A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct...A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra since it is graded connected. The main conclusions are that the vector space spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf algebra and that there is a Hopf homomorphism from permutations to label simple graphs.展开更多
In this paper, we categorify a Hom-associative algebra by imposing the Homassociative law up to some isomorphisms on the multiplication map and requiring that these isomorphisms satisfy the Pentagon axiom, and obtain ...In this paper, we categorify a Hom-associative algebra by imposing the Homassociative law up to some isomorphisms on the multiplication map and requiring that these isomorphisms satisfy the Pentagon axiom, and obtain a 2-Hom-associative algebra. On the other hand, we introduce the dual Hom-quasi-Hopf algebra and show that any dual Homquasi-Hopf algebras can be viewed as a 2-Hom-associative algebra.展开更多
In this paper we generalize the notions of crossed products and L-R smash products in the context of multiplier Hopf algebras. We use comodule algebras to define generalized diagonal crossed products, L-R smash produc...In this paper we generalize the notions of crossed products and L-R smash products in the context of multiplier Hopf algebras. We use comodule algebras to define generalized diagonal crossed products, L-R smash products, two-sided smash products and two-sided crossed products and prove that they are all associative algebras. Then we show the isomorphic relations of them.展开更多
This paper gives a sufficient and necessary condition for twisted products tobe weak Hopf algebras, moreover, gives a description for smash products to be weak Hopfalgebras. It respectively generalizes R.K.Molnar′s m...This paper gives a sufficient and necessary condition for twisted products tobe weak Hopf algebras, moreover, gives a description for smash products to be weak Hopfalgebras. It respectively generalizes R.K.Molnar′s major result and I.Boca′s result.展开更多
Some basic properties of the antipode of an NoI-graded χ-Hopf algebra arestudied. Also, several equivalent conditions of an NoI-graded χ-bialgebra (χ-Hopf algebra)are given.
We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on theco-multiplication map and requiring that these isomorphisms satisfy certain law of their own,which is called the co...We categorify the notion of coalgebras by imposing a co-associative law up to some isomorphisms on theco-multiplication map and requiring that these isomorphisms satisfy certain law of their own,which is called the copentagonidentity.We also set up a 2-category of 2-coalgebras.The purpose of this study is from the idea of reconsideringthe quasi-Hopf algebras by the categoriScation process,so that we can study the theory of quasi-Hopf algebras and theirrepresentations in some new framework of higher category theory in naturai ways.展开更多
The purpose of this paper is to present some dual properties of dual comodule.It turns out that dual comodule has universal property (cf.Theorem 2).Since(()*,()°)is an adjoint pair(cf.Theorem 3),some nice propert...The purpose of this paper is to present some dual properties of dual comodule.It turns out that dual comodule has universal property (cf.Theorem 2).Since(()*,()°)is an adjoint pair(cf.Theorem 3),some nice properties of functor( )° are obtained.Finely Theoram 4 provides that the cotensor product is the dual of the tensor product by(M A N)°≌ M°□A.N°.Moreover,the result HomA(M,N)≌ Coma.(N°,M°)is proved for fite related modules M,N over a refiexive algebra A.展开更多
The Yetter-Drinfeld category of the Hopf algebra over a field is a pre braided category. In this paper we prove this result for the weak Hopf algebra. We study the smash product and smash coproduct, weak biproducts in...The Yetter-Drinfeld category of the Hopf algebra over a field is a pre braided category. In this paper we prove this result for the weak Hopf algebra. We study the smash product and smash coproduct, weak biproducts in the weak Hopf algebra over a field k. For a weak Hopf algebra A in left Yetter-Drinfeld category HHYD. we prove that the weak biproducts of A and H is a weak Hopf algebra.展开更多
Let(C, α) and(H, β) be Hom-bialgebras and ω : C H→ H C a linear map. We introduce a Hom-ω-smash coproduct(Cω■H, γ) and give necessary and sufficient conditions for(Cω■H, γ) to be a Hom-bialgebra. We stu...Let(C, α) and(H, β) be Hom-bialgebras and ω : C H→ H C a linear map. We introduce a Hom-ω-smash coproduct(Cω■H, γ) and give necessary and sufficient conditions for(Cω■H, γ) to be a Hom-bialgebra. We study the quasi-triangular structures over(Cω■H, γ)and show the necessary and sufficient conditions for(Cω■H, γ, R) to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D(H)*and construct quasi-triangular structures over D(H)*.展开更多
