Fusion category

Remarks

Example: for a finite group $G$, the category $\mathrm{Rep}(G)$ of finite-dimensional complex representations of $G$ is a fusion category over $\C$. It is semisimple by Maschke's theorem, has finitely many irreducible representations up to isomorphism, the dual of a representation gives rigidity, and the trivial representation is simple. Non-example: the category of finite-dimensional complex representations of the group $\Z$ satisfies clauses 1, 2, 5 and 6. It is not semisimple, and it has infinitely many simple objects, one for each $\lambda \in \C^\times$. Non-example: for a field $k$ of characteristic $p$ and a group $G$ of order divisible by $p$, $\mathrm{Rep}_k(G)$ fails clause 3 by the failure of Maschke's theorem. If $X_1, \ldots, X_n$ are the simple objects, the fusion rules are the integers $N_{ij}^m$ with $X_i \otimes X_j \cong \bigoplus_m N_{ij}^m X_m$.