Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^\*$, we know that the decomposition of $V \otimes V^\*$ must contain a copy of the trivial module. Is it true that it will only ever contain a single copy of the trivial representation?


Does this generalise to the setting of suitably qualified ***semisimple rigid monodial categories***? A specific guess: 


*For a semisimple rigid braided monoidal category $\mathcal{C}$, with $V$ a simple object in $\mathcal{C}$, and $V^\*$ its dual, the decomposition
of $V \otimes V^\*$ into simple objects contains one and only one copy of the unit object of $\mathcal{C}$.* 



Abstracting the properties of the category $\frak{g}$-modules
$\newcommand{\C}{\mathcal{C}}$
To expand on Victor's comment: you don't actually need $\C$ to be semi-simple (let's assume it's abelian to be safe). By definition the multiplicity of $1\_C$ in $V\otimes V^\*$ is the dimension of $$Hom\_\C(1\_\C,V\otimes V^\*)\cong Hom\_\C(V,V).$$


Now Schur Lemma is valid at this level of generality, so if $V$ is simple this space is 1-dimensional.


