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Keywords:
Grothendieck ring; simple module; quantum double; quaternion group
Grothendieck ring; simple module; quantum double; quaternion group
Summary:
Let $\Bbbk $ be an algebraically closed field of characteristic $p\neq 2$, and let $Q_8$ be the quaternion group. We describe the structures of all simple modules over the quantum double $D(\Bbbk Q_8)$ of group algebra $\Bbbk Q_8$. Moreover, we investigate the tensor product decomposition rules of all simple $D(\Bbbk Q_8)$-modules. Finally, we describe the Grothendieck ring $G_0(D(\Bbbk Q_8))$ by generators with relations.
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