Double Biproduct Hom-Bialgebra and Related Quasitriangular Structures

Let $(H, \b)$ be a Hom-bialgebra such that $\b^2={\rm id}_H$. $(A, \aa)$ is a Hom-bialgebra in the left-left Hom-Yetter-Drinfeld category $_H^H{\mathbb{YD}}$ and $(B, \ab)$ is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category ${\mathbb{YD}}_H^H$. The authors define the two-sided smash product Hom-algebra $(A\natural H\natural B, \aa\o \b\o \ab)$ and the two-sided smash coproduct Hom-coalgebra $(A\diamond H\diamond B, \aa\o \b\o \ab)$. Then the necessary and sufficient conditions for $(A\natural H\natural B, \aa\o \b\o \ab)$ and $(A\diamond H\diamond B, \aa\o \b\o \ab)$ to be a Hom-bialgebra (called the double biproduct Hom-bialgebra and denoted by $(A^{\natural}_{\diamond} H^{\natural}_{\diamond} B,\aa\o \b\o \ab)$) are derived. On the other hand, the necessary and sufficient conditions for the smash coproduct Hom-Hopf algebra $(A\diamond H,\aa\o \b)$ to be quasitriangular are given.