We say that an abelian category is **semisimple** if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects. 


Let $({\cal C},\otimes,\*)$ be a semisimple monoidal category with duals. If $V$ is a simple object in ${\cal C}$, then can $V \otimes V$ ever contain $V^\*$ as a summand?



Semisimple monoidal category with duals
Yes it can.
Example: $\mathcal{C} = Rep-S\_3$, and $V$ the two dimensional irreducible representation.
Then $V\otimes V$ splits as the direct sum $V\oplus \mathbb{1}\oplus \chi$ where $\mathbb{1}$ is the trivial representation and $\chi$ is the sign representation. The dual of $V$ is $V$ itself, which is a direct summand of $V\otimes V$. 


