Mdtryhht

Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $mathcalC$ equipped with a fiber functor $F: mathcalC to Vect_k$ for a field $k$ (of characteristic $0$) there exists a reductive algebraic group $G cong Aut(F)$ such that $mathcalC cong Rep(G)$. This means that any such category is associated with a root datum.

Is there a version of this reconstruction theorem that will tell us when a category $mathcalC$ is the category of finite dimensional representations of a semisimple group? I would like to be able to associate with a Tannakian category a root system, and not just a root datum.

In order for $mathcal C$ to come from an algebraic group rather than a pro-algebraic one, you want $mathcal C$ to be finitely generated. And for semisimplicity, you want the group to have finite center. The center can be read off from the category. Cf. my paper “On the center of a compact group”, Intern. Math. Res. Notes. 2004:51, 2751-2756 (2004) or math.CT/0312257.

Another criterion is that there should be only finitely many objects of bounded dimension. This condition might be easy to check in practice from abstract finiteness theorems. The proof is that, if the group is not semi simple, you can take any 1-dimensional character of the identity component and induce up to the main group. Because there are infinitely many characters, infinitely many representations.