nLab
graded fusion category

Contents

Context

Monoidal categories

Fusion categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Definition

Definition

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
Definition

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]).

Terminology

  • 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^*.

Examples

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

or

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).

References

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