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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?

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    $\begingroup$ I would say that an abelian category is semisimple if every short exact sequence splits. So I would regard $\text{Vect}$, the abelian category of not-necessarily-finite-dimensional vector spaces, as semisimple. I would also say that an object is semisimple if it's a direct sum, not necessarily finite, of simples. $\endgroup$
    – Qiaochu Yuan
    Commented Aug 5, 2015 at 7:23

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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 $V\otimes V$.

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    $\begingroup$ An even simpler example: $1 \otimes 1 \simeq 1$, and $1$ is self dual. $\endgroup$
    – Sam Gunningham
    Commented Aug 4, 2015 at 18:06

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