Tannaka duality for semisimple groups Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $\mathcal{C}$ equipped with a fiber functor $F: \mathcal{C} \to Vect_k$ for a field $k$ (of characteristic $0$) there exists a reductive algebraic group $G \cong Aut(F)$ such that $\mathcal{C} \cong Rep(G)$. 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 $\mathcal{C}$ 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. 
 A: 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.
A: 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.
A: I've decided to turn my comments into an answer.
(a) The conditions characterizing the Tannakian categories attached to connected reductive groups can be found in Chapter 2 (2.20, 2.22, 2.23) of the notes by Deligne and Milne on Tannakian categories.
(b) The center of $G$ is reflected in the gradations on the Tannakian category $Rep(G)$. For example, let $D$ be a diagonalizable algebraic group with character group $M$. To give a homomorphism $D\to Z(G)$ is the same as giving an $M$-gradation on $Rep(G)$. See 5.1 of the notes by Deligne and Milne. 
(c) To attach a Tannakian category to a root system, choose a semisimple Lie algebra $L$ with the given root system. Then $Rep(L)$ is a Tannakian category with corresponding group $G$ the simply connected semisimple algebraic group with Lie algebra $L$. The category has a natural gradation by $P/Q$ from which it is possible to read off the category corresponding to any lattice $X$ in $P$ containing $Q$. This gives a complete description of the Tannakian categories corresponding to root systems (better, diagrams) without using algebraic groups. See arXiv:0705.1348 –
