Let （C,α） and （H, β） be Hom-bialgebras and ω ： C × H → H × C a linear map. We introduce a Horn-ω-smash coproduct （Cω H, γ） and give necessary and sufficient conditions for （Cω H, γ） to be...Let （C,α） and （H, β） be Hom-bialgebras and ω ： C × H → H × C a linear map. We introduce a Horn-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
Let （H, a） be a monoidal Hom-bialgebra and （B,p） be a left （H, a）-Hom-comodule coalgebra. The new monoidal Hom-algebra B#y H is constructed with a Hom-twisted product Ba[H] and a. B × H Hom-smash coproduc...Let （H, a） be a monoidal Hom-bialgebra and （B,p） be a left （H, a）-Hom-comodule coalgebra. The new monoidal Hom-algebra B#y H is constructed with a Hom-twisted product Ba[H] and a. B × H Hom-smash coproduct. Moreover, a sufficient and necessary condition for B#y / to be a monoidal Hom-bialgebra is given. In addition, let （H, a） be a Hom-σ- Hopf algebra with Hom-〇 --antipode SH, and a sufficient condition for this new monoidal Hom-bialgebra B#y H with the antipode S defined by S（b×h）=（1B×SH（a^-1）b（-1）））（SB（b（0））×1H to be a monoidal Hom-Hopf algebra is derived.展开更多
Let（C, α） and（H, β） be Hom-bialgebras and ω ： C H → H C a linear map.We introduce the concept of a Hom-ω-crossed coproduct（Cω σ H, γ） and we give necessary and sufficient conditions for the new object ...Let（C, α） and（H, β） be Hom-bialgebras and ω ： C H → H C a linear map.We introduce the concept of a Hom-ω-crossed coproduct（Cω σ H, γ） and we give necessary and sufficient conditions for the new object to be a Hom-Hopf algebra.展开更多
We show that two module homomorphisms for groups and Lie algebras established by Xi(2012)can be generalized to the setting of quasi-triangular Hopf algebras.These module homomorphisms played a key role in his proof of...We show that two module homomorphisms for groups and Lie algebras established by Xi(2012)can be generalized to the setting of quasi-triangular Hopf algebras.These module homomorphisms played a key role in his proof of a conjecture of Yau(1998).They will also be useful in the problem of decomposition of tensor products of modules.Additionally,we give another generalization of result of Xi(2012)in terms of Chevalley-Eilenberg complex.展开更多
Let （H, α） be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules H^H HYD is isomorphic t...Let （H, α） be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules H^H HYD is isomorphic to the center of the category of left （H, α）-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.展开更多
Let m,n be two positive integers,k be an algebraically closed field with char(k)■mn.Radford constructed an mn^(2)-dimensional Hopf algebra R_(mn)(q)such that its Jacobson radical is not a Hopf ideal.We show that the ...Let m,n be two positive integers,k be an algebraically closed field with char(k)■mn.Radford constructed an mn^(2)-dimensional Hopf algebra R_(mn)(q)such that its Jacobson radical is not a Hopf ideal.We show that the Drinfeld double D(R_(mn)(q))of Radford Hopf algebra R_(mn)(q)has ribbon elements if and only if n is odd.Moreover,if m is even and n is odd,then D(R_(mn)(q))has two ribbon elements,if both m and n are odd,then D(R_(mn)(q))has only one ribbon element.Moreover,we compute explicitly all ribbon elements of D(R_(mn)(q)).展开更多
In this paper,we study the category of corepresentations of a monoidal comonad.We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle(coseparable)comonad,and it is a braided cat...In this paper,we study the category of corepresentations of a monoidal comonad.We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle(coseparable)comonad,and it is a braided category if and only if the monoidal comonad admit a cobraided structure.At last,as an application,the braided structure and the semisimplicity of the Hom-comodule category of a monoidal Hom-bialgebra are discussed.展开更多
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 Horn-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The National Natural Science Foundation of China(No.11371088,10871042,11571173)the Fundamental Research Funds for the Central Universities(No.KYLX15_0105)
文摘Let （H, a） be a monoidal Hom-bialgebra and （B,p） be a left （H, a）-Hom-comodule coalgebra. The new monoidal Hom-algebra B#y H is constructed with a Hom-twisted product Ba[H] and a. B × H Hom-smash coproduct. Moreover, a sufficient and necessary condition for B#y / to be a monoidal Hom-bialgebra is given. In addition, let （H, a） be a Hom-σ- Hopf algebra with Hom-〇 --antipode SH, and a sufficient condition for this new monoidal Hom-bialgebra B#y H with the antipode S defined by S（b×h）=（1B×SH（a^-1）b（-1）））（SB（b（0））×1H to be a monoidal Hom-Hopf algebra is derived.
文摘Let（C, α） and（H, β） be Hom-bialgebras and ω ： C H → H C a linear map.We introduce the concept of a Hom-ω-crossed coproduct（Cω σ H, γ） and we give necessary and sufficient conditions for the new object to be a Hom-Hopf algebra.
基金supported by National Natural Science Foundation of China (Grant No. 11501546)
文摘We show that two module homomorphisms for groups and Lie algebras established by Xi(2012)can be generalized to the setting of quasi-triangular Hopf algebras.These module homomorphisms played a key role in his proof of a conjecture of Yau(1998).They will also be useful in the problem of decomposition of tensor products of modules.Additionally,we give another generalization of result of Xi(2012)in terms of Chevalley-Eilenberg complex.
基金Acknowledgements The authors sincerely thank the referees for their valuable suggestions and comments on this paper. This work was supported by the National Natural Science Foundation of China （Grant Nos. 11601486, 61272007. 11401534）.
文摘Let （H, α） be a monoidal Hom-Hopf algebra. In this paper, we will study the category of Hom-Yetter-Drinfeld modules. First, we show that the category of left-left Hom-Yetter-Drinfeld modules H^H HYD is isomorphic to the center of the category of left （H, α）-Hom-modules. Also, by the center construction, we get that the categories of left-left, left-right, right-left, and right-right Hom-Yetter-Drinfeld modules are isomorphic as braided monoidal categories. Second, we prove that the category of finitely generated projective left-left Hom-Yetter-Drinfeld modules has left and right duality.
基金Supported by NNSF of China(Grant Nos.12371041,12201545)Natural Science Foundation of the Jiangsu Higher Education Institutions of China(Grant No.22KJD110006)。
文摘Let m,n be two positive integers,k be an algebraically closed field with char(k)■mn.Radford constructed an mn^(2)-dimensional Hopf algebra R_(mn)(q)such that its Jacobson radical is not a Hopf ideal.We show that the Drinfeld double D(R_(mn)(q))of Radford Hopf algebra R_(mn)(q)has ribbon elements if and only if n is odd.Moreover,if m is even and n is odd,then D(R_(mn)(q))has two ribbon elements,if both m and n are odd,then D(R_(mn)(q))has only one ribbon element.Moreover,we compute explicitly all ribbon elements of D(R_(mn)(q)).
基金the National Natural Science Foundation of China(Nos.11626138,11626139)the Natural Science Foundation of Shandong Province(No.ZR2016AQ03).
文摘In this paper,we study the category of corepresentations of a monoidal comonad.We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle(coseparable)comonad,and it is a braided category if and only if the monoidal comonad admit a cobraided structure.At last,as an application,the braided structure and the semisimplicity of the Hom-comodule category of a monoidal Hom-bialgebra are discussed.
