scholarly journals GLOBALIZATION OF TWISTED PARTIAL HOPF ACTIONS

2016 ◽  
Vol 101 (1) ◽  
pp. 1-28 ◽  
Author(s):  
MARCELO M. S. ALVES ◽  
ELIEZER BATISTA ◽  
MICHAEL DOKUCHAEV ◽  
ANTONIO PAQUES
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Partial Action ◽  
Necessary And Sufficient

In this work, we review some properties of twisted partial actions of Hopf algebras on unital algebras and give necessary and sufficient conditions for a twisted partial action to have a globalization. We also elaborate a series of examples.

Cleft comodules over Hopf quasigroups

2015 ◽  
Vol 17 (06) ◽  
pp. 1550007 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez ◽  
C. Soneira Calvo
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Product ◽  
Comodule Algebra ◽  
Hopf Quasigroup ◽  
Necessary And Sufficient

In this paper, we provide necessary and sufficient conditions for a cleft right H-comodule algebra (A, ϱA) over a Hopf quasigroup H to be isomorphic as an algebra to the crossed product AH♯σAHH, where AH is the coinvariants subalgebra of A and σAH is a morphism between H ⊗ H and AH. As a consequence, we obtain the corresponding version in the nonassociative setting of the result given by Blattner, Cohen and Montgomery for projections of Hopf algebras with coalgebra splitting. Concrete examples satisfying the obtained conditions are provided.

(𝜃,ω)-Twisted Radford’s Hom-biproduct and ϖ-Yetter–Drinfeld modules for Hom-Hopf algebras

2020 ◽  
Vol 19 (03) ◽  
pp. 2050046
Author(s):  
Xiao-Li Fang ◽  
Tae-Hwa Kim
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Tensor Products ◽  
Product Formula ◽  
Necessary And Sufficient Conditions ◽  
Drinfeld Modules ◽  
Associative Algebras ◽  
Necessary And Sufficient

To unify different definitions of smash Hom-products in a Hom-bialgebra [Formula: see text], we firstly introduce the notion of [Formula: see text]-twisted smash Hom-product [Formula: see text]. Secondly, we find necessary and sufficient conditions for the twisted smash Hom-product [Formula: see text] and the twisted smash Hom-coproduct [Formula: see text] to afford a Hom-bialgebra, which generalize the well-known Radford’s biproduct and the Hom-biproduct obtained in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65]. Furthermore, we introduce the notion of the category of [Formula: see text]-Yetter-Drinfeld modules which unifies the ones of Hom-Yetter Drinfeld category appeared in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65] and [A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, J. Math. Glasgow 513–538 (2016) 58]. Finally, we prove that the [Formula: see text]-twisted Radford’s Hom-biproduct [Formula: see text] is a Hom-bialgebra if and only if [Formula: see text] is a Hom-bialgebra in the category of [Formula: see text]-Yetter–Drinfeld modules [Formula: see text], generalizing the well-known Majid’s conclusion.

scholarly journals POINTED HOPF ALGEBRAS WITH CLASSICAL WEYL GROUPS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250066
Author(s):  
SHOUCHUAN ZHANG ◽  
YAO-ZHONG ZHANG
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weyl Groups ◽  
Drinfeld Modules ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Necessary And Sufficient

We prove that Nichols algebras of irreducible Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by 𝕊nare infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter–Drinfeld modules over classical Weyl groups A ⋊ 𝕊nsupported by A to be finite dimensional.

THE QUASITRIANGULAR STRUCTURES FOR ω-SMASH COPRODUCT HOPF ALGEBRAS

2009 ◽  
Vol 08 (05) ◽  
pp. 673-687 ◽  
Author(s):  
ZHENGMING JIAO
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Necessary And Sufficient

In this paper, the quasitriangular structures of ω-smash coproduct Hopf algebras Bω ⋈ H as constructed by Caenepeel, Ion, Militaru and Zhu were studied. Necessary and sufficient conditions for ω-smash coproduct Hopf algebras to be quasitriangular Hopf algebras are given in terms of properties of their components. As applications of our results, some special cases are discussed. Especially, The quasitriangular structures for D(H)* and H4ω ⋈ kℤ2 are constructed.

