Fusion category

In mathematics, a fusion category is a category that is abelian, $k$-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 $k$ is algebraically closed, then the latter is equivalent to $\mathrm{H}\mathrm{o}\mathrm{m}(1,1)\cong k$ by Schur's lemma.

Examples

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

Reconstruction

• 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).

References

• Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor (2005). "On Fusion Categories". Annals of Mathematics 162 (2): 581–642. doi:10.4007/annals.2005.162.581. ISSN 0003-486X. https://annals.math.princeton.edu/2005/162-2/p01. 

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor (2005). Tensor Categories.