scholarly journals A method of constructing braided Hopf algebras

Filomat ◽  
2010 ◽  
Vol 24 (2) ◽  
pp. 53-66
Author(s):  
Tianshui Ma ◽  
Shuanhong Wang ◽  
Shaoxian Xu
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Recent Work ◽  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Braided Hopf Algebra ◽  
Subject Classifications ◽  
Mathematics Subject Classifications ◽  
Twisted Tensor Product ◽  
Necessary And Sufficient

Let A and B be two Hopf algebras and R ( Hom(B ( A, A ( B), the twisted tensor product Hopf algebra A#RB was introduced by S. Caenepeel et al in [3] and further studied in our recent work [6]. In this paper we give the necessary and sufficient conditions for A#RB to be a Hopf algebra with a projection. Furthermore, a braided Hopf algebra A is constructed by twisting the multiplication of A through a (?, R)-pair (A, B). Finally we give a method to construct Radford's biproduct directly by defining the module action and comodule action from the twisted tensor biproduct. 2010 Mathematics Subject Classifications. 16W30. .

QUASITRIANGULARITY OF BRZEZIŃSKI'S CROSSED COPRODUCTS

2011 ◽  
Vol 10 (02) ◽  
pp. 241-255 ◽  
Author(s):  
TIANSHUI MA ◽  
HAIYING LI ◽  
SHUANHONG WANG
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Recent Work ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Algebra Structure ◽  
Product Algebra ◽  
Necessary And Sufficient ◽  
Crossed Coproducts ◽  
Quasitriangular Hopf Algebra

In continuation of our recent work about the quasitriangular structures for the twisted tensor biproduct, we give the necessary and sufficient conditions for Brzeziński crossed coproduct coalgebra, including the twisted tensor coproduct introduced by Caenepeel, Ion, Militaru and Zhu and crossed coproduct as constructed by Lin, equipped with the usual tensor product algebra structure to be a Hopf algebra. Furthermore, the necessary and sufficient conditions for Brzeziński crossed coproduct to be a quasitriangular Hopf algebra are obtained.

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 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,?).

scholarly journals Browder and Weyl spectra of upper triangular operator matrices

Filomat ◽  
2010 ◽  
Vol 24 (2) ◽  
pp. 111-130 ◽  
Author(s):  
B.P. Duggal
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Weyl’S Theorem ◽  
Single Valued Extension Property ◽  
Browder’S Theorem ◽  
Subject Classifications ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Mathematics Subject Classifications ◽  
Necessary And Sufficient

Let MC = (A/0 C/B) ( B(X ( X ) be an upper triangulat Banach space operator. The relationship between the spectra of MC and M0, and their various distinguished parts, has been studied by a large number of authors in the recent past. This paper brings forth the important role played by SVEP, the single-valued extension property, in the study of some of these relations. Operators MC and M0 satisfying Browder's, or a-Browder's, theorem are characterized, and we prove necessary and sufficient conditions for implications of the type 'M0 satisfies a-Browder's (or a-Weyl's) theorem ( MC satisfies a-Browder's (resp., a-Weyl's) theorem' to hold. 2010 Mathematics Subject Classifications. Primary 47B47, 47A10, 47A11. .

scholarly journals (r,p)-absolutely summing operators on the spaceC (T,X)and applications

2001 ◽  
Vol 6 (5) ◽  
pp. 309-315 ◽  
Author(s):  
Dumitru Popa
Keyword(s):  
Integral Operator ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Summing Operator ◽  
Absolutely Summing Operators ◽  
Injective Tensor Product ◽  
Absolutely Summing Operator ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for an operator on the spaceC (T,X)to be(r,p)-absolutely summing. Also we prove that the injective tensor product of an integral operator and an(r,p)-absolutely summing operator is an(r,p)-absolutely summing operator.

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.

A Family of Non-weight Modules over the Virasoro Algebra

2020 ◽  
Vol 27 (04) ◽  
pp. 807-820
Author(s):  
Guobo Chen
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Necessary And Sufficient Conditions ◽  
Weight Module ◽  
Highest Weight Module ◽  
Isomorphism Classes ◽  
Highest Weight ◽  
Necessary And Sufficient ◽  
Weight Modules

In this paper, we consider the tensor product modules of a class of non-weight modules and highest weight modules over the Virasoro algebra. We determine the necessary and sufficient conditions for such modules to be simple and the isomorphism classes among all these modules. Finally, we prove that these simple non-weight modules are new if the highest weight module over the Virasoro algebra is non-trivial.

A class of simple non-weight modules over the virasoro algebra

2020 ◽  
Vol 63 (4) ◽  
pp. 956-970 ◽  
Author(s):  
Haibo Chen ◽  
JianZhi Han
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Virasoro Algebra ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Infinite Dimensional ◽  
Highest Weight ◽  
Alpha Omega ◽  
Necessary And Sufficient ◽  
Infinite Dimensional Lie Algebra

AbstractThe Virasoro algebra $\mathcal {L}$ is an infinite-dimensional Lie algebra with basis {Lm, C| m ∈ ℤ} and relations [Lm, Ln] = (n − m)Lm+n + δm+n,0((m3 − m)/12)C, [Lm, C] = 0 for m, n ∈ ℤ. Let $\mathfrak a$ be the subalgebra of $\mathcal {L}$ spanned by Li for i ≥ −1. For any triple (μ, λ, α) of complex numbers with μ ≠ 0, λ ≠ 0 and any non-trivial $\mathfrak a$-module V satisfying the condition: for any v ∈ V there exists a non-negative integer m such that Liv = 0 for all i ≥ m, non-weight $\mathcal {L}$-modules on the linear tensor product of V and ℂ[∂], denoted by $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))\ (\Omega (\lambda ,\alpha )=\mathbb {C}[\partial ]$ as vector spaces), are constructed in this paper. We prove that $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ is simple if and only if μ ≠ 1, λ ≠ 0, α ≠ 0. We also give necessary and sufficient conditions for two such simple $\mathcal {L}$-modules being isomorphic. Finally, these simple $\mathcal {L}$-modules $\mathcal {M}(V,\mu ,\Omega (\lambda ,\alpha ))$ are proved to be new for V not being the highest weight $\mathfrak a$-module whose highest weight is non-zero.

Composition results for strongly summing and dominated multilinear operators

2014 ◽  
Vol 12 (10) ◽  
Author(s):  
Dumitru Popa
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bilinear Operator ◽  
Multilinear Operators ◽  
Weakly Compact ◽  
Necessary And Sufficient ◽  
Composition Theorem

AbstractIn this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.

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