The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra At =n i =0Aαi tfrom a Clifford monoid S = [Y; Gα, φα,β ]by Ore-extensions, and to obtain a ...The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra At =n i =0Aαi tfrom a Clifford monoid S = [Y; Gα, φα,β ]by Ore-extensions, and to obtain a co-Frobenius semilattice graded weak Hopf algebra H(S, n, c, χ, a, b) through factoring At by a semilattice graded weak Hopf ideal.展开更多
In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A ...In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A's diagram equaling the diagram of its maximal pointed Hopf subalgebra.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
Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H...Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context <A^H,A#H,A,A,τ,μ> connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.展开更多
In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra H.Although the category of all left H-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Dri...In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra H.Although the category of all left H-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra H, and a series of weak Hopf algebras. Some results of [8] are generalized.展开更多
This article is devoted to the study of the symmetry in the Yetter-Drinfeld category of a finite-dimensional weak Hopf algebra.It generalizes the corresponding results in Hopf algebras.
The concept of (f,σ)-pair (B,H)is introduced, where B and H are Hopf algebras. A braided tensor category which is a tensor subcategory of the category HM of left H-comodules through an (f,σ)-pair is constructed. In ...The concept of (f,σ)-pair (B,H)is introduced, where B and H are Hopf algebras. A braided tensor category which is a tensor subcategory of the category HM of left H-comodules through an (f,σ)-pair is constructed. In particularly, a Yang-Baxter equation is got. A Hopf algebra is constructed as well in the Yetter-Drinfel’d category HHYD by twisting the multiplication of B.展开更多
文摘A lot of combinatorial objects have a natural bialgebra structure. In this paper, we prove that the vector space spanned by labeled simple graphs is a bialgebra with the conjunction product and the unshuffle coproduct. In fact, it is a Hopf algebra since it is graded connected. The main conclusions are that the vector space spanned by labeled simple graphs arising from the unshuffle coproduct is a Hopf algebra and that there is a Hopf homomorphism from permutations to label simple graphs.
基金Supported by the National Natural Science Foundation of China（11047030, 11171055） Supported by the Grant from China Scholarship Counci1（2011841026）
文摘In this paper, we categorify a Hom-associative algebra by imposing the Homassociative law up to some isomorphisms on the multiplication map and requiring that these isomorphisms satisfy the Pentagon axiom, and obtain a 2-Hom-associative algebra. On the other hand, we introduce the dual Hom-quasi-Hopf algebra and show that any dual Homquasi-Hopf algebras can be viewed as a 2-Hom-associative algebra.
基金Foundation item： Supported by the Scientific Research Foundation for Doctoral Scientists of Henan University of Science and Technology（09001303） Supported by the National Natural Science Foundation of China（11101128）
文摘In this paper we generalize the notions of crossed products and L-R smash products in the context of multiplier Hopf algebras. We use comodule algebras to define generalized diagonal crossed products, L-R smash products, two-sided smash products and two-sided crossed products and prove that they are all associative algebras. Then we show the isomorphic relations of them.
基金This work is supported by National Natural Science Foundation of Chinaby the excellent doctorate fund of Nanjing agricultural university
文摘This paper gives a sufficient and necessary condition for twisted products tobe weak Hopf algebras, moreover, gives a description for smash products to be weak Hopfalgebras. It respectively generalizes R.K.Molnar′s major result and I.Boca′s result.
基金This paper is supported by the Chinese National Natural Science Foundation
文摘Some basic properties of the antipode of an NoI-graded χ-Hopf algebra arestudied. Also, several equivalent conditions of an NoI-graded χ-bialgebra (χ-Hopf algebra)are given.
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 theco-multiplication map and requiring that these isomorphisms satisfy certain law of their own,which is called the copentagonidentity.We also set up a 2-category of 2-coalgebras.The purpose of this study is from the idea of reconsideringthe quasi-Hopf algebras by the categoriScation process,so that we can study the theory of quasi-Hopf algebras and theirrepresentations in some new framework of higher category theory in naturai ways.
