Symmetric Categories

Christopher Ryba

August 21, 2017

Christopher Ryba Symmetric Tensor Categories 1 of 1 Monoidal Categories

Definition A monoidal category is a category C with a bifunctor ⊗ : C × C → C, together with the following: (Unit object) A distinguished object 1. (Unitor) For objects X of C, natural isomorphisms X ⊗ 1 → X and 1 ⊗ X → X. (Associator) For objects X,Y,Z of C, a natural isomorphism αX,Y,Z :(X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z). Coherency conditions for the unitor and associator.

Christopher Ryba Symmetric Tensor Categories 2 of 1 Braided and Symmetric Structure

Definition A monoidal category C is braided if it has the following data: (Braiding) For objects X,Y of C, a natural isomorphism CX,Y : X ⊗ Y → Y ⊗ X. Coherence and compatibility conditions with the unitor and associator.

The conditions imply that if si is CX,X applied to the i-th and ⊗n (i + 1)-th factors of X , then the si satisfy the braid relations, hence define an action of the braid group Bn.

If CX,Y ◦ CY,X = IdY ⊗X , we say C is a symmetric monoidal category. In this case, the braid group action factors through the Sn, and (with extra structure) allows the construction of symmetric and exterior powers of X.

Christopher Ryba Symmetric Tensor Categories 3 of 1 Rigid Structure

Definition A symmetric monoidal category C is rigid if every object X has a dual object X∗ such that: ∗ (Evaluation) There is a distinguished map evX : X ⊗ X → 1. (Coevaluation) There is a distinguished map ∗ coevX : 1 → X ⊗ X . (Triangle Identities) The following compositions are the identity maps (unitor and associator maps omitted):

coev ⊗Id Id ⊗ev X −−−−−−−−→X X X ⊗ X∗ ⊗ X −−−−−−→X X X

Id ∗ ⊗coev ev ⊗Id ∗ X∗ −−−−−−−−−→X X X∗ ⊗ X ⊗ X∗ −−−−−−−→X X X∗

Christopher Ryba Symmetric Tensor Categories 4 of 1 Main Example: Finite-Dimensional Vector Spaces

Let vectk be the category of finite-dimensional vector spaces over the field k. The usual makes it into a monoidal category with unit object k (1-dimensional ).

It has a symmetric braiding via CU,V (u ⊗ v) = v ⊗ u. Evaluation is literal evaluation; if f ∈ V ∗ and v ∈ V , then f ⊗ v 7→ f(v).

∗ If V has vi, and dual basis vi , then coevX takes 1 ∈ k = 1 P ∗ P to i vi ⊗ vi . If we write j λjvj for an element of V and P ∗ ∗ j µjvj for an element of V , the triangle identities reduce to: ! X X ∗ X X λjvj 7→ vi ⊗ vi ⊗ λjvj 7→ λivi j i j i ! X ∗ X ∗ X ∗ X ∗ µjvj 7→ µjvj ⊗ vi ⊗ vi 7→ µivi j j i i

Christopher Ryba Symmetric Tensor Categories 5 of 1 Symmetric Tensor Categories

Definition A symmetric tensor category is a rigid symmetric monoidal category C with the following properties: It is abelian. It is k-linear and ⊗ is bilinear on morphism spaces. It is locally finite.

EndC(1) = k.

Finite dimensional vector spaces over k are the main example. This generalises to finite dimensional modules over cocommutative Hopf (e.g. kG − mod for a finite group G).

We aim to present the Verlinde category Verp as an example that cannot be realised as vector spaces.

Christopher Ryba Symmetric Tensor Categories 6 of 1 Trace of Endomorphisms

Henceforth let C be a symmetric tensor category, and X an object of C. Definition

Given f ∈ EndC(X), we may define the trace of f, an element of k = EndC(1):

trX (f) = evX ◦ CX,X∗ ◦ (f ⊗ IdX∗ ) ◦ coevX

We define the of X to be dimC(X) = trX (IdX ).

Figure: Diagrammatic Description of Trace

Christopher Ryba Symmetric Tensor Categories 7 of 1 Negligible Morphisms

Definition A morphism f : X → Y is said to be negligible if for any g : Y → X, trY (f ◦ g) = 0. Let N (X,Y ) be the space of such morphisms.

Theorem Negligible morphisms form a tensor ideal. This means one may define a category whose objects are the same as those of C, and morphisms are homC(X,Y )/N (X,Y ), and the tensor structure descends to this category. We call this the semisimplification of C, and denote it C. The obvious functor ¯: C → C is a tensor functor.

Christopher Ryba Symmetric Tensor Categories 8 of 1 Semisimplification

Theorem The semisimplification C is semisimple. The simple objects are X for X an indecomposable object of C of nonzero dimension.

Note that if X has dimension zero, then IdX is negligible. Then any composition of a morphism with IdX is negligible, forcing ∼ EndC(X) = 0, making X = 0.

Christopher Ryba Symmetric Tensor Categories 9 of 1 Cyclic Group Representations

Let k have characteristic p > 0. Consider kCp − mod, where Cp is the cyclic group with p elements. Note that:

∼ p ∼ p kCp = k[x]/(x − 1) = k[x]/(x − 1)

The indecomposable objects of C are Li where i = 1, 2, ··· , p, where Li is an i-dimensional vector space on which x − 1 acts by a single nilpotent Jordan block.

We have the following tensor product rules:

L2 ⊗ L1 = L2

L2 ⊗ Li = Li−1 ⊕ Li+1 (for 1 < i < p)

L2 ⊗ Lp = Lp ⊕ Lp

Christopher Ryba Symmetric Tensor Categories 10 of 1 The Verlinde Category

Definition

The Verlinde Category Verp is the semsimplification of kCp − mod.

It has p − 1 simple objects, Li for i = 1, 2, ··· , p − 1 (since dim (L ) = 0, L = 0). kCp−mod p p

We have the following tensor product rules (assuming p ≥ 3):

L2 ⊗ L1 = L2

L2 ⊗ Li = Li−1 ⊕ Li+1 (for 1 < i < p − 1)

L2 ⊗ Lp−1 = Lp−2

Christopher Ryba Symmetric Tensor Categories 11 of 1 The Verlinde Category (Cont.)

Theorem

There category Verp is not a subcategory of vectk if p ≥ 5.

If not, one can choose an embedding and associate to each Li its dimension as a vector space (a natural number). Let v be the vector whose i-th entry is dimk(Li). The previous tensor product rules (which assumed p ≥ 3) give the equation:

Mv = dimk(L2)v Here M is the whose nonzero entries are 1 on the superdiagonal and subdiagonal. This is an eigenvalue equation. One can show that the eigenvalues of M are 2 cos(πr/p), where r = 1, 2, ··· , p − 1. This means dimk(L2) can only be a nonnegative integer if r = 1, p = 3 (not if p ≥ 5). ∼ ∼ Note that Ver2 = vectk and Ver3 = svectk. Christopher Ryba Symmetric Tensor Categories 12 of 1