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,?).

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.

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.

ON CROSSED DOUBLE BIPRODUCT

2013 ◽  
Vol 12 (05) ◽  
pp. 1250211 ◽  
Author(s):  
TIANSHUI MA ◽  
ZHENGMING JIAO ◽  
YANAN SONG
Keyword(s):  
Hopf Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Product ◽  
Special Cases ◽  
Coalgebra Structure ◽  
The One ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
Necessary And Sufficient

Let H be a bialgebra. Let σ : H ⊗ H → A be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action. Let B be a right H-module algebra and also a comodule coalgebra. In this paper, we provide necessary and sufficient conditions for the one-sided crossed product algebra A#σ H # B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the crossed double biproduct. Majid's double biproduct is recovered from this. Moreover, necessary and sufficient conditions are given for Brzeziński's crossed product equipped with the smash coproduct coalgebra structure to be a bialgebra. The celebrated Radford's biproduct in [The structure of Hopf algebra with a projection, J. Algebra92 (1985) 322–347], the unified product defined by Agore and Militaru in [Extending structures II: The quantum version, J. Algebra336 (2011) 321–341] and the Wang–Jiao–Zhao's crossed product in [Hopf algebra structures on crossed products Comm. Algebra26 (1998) 1293–1303] are all derived as special cases.

Cleft extensions for quasi-entwining structures

2018 ◽  
Vol 68 (2) ◽  
pp. 339-352
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Cleft Extension ◽  
Cleft Extensions ◽  
Necessary And Sufficient

Abstract In this paper we introduce the notions of quasi-entwining structure and cleft extension for a quasi-entwining structure. We prove that if (A, C, ψ) is a quasi-entwining structure and the associated extension to the submagma of coinvariants AC is cleft, there exists an isomorphism ωA between AC ⊗ C and A. Moreover, we define two unital but not necessarily associative products on AC ⊗ C. For these structures we obtain the necessary and sufficient conditions to assure that ωA is a magma isomorphism, giving some examples fulfilling these conditions.

scholarly journals Comparison of two notions of weak crossed product

2020 ◽  
pp. 2150059
Author(s):  
Jorge A. Guccione ◽  
Juan J. Guccione
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Crossed Product ◽  
Weak Hopf Algebra ◽  
Crossed Products ◽  
Hopf Algebroid ◽  
Hopf Algebroids ◽  
Weak Crossed Product ◽  
Cleft Extensions ◽  
Weak Hopf Algebras

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

(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 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.

HOMOGENEOUS SEMILOCAL GROUP RINGS AND CROSSED PRODUCTS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250145 ◽  
Author(s):  
M. H. FAHMY ◽  
SUSAN F. EL-DEKEN ◽  
S. M. ABDELWAHAB
Keyword(s):  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Rings ◽  
Crossed Products ◽  
Semilocal Ring ◽  
Necessary And Sufficient

Let J(R) be the Jacobson radical of a ring R. Then R is called homogeneous semilocal if R/J(R) is simple artinian. The aim of this paper is to find necessary and sufficient conditions for the group rings and the crossed products to be homogeneous semilocal ring.

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