Rigid tensor categories (with an associative and commutative tensor product) can be viewed as a generalization of algebraic groups (the case where a fiber functor exists). Super groups give other examples. Over an algebraically closed field and under the finiteness condition ``for some self dual generator $X$, the length of the $N^{\text{\rm th}}$ tensor power of $X$ is at most polynomial in $N$'', no other example is known. We will consider the case where the characteristic is zero, the category is semi-simple, and there are only finitely many simple objects.
