This paper gives a sufficient and necessary condition for twisted products to be weak Hopf algebras, moreover, gives a description for smash products to be weak Hopf algebras. It respectively generalizes R.K.Molnar...This paper gives a sufficient and necessary condition for twisted products to be weak Hopf algebras, moreover, gives a description for smash products to be weak Hopf algebras. It respectively generalizes R.K.Molnar's major result and I.Boca's result.展开更多
The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the （braided） monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf a...The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the （braided） monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl （2000）.展开更多
The Yetter-Drinfeld category of the Hopf algebra over a field is a prebraided 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 prebraided 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 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
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 main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra from a Clifford monoid S =[Y; Gα. φα,β]by Ore-extensions, and to obtain a co-Frobenius semilat...The main work of this article is to give a nontrivial method to construct pointed semilattice graded weak Hopf algebra from 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, x, a, b） through factoring At by a semilattice graded weak Hopf ideal.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
Weak Hopf algebra is defined. The aim of this work is to study the relations between the weak antipode of a weak Hopf algelra and the monoid of group like elements. At first, the author shows some properties of weak ...Weak Hopf algebra is defined. The aim of this work is to study the relations between the weak antipode of a weak Hopf algelra and the monoid of group like elements. At first, the author shows some properties of weak Hopf algebras. And, for some weak Hopf algebras, some properties of the weak antipodes are given if the monoids of group like elements are inverse or orthodox semigroups. At end, the author gets a weak Hopf algebra whose monoid of group like elements is regular.展开更多
The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective corre- spondence between the isomorphisms classes of H-cleft ...The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective corre- spondence between the isomorphisms classes of H-cleft extensions AH → A, where AH is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over AH. Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group H2φZ（AH） （H, Z（AH））, where Z（AH） is the center of AH.展开更多
Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study t...Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.展开更多
In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. A...In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.展开更多
In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually s...In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.展开更多
We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying jm = j for some integer m. We denote this algebra by wUqT（A）. This algebra is a weak Hopf ...We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying jm = j for some integer m. We denote this algebra by wUqT（A）. This algebra is a weak Hopf algebra if and only if m = 2,3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra Uq （A） of a generalized Kac-Moody algebra A.展开更多
Let H be a semisimple weak Hopf algebra, A a left H-module algebra. We prove that the injective dimension of the regular A-module is equal to the one of A as the module over the smash product A#H,nd equal to the one o...Let H be a semisimple weak Hopf algebra, A a left H-module algebra. We prove that the injective dimension of the regular A-module is equal to the one of A as the module over the smash product A#H,nd equal to the one of the regular A#H-module. Also, we give a necessary and sufficient condition for A being a Gorenstein algebra, in terms of the fixed subalgebra of A under the action of H on A.展开更多
The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussio...The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi- R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.展开更多
This paper gives a Maschke-type theorem over semisimple weak Hopf algebras,extends the well-known Maschke-type theorem given by Cohen and Fishman and constructs a Morita context over weak Hopf algebras.
The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak righ...The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak right H-comodule algebra and B the H-coinvariant subalgebra of A.First,some properties of Gorenstein projective H-modules in the representation category are studied,and the fact that Gorenstein global dimension of H is the same as the Gorenstein projective dimension of its left unital subalgebra is demonstrated.Secondly,by applying the integral theory of weak Hopf algebras,on the one hand,a sufficient and necessary condition that a projective A-module is a projective B-module is given;on the other hand,the separability of the functor AB-and that of the restriction of scalar function B(-)are described,respectively.Finally,as a mean result,the Gorenstein global dimension of a weak Hopf-Galois extension is investigated under the condition that H is both semisimple and cosemisimple.展开更多
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 to be weak Hopf algebras, moreover, gives a description for smash products to be weak Hopf algebras. It respectively generalizes R.K.Molnar's major result and I.Boca's result.
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).
文摘The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the （braided） monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl （2000）.
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 prebraided 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 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 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 from 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, x, a, b） through factoring At by a semilattice graded weak Hopf ideal.
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.
文摘Weak Hopf algebra is defined. The aim of this work is to study the relations between the weak antipode of a weak Hopf algelra and the monoid of group like elements. At first, the author shows some properties of weak Hopf algebras. And, for some weak Hopf algebras, some properties of the weak antipodes are given if the monoids of group like elements are inverse or orthodox semigroups. At end, the author gets a weak Hopf algebra whose monoid of group like elements is regular.
