General Constructions
Here we list some general constructions of fusion rings/categories and their properties.
Group Rings of Finite Groups
The group ring of any finite group $G$ is a fusion ring. For such rings:
- all objects are invertible and have quantum dimension equal to $1$,
- the rank is the order of the group.
Every group ring is categorifiable to a fusion category, whose $F$-symbols correspond to $3$-cocycles. $G$ is Abelian iff all its categorifications admit a braiding.
$\text{Rep}(G)$
The irreducible representations of any finite group with the tensor product as ring product form a fusion ring.
Quantum Double Constructions
Fusion Rings Coming From Lie Theory
$\text{SU}(2)_k$
Properties:
- quantum dimensions: $2\cos{\frac{\pi}{k+2}}$