Abstract
Let (H, β) be a Hom-bialgebra such that β 2 = id H . (A, α A ) is a Hom-bialgebra in the left-left Hom-Yetter-Drinfeld category HH YD and (B, α B ) is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category YD HH . The authors define the two-sided smash product Hom-algebra \(\left( {A\natural H\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\) and the two-sided smash coproduct Homcoalgebra \(\left( {A\diamondsuit H\diamondsuit B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\). Then the necessary and sufficient conditions for \(\left( {A\natural H\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\) and \(\left( {A\diamondsuit H\diamondsuit B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)\) to be a Hom-bialgebra (called the double biproduct Hom-bialgebra and denoted by \(\left( {A_\diamondsuit ^\natural H_\diamondsuit ^\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B})} \right)\) are derived. On the other hand, the necessary and sufficient conditions for the smash coproduct Hom-Hopf algebra \(\left( {A\diamondsuit H,{\alpha _A} \otimes \beta } \right)\) to be quasitriangular are given.
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This work was supported by the Henan Provincial Natural Science Foundation of China (No. 17A110007) and the Foundation for Young Key Teacher by Henan Province (No. 2015GGJS-088).
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Ma, T., Li, H. & Liu, L. Double biproduct Hom-bialgebra and related quasitriangular structures. Chin. Ann. Math. Ser. B 37, 929–950 (2016). https://doi.org/10.1007/s11401-016-1001-5
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DOI: https://doi.org/10.1007/s11401-016-1001-5