基金supported by the project of Ministerio de Ciencia e Innovación(No.MTM2010-15634)Fondo Europeo de Desarrollo Regional
文摘The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H. For a right H-comodule algebra, they obtain a bijective corre- spondence between the isomorphisms classes of H-cleft extensions AH → A, where AH is the subalgebra of coinvariants, and the equivalence classes of crossed systems for H over AH. Finally, they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group H2φZ（AH） （H, Z（AH））, where Z（AH） is the center of AH.
基金Supported in part by the Scientific Research Foundation of Zhejiang Provincial Education Department under grant number 20040322It is also sponsored by SRF for ROCS,SEM
文摘Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.
基金Ministerio de Educacidn y Ciencia Projects MTM2006-14908-C02-01,MTM2007-62427FEDER
文摘In this paper, we give a necessary and sufficient condition for a comodule algebra over a weak Hopf algebra to have a total integral, thus extending the classical theory developed by Doi in the Hopf algebra setting. Also, from these results, we deduce a version of Maschke＇s Theorem for （H, B）-Hopf modules associated with a weak Hopf algebra H and a right H-comodule algebra B.
基金the Program for New Century Excellent Talents in University (No 04-0522)the National Natural Science Foundation of China (No.10571153)
文摘In this paper, we define the notion of self-dual graded weak Hopf algebra and self-dual semilattice graded weak Hopf algebra. We give characterization of finite-dimensional such algebras when they are in structually simple forms in the sense of E. L. Green and E. N. Morcos. We also give the definition of self-dual weak Hopf quiver and apply these types of quivers to classify the finite- dimensional self-dual semilattice graded weak Hopf algebras. Finally, we prove partially the conjecture given by N. Andruskiewitsch and H.-J. Schneider in the case of finite-dimensional pointed semilattice graded weak Hopf algebra H when grH is self-dual.
文摘We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying jm = j for some integer m. We denote this algebra by wUqT（A）. This algebra is a weak Hopf algebra if and only if m = 2,3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra Uq （A） of a generalized Kac-Moody algebra A.
基金Acknowledgement I should like to thank Professor Zhang Pu＇s remarks and many helpful suggestions which improve the writing in English. the National Natural Science Foundation of China （No. 10301033 10271113）.
文摘Let H be a semisimple weak Hopf algebra, A a left H-module algebra. We prove that the injective dimension of the regular A-module is equal to the one of A as the module over the smash product A#H,nd equal to the one of the regular A#H-module. Also, we give a necessary and sufficient condition for A being a Gorenstein algebra, in terms of the fixed subalgebra of A under the action of H on A.
基金Project supported by the National Natural Science Foundation of China(No.19971074)by the Natural Science Foundation of Zhejiang Province(No.102028)
文摘The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi- R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.
基金supported by the National Natural Science Foundation of China(Grant No.10571153)the Postdoctoral Science Foundation of China(Grant No.2005037713)the Postdoctoral Science Foundation of Jiangsu(Grant No.0203003403).
文摘This paper gives a Maschke-type theorem over semisimple weak Hopf algebras,extends the well-known Maschke-type theorem given by Cohen and Fishman and constructs a Morita context over weak Hopf algebras.
基金The National Natural Science Foundation of China(No.11601203)the China Postdoctoral Science Foundation(No.2018M642128)Qing Lan Project of Jiangsu Province,the Natural Science Foundation of Jiangsu Province(No.BK20150113).
文摘The representation of weak Hopf algebras is studied by investigating the Gorenstein dimensions of weak Hopf algebras and weak Hopf-Galois extensions.Let H be a weak Hopf algebra with a bijective antipode,A a weak right H-comodule algebra and B the H-coinvariant subalgebra of A.First,some properties of Gorenstein projective H-modules in the representation category are studied,and the fact that Gorenstein global dimension of H is the same as the Gorenstein projective dimension of its left unital subalgebra is demonstrated.Secondly,by applying the integral theory of weak Hopf algebras,on the one hand,a sufficient and necessary condition that a projective A-module is a projective B-module is given;on the other hand,the separability of the functor AB-and that of the restriction of scalar function B(-)are described,respectively.Finally,as a mean result,the Gorenstein global dimension of a weak Hopf-Galois extension is investigated under the condition that H is both semisimple and cosemisimple.
