Fusion category

In mathematics, a fusion category is a category that is abelian, ${\displaystyle k}$-linear, semisimple, monoidal, and rigid, and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If the ground field ${\displaystyle k}$ is algebraically closed, then the latter is equivalent to ${\displaystyle \mathrm {Hom} (1,1)\cong k}$ by Schur's lemma.

The Representation Category of a finite group ${\displaystyle G}$ of cardinality ${\displaystyle n}$ over a field ${\displaystyle \mathbb {K} }$ is a fusion category if and only if ${\displaystyle n}$ and the characteristic of ${\displaystyle \mathbb {K} }$ are coprime. This is because of the condition of semisimplicity which needs to be checked by the Maschke's theorem.

• Under Tannaka–Krein duality, every fusion category arises as the representations of a weak Hopf algebra.
• Every fusion category admits a skeletonization, and so a fusion category can be specified simply by specifying the fusion rules of the underlying fusion ring (note that due to Ocneanu Rigidity, this is not a unique specification in general).