Let $\mathcal{C}$ be a unitary fusion category. Is it true that the tensor product of any two simple objects is simple ? If not, are there interesting (nontrivial) examples of such a $\mathcal{C}$ ?
1 Answer
$\begingroup$
$\endgroup$
Typically the tensor product of two simple objects is not simple. The smallest example is $\text{Fib}$, the Fibonacci category (also called the Yang-Lee category), that has two simple objects: $\mathbf 1$ and $\tau$. The interesting fusion rule for this category is that $$\tau\otimes \tau=\mathbf 1\oplus\tau\,.$$ Since the right-hand side is a nontrivial direct sum, this object must be non-simple.
The condition that the tensor product of simples is always simple is very strict. Using rigidity, every simple object $X$ has a dual $X^*$, and this dual object is also necessarily simple. Duality forces there to be a nonzero map $\mathbf{1}\to X\otimes X^*$, and if the right-hand side is simple, then this map must be an isomorphism by Schur's lemma. This means that all simple objects are invertible objects, and hence form a group under tensor product, when taken up to isomorphism.
Thus these categories must have all simple objects invertible, and this is what's known as a pointed fusion category. It turns out that pointed fusion categories can be completely classified up to monoidal equivalence, and are all of the form $\text{Vec}_G^\omega$, where $\omega\in H^3(G;\mathbb C^\times)$. For more details on this classification, see e.g. Examples 2.3.6 & 2.3.8 and Section 2.6 of Etingof, Gelaki, Nikshych, and Ostrik.
One further comment: All of these observations are about fusion categories (over $\mathbb C$) generally. All of the categories discussed above admit unitary structure, so the unitary assumption is not restrictive for the purposes of this question.
