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.

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.

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

2009 ◽  
Vol 16 (02) ◽  
pp. 293-308 ◽  
Author(s):  
Qingwen Wang ◽  
Guangjing Song ◽  
Xin Liu
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Common Solution ◽  
Special Cases ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

The Projective Antecedent of the Three Reflection Theorem

1991 ◽  
Vol 34 (2) ◽  
pp. 265-274
Author(s):  
F. A. Sherk
Keyword(s):  
Projective Geometry ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Metric Geometry ◽  
Natural Question ◽  
Complete Answer ◽  
Special Cases ◽  
Reflection Theorem ◽  
Necessary And Sufficient

AbstractA complete answer is given to the question: Under what circumstances is the product of three harmonic homologies in PG(2, F) again a harmonic homology ? This is the natural question to ask in seeking a generalization to projective geometry of the Three Reflection Theorem of metric geometry. It is found that apart from two familiar special cases, and with one curious exception, the necessary and sufficient conditions on the harmonic homologies produce exactly the Three Reflection Theorem.

(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.

scholarly journals H-function with complex parameters I: existence

2001 ◽  
Vol 25 (9) ◽  
pp. 571-586
Author(s):  
Fadhel A. Al-Musallam ◽  
Vu Kim Tuan
Keyword(s):  
Sufficient Conditions ◽  
Integral Transforms ◽  
Necessary And Sufficient Conditions ◽  
Special Cases ◽  
Type Integral ◽  
Necessary And Sufficient

AnH-function with complex parameters is defined by a Mellin-Barnes type integral. Necessary and sufficient conditions under which the integral defining theH-function converges absolutely are established. Some properties, special cases, and an application to integral transforms are given.

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.

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.

scholarly journals A note on the intersection of free maximal ideals

1969 ◽  
Vol 10 (1-2) ◽  
pp. 204-206 ◽  
Author(s):  
Stewart M. Robinson
Keyword(s):  
Compact Support ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Uniform Structure ◽  
Maximal Ideals ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Ring Of Functions

In [2] we proved that if X admits a complete uniform structure, the intersection of the free maximal ideals in C(x) is precisely Ck(X), the ring of functions with compact support. In the present paper we are able to sharpen this result somewhat and give necessary and sufficient conditions on a space X so that this conclusion holds. Both our previous result and that of Kohls for p-spaces follow as special cases of our theorem.

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