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Gabriel's theorem [5] (cf. below for precise statements) was generalized by Dlab-Ringel [3], [4] where Dynkin graphs of type Bn, Cn, Fit G2 also enter in the classificationtogether with the graphs of type An, Dn, En in [5]. We give in this note another generalization of [53 using the fact that Bn, Cn, Fif G2 are obtained by the so-called folding-operation from An, Dn, E6. Our formulation is rather similar to the original formulation in [5]. Let F be a finitegraph. We denote its set of vertices by FQ and its set of edges by I\ (there may be several edges between two vertices and loops joining a vertex to itself). Let A be an orientation of f. For each /e/i we denote its starting-point by ≪(/)and its end-point by /3(/). For a fixed fieldk we define a category X(F, A) after Gabriel F51 as follows.

Foldings of root systems and Gabriel's theorem