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.

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.

Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor (2005). Tensor Categories. ISSN 0885-4653.