Tensor categories

Victor Ostrik
University of Oregon

Abstract: This series of talks will be devoted to the theory of tensor categories which is a mathematical side of quantum symmetries. We will discuss basic definitions, examples and constructions of such categories, as well as some structure theory and classification results.

Suggested lectures as prerequisites:

1) Categories, functors, natural transformations, some idea about additive and abelian categories

2) Finite dimensional algebras over (alg closed) fields: Wedderburn-Artin theory and Jacobson radical; projective and injective modules

3) Representations of groups

• Finite groups.

• Understanding of examples of small order.

• Compact groups.

• Representation of $SU(2)$.

4) Cohomology of groups: cocycles and coboundaries. (Seccions 1.1 and 1.2 of the document.)

Lecture notes

Notes by Harshit Yadav

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Problem session

In charge of Paul Gustafson

Problems 1

1. Make precise and prove the following statement: Functors and multi-functors between semisimple categories are determined (up to natural isomorphism of functors) by their values.

2. Let $(\mathcal{C},\otimes,a)$ be a category with a tensor product and an associativity constraint satisfying a pentagon axiom. Let $S\in \mathcal{C}$ be an object such $S \otimes S \simeq S$ and the functors $X\mapsto X\otimes S$ and $X\mapsto S\otimes X$ are fully faithful. Prove that $S$ is the unit object of $\mathcal{C}$. (cf Lemma 4.2.6 of [KS]).

3. Let $\mathcal{C}$ be a monoidal category and $X\in \mathcal{C}$. Prove that the (right) dual objects of $X$ are all canonically isomorphic. (see Proposition 2.10.5 in [EGNO]).

4. Let $\mathcal{C}$ be a monoidal category.

   (a) Define “most obvious” monoidal structure on the category $\mathcal{C} \times \mathcal{C}$. Show that $\mathcal{C}$ is braided if and only if the tensor product functor $\mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a tensor functor (this means that braidings are in bijection with the structures of tensor functor).

   (b) Assume $\mathcal{C}$ is braided. Define ``most obvious’’ braided structure on the category $\mathcal{C} \times \mathcal{C}$. Show that $\mathcal{C}$ is symmetric if and only if the tensor product functor $\mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is braided.

5. Prove that the two hexagon axioms in the definition of a braided monoidal category are independent, that is none of them imply the other one. Hint: think about pointed categories with trivial associator.

6. Let $X,Y\in \mathcal{C}$ be two objects such that $X\otimes Y\simeq Y\otimes X\simeq \bf{1}$. Prove that $X$ is invertible. (see Ex 4.12 from [KS]).

7. Give a description of pointed categories in terms of normalized cocycles (see e.g. Exercise 2.3.9 in [EGNO]).

8. Give a complete classification of pointed fusion categories with the underlying group $C_2$. Include the description in the case of tensor, braided tensor, and symmetric categories. What are possible pivotal/spherical structures? (see Problem 4.5 (v) from [KS]).

9. Replace the group $C_2$ by $C_3$ in the previous exercise.

10. (See Problem 4.13 from [KS]) Let $X\in \mathcal{C}$ and assume that there is $n\in \mathbb Z_{>0}$ and an isomorphism $\lambda: X^{\otimes n}\simeq \bf{1}$. Then we have two isomorphisms $X^{\otimes (n+1)}\to X$, namely $\lambda \otimes \operatorname{id}$ and $\operatorname{id} \otimes \lambda$.

    (a) These two isomorphisms are not necessarily the same.

    (b) These two isomorphisms are or are not the same independently of the choice of $\lambda$.

    (c) If the isomorphisms are the same the object $X$ generates a subcategory of $\mathcal{C}$ equivalent to $\operatorname{Vec}_{C_n}$.

    (d) The ratio of the two isomorphisms is always a root of 1 of degree $n$.

11. Show that there exists a surjective tensor functor from $\operatorname{Vec}_{C_4}$ to $\operatorname{Vec}_{C_2}^\omega$ where $\omega$ is nontrivial.

12. Compute explicitly the associativity constraint for the Fibonacci category.

13. Describe braided pointed fusion categories in terms of abelian cocycles (see Exercises 8.4.3-8.4.6 in [EGNO]).

14. Compute the action of $\operatorname{Aut}(C_n)$ on $H^3(C_n, k^\times )$. (See Example 2.6.4 in [EGNO]).

15. Let $a(n)=$ number of pointed fusion categories such that the isomorphism classes of simple objects form a cyclic group of order $n$ up to tensor equivalence. Compute $a(n)$ as much as possible.

Bibliography

[KS] = M.Kashiwara, P.Schapira “Categories and sheaves”.

[EGNO] = P.Etingof, S.Gelaki, D.Nikshych, V.Ostrik “Tensor categories”.