I'm ultimately trying to construct a Hopf algebra based on a finite non-abelian group $G$ such that its irreducible representations:
- are labeled by $g\in G$ and $\rho\in\text{irr}(G)$
- have the fusion rules $$(g,\rho_i)(h,\rho_j)=\sum_kN_{i,j}^k(gh,\rho_k),$$ where the coefficients $N_{i,j}^k$ are the fusion rules of the irreducible representations of $G$.
I believe such a Hopf algebra exists because of the statement that non-anomalous fusion categories are equivalent to representation categories of finite semisimple Hopf algebras.
The quantum double $\mathcal{D}(G)$ has irreducible representations labeled by conjugacy classes of elements $\bar a$, $a\in G$ and irreps of the centralizer $Z(a)$ of $a\in G$. This is similar to my first requirement, but with less data necessary to specify the irreducible representations. For this reason I suspect that the Hopf algebra I'm interested in is defined over $\mathbb{C}G\otimes\mathbb{C}^G$ as a vector space, as is the quantum double, but with a different multiplication.
I think it would be instructive to understand how to construct irreducible representations of the Hopf algebra of functions on a finite group $\mathbb{C}^G$, towards the goal of constructing a Hopf algebra satisfying 1 and 2. In general, I've found very few examples of explicit constructions of irreps of Hopf algebras, but one helpful resource has been section 2.2 of this paper. I would appreciate any help or resources related to this smaller problem, or the larger problem above.