In mathematics, a fusion category is a category that is abelian, formula_1-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 formula_1 is algebraically closed, then the latter is equivalent to formula_3 by Schur's lemma.
Examples.
The Representation Category of a finite group formula_4 of cardinality formula_5 over a field formula_6 is a fusion category if and only if formula_5 and the characteristic of formula_6 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.
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