We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it...We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧C C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coal-gebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.展开更多
Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the st...Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded A-Hopf algebra is invertible.展开更多
A duality theorem for Hopf crossed coproduct is proved. This theorem plays a role similar to that appearing in the work of Koppinen (which generalized the corresponding results of group grraded ring).
In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple grap...In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs. In this paper, we generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices. Then we prove that the vector space spanned by (0,1)-matrices with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.展开更多
Some basic properties of the antipode of an N0I-graded x-Hopf algebra are studied. Also, several equivalent conditions of an N0I-graded x-bialgebra (x-Hopf algebra) are given.
Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C（H^＊）^＋ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple...Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C（H^＊）^＋ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple coextension and C is an injective right C-comodule, then gl. dim（the smash coproduct of C and H） = gl. dim（C） = gl. dim（C）, where gl. dim（C） denotes the global dimension of coalgebra C.展开更多
We study cotilting comodules and f-cotilting comodules and give a description of localization of f-cotilting comodules and classical tilting comodules. First, we introduce T-cotilting injective comodules and their dim...We study cotilting comodules and f-cotilting comodules and give a description of localization of f-cotilting comodules and classical tilting comodules. First, we introduce T-cotilting injective comodules and their dimensions which are important for researching cotilting comodules. Then we characterize the localization in f-cotilting comodules, finitely copresented comodules, and classical tilting comodules. In particular, we obtain a localizing property about finitely copresented comodules.展开更多
In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, a...In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n)are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.展开更多
R. J. Blattner and S. Montgomery have proved the duality theorem of Hopf module algebras in Ref. [1]. This theorem contains duality for crossed product of von Neumann algebras. In 1977, R. K. Molnar introduced the con...R. J. Blattner and S. Montgomery have proved the duality theorem of Hopf module algebras in Ref. [1]. This theorem contains duality for crossed product of von Neumann algebras. In 1977, R. K. Molnar introduced the concept of Hopf comodule coalgebras which is a dual notation of Hopf module algebra, and discussed their properties. However, the duality theorem of Hopf comodule coalgebras has not been proved yet. In this note we shall deal with this situation by defining the展开更多
In this paper, we introduce the notion of （＊）-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of （＊）-serial coalgebras and their comodules, and obtain sufficient a...In this paper, we introduce the notion of （＊）-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of （＊）-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be （＊）-serial.展开更多
The embedding theorem ofZ-graded Lie superalgebras is given and proved. As a subsidiary result it is proved that a transitiveZ-graded restricted lie superalgebm $G = \mathop \oplus \limits_{i \geqslant - 1} G_i $ must...The embedding theorem ofZ-graded Lie superalgebras is given and proved. As a subsidiary result it is proved that a transitiveZ-graded restricted lie superalgebm $G = \mathop \oplus \limits_{i \geqslant - 1} G_i $ must be isomorphic toW(m,n, 1) if the dimension ofG i satisfies a certain condition.展开更多
Let H be a Hopf algebra,H<sub>1</sub> be a sub-Hopf algebra of H,H<sub>2</sub> be the quotient Hopt algebra of H modular H<sub>1</sub>.This paper gives a simplified complex by defin...Let H be a Hopf algebra,H<sub>1</sub> be a sub-Hopf algebra of H,H<sub>2</sub> be the quotient Hopt algebra of H modular H<sub>1</sub>.This paper gives a simplified complex by defining a new base for the cobar complex and proves that the cobar complex of H has the same cohomology algebra with a twisted product of the cobar complexes of H<sub>1</sub> and H<sub>2</sub>.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.10271113&10301033)the Doctoral Foundation of the Chinese Education Ministry.
文摘We introduce the quiver of a bicomodule over a cosemisimple coalgebra. Applying this to the coradical C0 of an arbitrary coalgebra C, we give an alternative definition of the Gabriel quiver of C, and then show that it coincides with the known Ext quiver of C and the link quiver of C. The dual Gabriel theorem for a coalgebra with a separable coradical is obtained, which generalizes the corresponding result for a pointed coalgebra. We also give a new description of C1 = C0 ∧C C0 of any coalgebra C, which can be regarded as a generalization of the first part of the well-known Taft-Wilson Theorem for pointed coal-gebras. As applications, we give a characterization of locally finite coalgebras via their Gabriel quivers, and a property of the Gabriel quiver of a quasi-coFrobenius coalgebra.
