear map, where [Formula: see text] is a left [Formula: see text]-module algebra, and a coalgebra with a left [Formula: see text]-weak coaction. Let [Formula: see text] be a linear map, where [Formula: see text] is a right [Formula: see text]-module algebra, and a coalgebra with a right [Formula: see text]-weak coaction. In this paper, we extend the construction of two-sided smash coproduct to two-sided crossed coproduct [Formula: see text]. Then we derive the necessary and sufficient conditions for two-sided smash product algebra [Formula: see text] and [Formula: see text] to be a bialgebra, which generalizes the Majid’s double biproduct in [Double-bosonization of braided groups and the construction of [Formula: see text], Math. Proc. Camb. Philos. Soc. 125(1) (1999) 151–192] and the Wang–Wang–Yao’s crossed coproduct in [Hopf algebra structure over crossed coproducts, Southeast Asian Bull. Math. 24(1) (2000) 105–113].

scholarly journals Quantum Groupoids Acting on Semiprime Algebras

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.

scholarly journals On Hopf DeMeyer-Kanzaki Galois extensions

2003 ◽  
Vol 2003 (26) ◽  
pp. 1627-1632
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Hopf Algebra ◽  
Galois Extension ◽  
Smash Product ◽  
Module Algebra ◽  
Galois Extensions ◽  
Finite Dimensional ◽  
Dual Hopf Algebra ◽  
Finite Dimensional Hopf Algebra ◽  
Algebra Over A Field

LetHbe a finite-dimensional Hopf algebra over a fieldk,Ba leftH-module algebra, andH∗the dual Hopf algebra ofH. For anH∗-Azumaya Galois extensionBwith centerC, it is shown thatBis anH∗-DeMeyer-Kanzaki Galois extension if and only ifCis a maximal commutative separable subalgebra of the smash productB#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.

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