Double biproduct Hom-bialgebra and related quasitriangular structures

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 H H YD and (B, α B ) is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category YD H H . 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.