基金Supported by the National Natural Science Foundation of ChinaYangzhou University Natural Science Foundation
文摘Let G be an abelian group, B the G-graded λ-Hopf algebra with A being a bicharacter on G. By introducing some new twisted algebras （coalgebras）, we investigate the basic properties of the graded antipode and the structure for B. We also prove that a G-graded λ-Hopf algebra can be embedded in a usual Hopf algebra. As an application, it is given that if G is a finite abelian group then the graded antipode of a finite dimensional G-graded A-Hopf algebra is invertible.
文摘A duality theorem for Hopf crossed coproduct is proved. This theorem plays a role similar to that appearing in the work of Koppinen (which generalized the corresponding results of group grraded ring).
文摘In 2014, Vargas first defined a super-shuffle product and a cut-box coproduct on permutations. In 2020, Aval, Bergeron and Machacek introduced the super-shuffle product and the cut-box coproduct on labeled simple graphs. In this paper, we generalize the super-shuffle product and the cut-box coproduct from labeled simple graphs to (0,1)-matrices. Then we prove that the vector space spanned by (0,1)-matrices with the super-shuffle product is a graded algebra and with the cut-box coproduct is a graded coalgebra.
基金This paper is supported by the Chinese National Natural Science Foundation
文摘Some basic properties of the antipode of an N0I-graded x-Hopf algebra are studied. Also, several equivalent conditions of an N0I-graded x-bialgebra (x-Hopf algebra) are given.
基金Supported by the Foundation of Key Research Program (No. 02021029)the NSF (No. 2004kj352) of Anhui Province, China
文摘Let H be a finite dimensional cosemisimple Hopf algebra, C a left H-comodule coalgebra and let C = C/C（H^＊）^＋ be the quotient coalgebra and the smash coproduct of C and H. It is shown that if C/C is a eosemisimple coextension and C is an injective right C-comodule, then gl. dim（the smash coproduct of C and H） = gl. dim（C） = gl. dim（C）, where gl. dim（C） denotes the global dimension of coalgebra C.
基金Acknowledgements The authors would like to thank the referees for the careful reading and valuable suggestions. This work was supported by National Natural Science Foundation of China （Grant Nos. 11271119, 11201314） and the Natural Science Foundation of Beijing （Grant No. 1122002）.
文摘We study cotilting comodules and f-cotilting comodules and give a description of localization of f-cotilting comodules and classical tilting comodules. First, we introduce T-cotilting injective comodules and their dimensions which are important for researching cotilting comodules. Then we characterize the localization in f-cotilting comodules, finitely copresented comodules, and classical tilting comodules. In particular, we obtain a localizing property about finitely copresented comodules.
基金supported by the Program for New Century Excellent Talents in University(Grant No.04-0522)the second author was supported by the National Natural Science Foundation of China(Grant Nos.10271113&10501041)the Doctoral Foundation of the Chinese Education Ministry.
文摘In this paper, we study the structures of monomial Hopf algebras over a field of positive characteristic. A necessary and sufficient condition for the monomial coalgebra Cd(n) to admit Hopf structures is given here, and if it is the case, all graded Hopf structures on Cd(n)are completely classified. Moreover, we construct a Hopf algebras filtration on Cd(n) which will help us to discuss a conjecture posed by Andruskiewitsch and Schneider. Finally combined with a theorem by Montgomery, we give the structure theorem for all monomial Hopf algebras.
文摘R. J. Blattner and S. Montgomery have proved the duality theorem of Hopf module algebras in Ref. [1]. This theorem contains duality for crossed product of von Neumann algebras. In 1977, R. K. Molnar introduced the concept of Hopf comodule coalgebras which is a dual notation of Hopf module algebra, and discussed their properties. However, the duality theorem of Hopf comodule coalgebras has not been proved yet. In this note we shall deal with this situation by defining the
文摘In this paper, we introduce the notion of （＊）-serial coalgebras which is a generalization of serial coalgebras. We investigate the properties of （＊）-serial coalgebras and their comodules, and obtain sufficient and necessary conditions for a basic coalgebra to be （＊）-serial.
文摘The embedding theorem ofZ-graded Lie superalgebras is given and proved. As a subsidiary result it is proved that a transitiveZ-graded restricted lie superalgebm $G = \mathop \oplus \limits_{i \geqslant - 1} G_i $ must be isomorphic toW(m,n, 1) if the dimension ofG i satisfies a certain condition.
基金Supported by National Natural Science Foundation of China
文摘Let H be a Hopf algebra,H<sub>1</sub> be a sub-Hopf algebra of H,H<sub>2</sub> be the quotient Hopt algebra of H modular H<sub>1</sub>.This paper gives a simplified complex by defining a new base for the cobar complex and proves that the cobar complex of H has the same cohomology algebra with a twisted product of the cobar complexes of H<sub>1</sub> and H<sub>2</sub>.
