TITLE: Tannaka duality for semisimple groups
QUESTION [9 upvotes]: 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.

REPLY [2 votes]: 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 –