Let G be a group and （A, B） be a pair of multiplier Hopf algebras, where B is regular G-cograded. Let π be a crossing action of G on B, D^π=A^cop∝B=＋p∈GDπ^p with Dπ^p=A^cop∝Bp, is the Drinfeld double of the ...Let G be a group and （A, B） be a pair of multiplier Hopf algebras, where B is regular G-cograded. Let π be a crossing action of G on B, D^π=A^cop∝B=＋p∈GDπ^p with Dπ^p=A^cop∝Bp, is the Drinfeld double of the pair （A, B）, and then the deformation D^π becomes a multiplier Hopf algebra. B×A can be considered as a subalgebra of M（D^π×D^π）, the image of element b×a in B×A is （1∝b）×（a∝1） in M（D^π×D^π）. Let W =∑αWα∈ M（B×A） be a π-canonical multiplier for the pair （A, B） with Wα∈M（Bα×A） for all α∈G. The image of W in M（D^π×D^π）is a π-quasitriangular structure over D^π.展开更多
We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certairl law of their own, which is called the...We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certairl law of their own, which is called the copentagon identity. We also set up a 2-category of 2-coalgebras. The purpose of this study is from the idea of reconsidering the quasi-Hopf algebras by the categorification process, so that we can study the theory of quasi-Hopf algebras and their representations in some new framework of higher category theory in natural ways.展开更多
The construction of the biproduct of Hopf algebras, which consists of smash product and the dual notion of smash coproduct, was first formulated by Radford. In this paper we study the quasitriangular structures over b...The construction of the biproduct of Hopf algebras, which consists of smash product and the dual notion of smash coproduct, was first formulated by Radford. In this paper we study the quasitriangular structures over biproduct Hopf algebras B*H. We show the necessary and sufficient conditions for biproduct Hopf algebras to be quasitriangular. For the case when they are, we determine completely the unique formula of the quasitriangular structures. And so we find a way to construct solutions of the Yang-Baxter equation over biproduct Hopf algebras in the sense of (Majid, 1990).展开更多
The notions of u-quasi-Hopf algebras and the quantum dimension dimuM of a representation M by u are introduced.It is shown that a u-quasi-Hopf algebra H is semisimple if and only if there is a finite-dimensional proje...The notions of u-quasi-Hopf algebras and the quantum dimension dimuM of a representation M by u are introduced.It is shown that a u-quasi-Hopf algebra H is semisimple if and only if there is a finite-dimensional projective H-module P such that dimu P is invertible.展开更多
In this paper,we show that if H is a finite dimensional Hopf algebra then H is quasitri-angular if and only if H is coquasi-triangular. As a consequentility ,we obtain a generalized result of Sauchenburg.
This paper introduces the concepts of cylinder coalgebras and cylinder coproducts for quasitriangular bialgebras, and points out that there exists an anti-coalgebra isomorphism （H,△^-）≌ （H,△^-）, where （H, △^-...This paper introduces the concepts of cylinder coalgebras and cylinder coproducts for quasitriangular bialgebras, and points out that there exists an anti-coalgebra isomorphism （H,△^-）≌ （H,△^-）, where （H, △^-） is the cylinder coproduct, and （H,△^-） is the braided coproduct given by Kass. For any finite dimensional Hopf algebra H, the Drinfel＇d double （D（H）,△^-D（H）） is proved to be the cylinder coproduct. Let （H, H, R） be copaired Hopf algebras. If R ∈ Z（H×H） with inverse R-1 and skew inverse R, then the twisted coalgebra （H^R）^R-1 is constructed via twice twists, whose comultiplication is exactly the cylinder coproduct. Moreover, for any generalized Long dimodule, some solutions for Yang-Baxter equations, four braid pairs and Long equations are constructed via cylinder twists.展开更多
基金Specialized Research Fund for the Doctoral Program of Higher Education(No20060286006)the National Natural Science Foundation of China(No10871042)
文摘Let G be a group and （A, B） be a pair of multiplier Hopf algebras, where B is regular G-cograded. Let π be a crossing action of G on B, D^π=A^cop∝B=＋p∈GDπ^p with Dπ^p=A^cop∝Bp, is the Drinfeld double of the pair （A, B）, and then the deformation D^π becomes a multiplier Hopf algebra. B×A can be considered as a subalgebra of M（D^π×D^π）, the image of element b×a in B×A is （1∝b）×（a∝1） in M（D^π×D^π）. Let W =∑αWα∈ M（B×A） be a π-canonical multiplier for the pair （A, B） with Wα∈M（Bα×A） for all α∈G. The image of W in M（D^π×D^π）is a π-quasitriangular structure over D^π.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10975102, 11031005 10871135, 10871227, and PHR201007107
文摘We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on the co-multiplication map and requiring that these isomorphisms satisfy certairl law of their own, which is called the copentagon identity. We also set up a 2-category of 2-coalgebras. The purpose of this study is from the idea of reconsidering the quasi-Hopf algebras by the categorification process, so that we can study the theory of quasi-Hopf algebras and their representations in some new framework of higher category theory in natural ways.
文摘The construction of the biproduct of Hopf algebras, which consists of smash product and the dual notion of smash coproduct, was first formulated by Radford. In this paper we study the quasitriangular structures over biproduct Hopf algebras B*H. We show the necessary and sufficient conditions for biproduct Hopf algebras to be quasitriangular. For the case when they are, we determine completely the unique formula of the quasitriangular structures. And so we find a way to construct solutions of the Yang-Baxter equation over biproduct Hopf algebras in the sense of (Majid, 1990).
文摘The notions of u-quasi-Hopf algebras and the quantum dimension dimuM of a representation M by u are introduced.It is shown that a u-quasi-Hopf algebra H is semisimple if and only if there is a finite-dimensional projective H-module P such that dimu P is invertible.
基金Partially supported by the National Natural Science Foundation of China.
文摘In this paper,we show that if H is a finite dimensional Hopf algebra then H is quasitri-angular if and only if H is coquasi-triangular. As a consequentility ,we obtain a generalized result of Sauchenburg.
基金the National Natural Science Foundation of China(10571153),and Postdoctoral Science Foundation of China(2005037713)
文摘This paper introduces the concepts of cylinder coalgebras and cylinder coproducts for quasitriangular bialgebras, and points out that there exists an anti-coalgebra isomorphism （H,△^-）≌ （H,△^-）, where （H, △^-） is the cylinder coproduct, and （H,△^-） is the braided coproduct given by Kass. For any finite dimensional Hopf algebra H, the Drinfel＇d double （D（H）,△^-D（H）） is proved to be the cylinder coproduct. Let （H, H, R） be copaired Hopf algebras. If R ∈ Z（H×H） with inverse R-1 and skew inverse R, then the twisted coalgebra （H^R）^R-1 is constructed via twice twists, whose comultiplication is exactly the cylinder coproduct. Moreover, for any generalized Long dimodule, some solutions for Yang-Baxter equations, four braid pairs and Long equations are constructed via cylinder twists.
基金Project supported by Guangxi Graduate Education Innovation Program(JGY2014092)Research Project and Youth Program of Guangdong University of Science&Technology(GKY-2016KYYB-15,GKY-2017KYQN-4)“Quality Project” of Guangdong University of Science and Technology in2016
