FUSION CATEGORIES AND MODULE CATEGORIES

EVAN JENKINS

k is an algebraically closed field of characteristic 0.

1. Finite semisimple categories Definition. A k-linear abelian category C is called semisimple if every object is a finite direct sum of simple objects.

Definition. A semisimple category is finite if it has finitely many iso- morphism classes of simple objects.

Observation: C ∼= Vect×n. Definition. Let C and D be categories. The tensor product of C and D is the category C  D such that Fun(C  D, E) consits of bilinear functors C × D → E.

2. Fusion categories Definition. Let C be a tensor/monoidal category. C is called a fusion category if (1) Every object has a left and right dual (2) 1 is simple.

Examples (G is a finite group) of fusion categories:

(1) VectG, the category of G-graded vector spaces (2) Rep G, representations of G. Let C be a fusion category.

Definition. The Drinfeld center of C is a braided fusion category Z1(C) defined as as follows: • Objects are pairs (X, c), where X ∈ C and c is the natural ∼ transformation cy : X ⊗ Y → Y ⊗ X, such that (1) c1 = id (2) The diagram 1 2 EVAN JENKINS

cy⊗z X ⊗ Y ⊗ Z Y ⊗ Z ⊗ X

cy ⊗ idz idy ⊗cz

Y ⊗ X ⊗ Z commutes. • Morphisms Hom((X, c), (X0, c0)) are morphisms between X and X0, which make the obvious diagram commute.

Example: C = VectG, the category of vector bundles on G.V = ⊕g∈GVg. Simple objects are kh for some h ∈ H. Consider ∼ ch : V ⊗ kh → kh ⊗ V. This is the same as ∼ ch : kh−1 ⊗ V ⊗ kh → V.

This means that we can identify Z1(C) with the category of Ad-equivariant vector bundles on G. Let C be a fusion category. Definition. A left C-module (left module category) is a ctegory M equipped with a functor . : C  M → M together with natural iso- morphisms (1) 1 . M →∼ M (2)( X ⊗ Y ) . M →∼ X. (Y. M). Examples: (1) C is always a left (and right) C-module. (2) If F : C → D is a tensor functor, then D becomes a left C- module by X.Y = F (X) ⊗ Y for any X ∈ C,Y ∈ D. Let M, N be C-modules. Definition. A morphism from M to N is a functor F : M → N equipped with a natural isomorphism F (X.M) →∼ X.F (M), satisfying some coherence axioms. Definition. Two modules are equivalent if there is a morphism M → N , which is an equivalence of categories. Definition. The direct sum of M and N is the category M ⊕ N with a natural defined composition. FUSION CATEGORIES AND MODULE CATEGORIES 3

Definition. A module is indecomposable if it is not equivalent to a direct sum of proper submodules.

3. Bimodules and Morita equivalence Let C, D be fusion categories. Definition. A(C, D)-bimodule is a category M with a left C- and right D-module structure together with a natural isomorphism (X.M)/Y →∼ X. (M /Y ), satisfying some coherence diagrams.

op This definition is equivalent to that of a C  D -module. Examples:

(1) CCC (2) For a functor C → D we get CDD.

Tensor product of CMD and DNE is defined in the following way. Consider the diagram

/,⊗,. / CMD  D  D  DNE ⇒ CMD  D  DNE ⇒ CMD  DNE . .

The bicolimit of this diagram is CM D NE . A morphism CM D NE → CPE is the same as a morphism

F : CM  NE → CPE together with ∼ F ((M/Y )  N) → F (M  (Y.N)).

Definition. A bimodule CMD is invertible if there exists DNC ushc that ∼ CM D NE = CCC and ∼ DN C MD = DDD. We also call M a Morita equivalence. ∼ Theorem 1. C and D are Morita equivalent iff Z1(C) = Z1(D). Exercise: ∼ (1) Show that Z1(C) = End(CCC). (2) Use this to prove the forward implication. Program: classify Morita equivalences between fusion categories. 4 EVAN JENKINS

4. Concordance

Consider an indecomposable CM. Definition. The dual of C with respect to M is the fusion category ∗ CM = End(CM). Theorem 2. (1) M has the structure of an invertible (C, C)-bimodule. (2) If M is an invertible bimodule, then D ∼ C∗ . C D = CM

Example: let C = VectG. We have the forgetful functor C → Vect. So, we get CM = C Vect . ∗ What is CM? Endofunctors of Vect are all given by tensoring with some vector space. What does a compatibility with the C-action give? For every g ∈ G we have ∼ φg : F (kg) ⊗ V → V. ∼ These give isomorphisms V → V . Moreover, we have φg ◦ φh = φhg. ∗ We conclude that V is a representation of G, i.e. CM = Rep G.