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 2 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 = ⊕g2GVg. Simple objects are kh for some h 2 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 2 C;Y 2 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 2 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..