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Keywords:
Hopf $\pi $-algebra; $H$-$\pi $-modules; braided monoidal category; braided monoidal functor
Hopf $\pi $-algebra; $H$-$\pi $-modules; braided monoidal category; braided monoidal functor
Summary:
Let $\pi $ be a group, and $H$ be a semi-Hopf $\pi $-algebra. We first show that the category $_H{\mathcal M}$ 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 $_H{\mathcal M}$ 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 $_H\mathcal M$ is a braided monoidal category and $F_{\alpha }$ is a strict braided monoidal functor for any $\alpha \in \pi $. Finally, we give two examples of Hopf $\pi $-algebras and describe the categories of modules over them.
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