A twisted generalization of quasitriangular Hopf algebras called quasitriangular Hom-Hopf algebras is introduced. We characterize these algebras in terms of certain morphisms. We also give their equivalent description via a braided monoidal category $̃{(}_Hℳ)$. Finally, we study the twisting structure of quasitriangular Hom-Hopf algebras by conjugation with Hom-2-cocycles.

Let $\pi$ be a group, and $H$ be a semi-Hopf $\pi$-algebra. We first show that the category ${}_Hℳ$ of left $\pi$-modules over $H$ is a monoidal category with a suitably defined tensor product and each element $\alpha$ in $\pi$ induces a strict monoidal functor $F_{\alpha }$ from ${}_Hℳ$ to itself. Then we introduce the concept of quasitriangular semi-Hopf $\pi$-algebra, and show that a semi-Hopf $\pi$-algebra $H$ is quasitriangular if and only if the category ${}_Hℳ$ is a braided monoidal category and $F_{\alpha }$ is a strict braided monoidal functor for any $\alpha \in \pi$. Finally,...