graded fusion category



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In higher category theory




Let 𝒞\mathcal{C} be a semisimple linear(Vect-enriched) (over a field kk) category and GG a finite group. A GG-grading on 𝒞\mathcal{C} is a direct sum decomposition of 𝒞\mathcal{C} into homogeneous components 𝒞 g\mathcal{C}_g, which are again kk-linear semisimple categories:

𝒞 gG𝒞 g\mathcal{C} \simeq \bigoplus_{g \in G} \mathcal{C}_g

A graded fusion category is a fusion category with a chosen GG-grading for some group, such that the monoidal product maps 𝒞 g×𝒞 h\mathcal{C}_g \times \mathcal{C}_h into 𝒞 gh\mathcal{C}_{g h}.

Alternative Definition

Let Λ 𝒞\Lambda_\mathcal{C} be the set of equivalence classes of simple objects in 𝒞\mathcal{C}. A GG-grading is a map deg:Λ 𝒞G\operatorname{deg}\colon \Lambda_\mathcal{C} \to G such that for any two simples XX and YY and any simple subobject ZXYZ \hookrightarrow X \otimes Y, we have deg([Z])=deg([X])deg([Y])\operatorname{deg}([Z]) = \operatorname{deg}([X]) \operatorname{deg}([Y]).


  • The trivial component 𝒞 1\mathcal{C}_1 of the grading is a full fusion subcategory.
  • A GG-extension of a fusion category 𝒟\mathcal{D} is a GG-graded fusion category 𝒞\mathcal{C} such that 𝒟𝒞 1\mathcal{D} \simeq \mathcal{C}_1.
  • A grading is faithful if 𝒞 g0\mathcal{C}_g \neq 0 for all gGg \in G. This is a standard assumption.
  • The adjoint category 𝒞 ad\mathcal{C}_{\text{ad}} is the full fusion subcategory of 𝒞\mathcal{C} spanned by objects of the form XX *X \otimes X^*.


Universal grading

From the definition, it is clear that gradings are covariant in the group. A group homomorphism ϕ:GH\phi\colon G \to H gives an obvious map of the set of gradings:

𝒞 hH gϕ 1(h)𝒞 g \mathcal{C} \simeq \bigoplus_{h \in H} \bigoplus_{g \in \phi^{-1}(h)} \mathcal{C}_g


deg H=ϕdeg G \operatorname{deg}_H = \phi \circ \operatorname{deg}_G

Morphisms of gradings are therefore simply group homomorphisms.

For every fusion category 𝒞\mathcal{C}, there exists a universal grading by a group U 𝒞U_\mathcal{C}. It has the following properties:

  • It is faithful.
  • The trivial component is the full fusion subcategory spanned by objects of the form XX *X \otimes X^*.
  • Every full fusion subcategory 𝒟𝒞\mathcal{D} \subset \mathcal{C} containing the adjoint category 𝒞 ad\mathcal{C}_{\text{ad}} is of the form 𝒟 hH𝒞 h\mathcal{D} \simeq \bigoplus_{h \in H} \mathcal{C}_h for some subgroup HU 𝒞H \subset U_\mathcal{C}.
  • The group of monoidal automorphisms of the identity functor is canonically isomorphic to Hom(U 𝒞,k ×)\operatorname{Hom}(U_\mathcal{C}, k^\times).


Last revised on August 29, 2017 at 09:21:59. See the history of this page for a list of all contributions to it.