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.

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 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. .

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.

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.

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.

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.

scholarly journals Property (Bb) and Tensor product

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1297-1303
Author(s):  
M.H.M. Rashid ◽  
T. Prasad
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Space Operator ◽  
Weyl Spectrum ◽  
Riesz Operators ◽  
Necessary And Sufficient ◽  
Banach Space Operators ◽  
Satisfy Property

In this paper, we find necessary and sufficient conditions for Banach Space operator to satisfy the property (Bb). Then we obtain, if Banach Space operators A ? B(X)and B ? B(Y) satisfy property (Bb) implies A x B satisfies property (Bb) if and only if the B-Weyl spectrum identity ?BW(A x B) = ?BW(A)?(B) U ?BW(B)?(A) holds. Perturbations by Riesz operators are considered.

scholarly journals Trivial action on the tensor product of finite groups

1988 ◽  
Vol 30 (3) ◽  
pp. 271-274 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Exact Sequence ◽  
Tensor Product ◽  
Direct Product ◽  
Semidirect Product ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Schur Multiplier ◽  
Necessary And Sufficient

Let G, H and K be finite groups such that K acts on both G and H. The action of K on G and H induces an action of K on their tensor product G ⊗ H, and we shall denote the K-stable subgroup of G ⊗ H by (G ⊗ H)K. In section 1 of this note we shall obtain necessary and sufficient conditions for (G ⊗ H)K = G ⊗ H. The importance of this result is that the direct product of G and H has Schur multiplier M(G × H) isomorphic to M(G) × M(H) × (G ⊗ H); moreover K: acts on M(G × H), and M(G × H)K is one of the terms contained in a fundamental exact sequence concerning the Schur multiplier of the semidirect product of K and G × H (see [3, (2.2.10) and (2.2.5)] for details). Indeed in section 2 we shall assume that G is abelian and use the fact that M(G) ≅ G ∧ G to find necessary and sufficient conditions for M(G)K = M(G).

Synchronization in mutual-coupled temporal Boolean networks

2017 ◽  
Vol 40 (7) ◽  
pp. 2211-2216 ◽  
Author(s):  
Qiang Wei ◽  
Cheng-jun Xie
Keyword(s):  
Theoretical Analysis ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Boolean Networks ◽  
Necessary And Sufficient Conditions ◽  
Complete Synchronization ◽  
Logical Relationship ◽  
Algebraic Form ◽  
Algebraic Forms ◽  
Necessary And Sufficient

In this paper, we first propose a mutual-coupled temporal Boolean networks model and then investigate complete synchronization in mutual-coupled temporal Boolean networks. The mutual-coupled temporal Boolean networks model with logical relationship is converted into an algebraic form based on a semi-tensor product. Necessary and sufficient conditions are derived to realize synchronization based on the algebraic forms. An example illustrates the effectiveness of the theoretical analysis.

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.

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