scholarly journals COQUASITRIANGULAR STRUCTURES FOR EXTENSIONS OF HOPF ALGEBRAS. APPLICATIONS

2012 ◽  
Vol 55 (1) ◽  
pp. 201-215 ◽  
Author(s):  
A. L. AGORE
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantum Double ◽  
Bijective Correspondence ◽  
Infinite Dimensional ◽  
Main Application ◽  
Split Extensions ◽  
Necessary And Sufficient

AbstractLet A ⊆ E be an extension of Hopf algebras such that there exists a normal left A-module coalgebra map π : E → A that splits the inclusion. We shall describe the set of all coquasitriangular structures on the Hopf algebra E in terms of the datum (A, E, π) as follows: first, any such extension E is isomorphic to a unified product A ⋉ H, for some unitary subcoalgebra H of E (A. L. Agore and G. Militaru, Unified products and split extensions of Hopf algebras, to appear in AMS Contemp. Math.). Then, as a main theorem, we establish a bijective correspondence between the set of all coquasitriangular structures on an arbitrary unified product A ⋉ H and a certain set of datum (p, τ, u, v) related to the components of the unified product. As the main application, we derive necessary and sufficient conditions for Majid's infinite-dimensional quantum double Dλ(A, H) = A ⋈τH to be a coquasitriangular Hopf algebra. Several examples are worked out in detail.

SIMPLICITY OF CROSSED PRODUCTS BY TWISTED PARTIAL ACTIONS

2019 ◽  
Vol 108 (2) ◽  
pp. 202-225
Author(s):  
ALEXANDRE BARAVIERA ◽  
WAGNER CORTES ◽  
MARLON SOARES
Keyword(s):  
Hausdorff Space ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Products ◽  
Topological Properties ◽  
Partial Action ◽  
Necessary And Sufficient ◽  
Skew Group Ring

In this article, we consider a twisted partial action $\unicode[STIX]{x1D6FC}$ of a group $G$ on an associative ring $R$ and its associated partial crossed product $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$. We provide necessary and sufficient conditions for the commutativity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ when the twisted partial action $\unicode[STIX]{x1D6FC}$ is unital. Moreover, we study necessary and sufficient conditions for the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ in the following cases: (i) $G$ is abelian; (ii) $R$ is maximal commutative in $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$; (iii) $C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$ is simple; (iv) $G$ is hypercentral. When $R=C_{0}(X)$ is the algebra of continuous functions defined on a locally compact and Hausdorff space $X$, with complex values that vanish at infinity, and $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ is the associated partial skew group ring of a partial action $\unicode[STIX]{x1D6FC}$ of a topological group $G$ on $C_{0}(X)$, we study the simplicity of $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ by using topological properties of $X$ and the results about the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$.

Twisted tensor biproduct monoidal Hom–Hopf algebras

2017 ◽  
Vol 10 (01) ◽  
pp. 1750011
Author(s):  
Tianshui Ma ◽  
Linlin Liu ◽  
Shaoxian Xu
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Smash Product ◽  
Linear Maps ◽  
Necessary And Sufficient

Let [Formula: see text] be a monoidal Hom-bialgebra, [Formula: see text] a monoidal Hom-algebra and a monoidal Hom-coalgebra. Let [Formula: see text] and [Formula: see text] be two linear maps. First, we construct the [Formula: see text]-smash product monoidal Hom-algebra [Formula: see text] and [Formula: see text]-smash coproduct monoidal Hom-coalgebra [Formula: see text]. Second, the necessary and sufficient conditions for [Formula: see text] and [Formula: see text] to be a monoidal Hom-bialgebra are obtained, which generalizes the results in [8, 11]. Lastly, we give some examples and applications.

scholarly journals Crossed products of Hom-Hopf algebras

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1295-1313
Author(s):  
Daowei Lu ◽  
Yizheng Li ◽  
Shuangjian Guo
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Product ◽  
Crossed Products ◽  
Cleft Extension ◽  
Necessary And Sufficient

Let (H,?) be a Hom-Hopf algebra and (A,?) be a Hom-algebra. In this paper we will construct the Hom-crossed product (A#?H???), and prove that the extension A ? A#?H is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make (A#?H???) into a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on (H,?).

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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