The Grothendieck ring of quantum double of quaternion group

Let \(\mathbb{k}\) $$\mathbb{k}$$ be an algebraically closed field of characteristic p ≠ 2, and let Q₈ be the quaternion group. We describe the structures of all simple modules over the quantum double \(D(\mathbb{k}Q_{8})\) $$D(\mathbb{k}{Q}_8)$$ of group algebra \(\mathbb{k}Q_{8}\) $$\mathbb{k}{Q}_8$$ . Moreover, we investigate the tensor product decomposition rules of all simple \(D(\mathbb{k}Q_{8})\) $$D(\mathbb{k}{Q}_8)$$ -modules. Finally, we describe the Grothendieck ring \(G_{0}(D(\mathbb{k}Q_{8}))\) $$G_0(D(\mathbb{k}{Q}_8))$$ by generators with relations.