基金the Nature Science Foundation of China(19901009),Nature Science oundation of Guangdong Province(970472000463)
文摘The purpose of this paper is to present some dual properties of dual comodule.It turns out that dual comodule has universal property (cf.Theorem 2).Since(()*,()°)is an adjoint pair(cf.Theorem 3),some nice properties of functor( )° are obtained.Finely Theoram 4 provides that the cotensor product is the dual of the tensor product by(M A N)°≌ M°□A.N°.Moreover,the result HomA(M,N)≌ Coma.(N°,M°)is proved for fite related modules M,N over a refiexive algebra A.
基金Partially supported by the Specialized Research Fund for the Doctoral Program of Higher Education(20060286006)the National Natural Science Foundation of China(10571026)the Southeast University Fund(XJ0707273).
基金The NSF (200510476001) of Education Department of Henan Province.
文摘The Yetter-Drinfeld category of the Hopf algebra over a field is a pre braided category. In this paper we prove this result for the weak Hopf algebra. We study the smash product and smash coproduct, weak biproducts in the weak Hopf algebra over a field k. For a weak Hopf algebra A in left Yetter-Drinfeld category HHYD. we prove that the weak biproducts of A and H is a weak Hopf algebra.
基金Supported by the National Natural Science Foundation of China(60873267)the Ningbo Natural Science Foundation of China(2011A610172)K.C.Wang Magna Fund in Ningbo University
文摘Let(C, α) and(H, β) be Hom-bialgebras and ω : C H→ H C a linear map. We introduce a Hom-ω-smash coproduct(Cω■H, γ) and give necessary and sufficient conditions for(Cω■H, γ) to be a Hom-bialgebra. We study the quasi-triangular structures over(Cω■H, γ)and show the necessary and sufficient conditions for(Cω■H, γ, R) to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D(H)*and construct quasi-triangular structures over D(H)*.
基金supported by the National Natural Science Foundation of China(11271318,11171296,and J1210038)the Specialized Research Fund for the Doctoral Program of Higher Education of China(20110101110010)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010001)
文摘The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra At =n i =0Aαi tfrom a Clifford monoid S = [Y; Gα, φα,β ]by Ore-extensions, and to obtain a co-Frobenius semilattice graded weak Hopf algebra H(S, n, c, χ, a, b) through factoring At by a semilattice graded weak Hopf ideal.
基金Supported by the National Natural Science Foundation of China(11271319,11301126)
文摘In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra A with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When A is of finite dimension, we provide a necessary and sufficient condition for A's diagram equaling the diagram of its maximal pointed Hopf subalgebra.
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).
基金Supported by the NSF of China(1097104910971052)+1 种基金the NSF of Hebei Province(A2008000135A2009000253)
文摘Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A^H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context <A^H,A#H,A,A,τ,μ> connecting the smash product A#H and the invariant subalgebra A^H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.
基金Supported by ZJNSF(LY17A010015,LZ14A010001)NNSF(11171296),CSC
文摘In this paper, we study a Yetter-Drinfeld module V over a weak Hopf algebra H.Although the category of all left H-modules is not a braided tensor category, we can define a Yetter-Drinfeld module. Using this Yetter-Drinfeld modules V, we construct Nichols algebra B(V) over the weak Hopf algebra H, and a series of weak Hopf algebras. Some results of [8] are generalized.
基金the National Natural Science Foundation of China(10301033 and 10271113)
文摘This article is devoted to the study of the symmetry in the Yetter-Drinfeld category of a finite-dimensional weak Hopf algebra.It generalizes the corresponding results in Hopf algebras.
基金Supported by the Zhejiang Provincial Natural Science Foundation (Y607075)
文摘The concept of (f,σ)-pair (B,H)is introduced, where B and H are Hopf algebras. A braided tensor category which is a tensor subcategory of the category HM of left H-comodules through an (f,σ)-pair is constructed. In particularly, a Yang-Baxter equation is got. A Hopf algebra is constructed as well in the Yetter-Drinfel’d category HHYD by twisting the multiplication of B.
