Gabriel's theorem

Remarks

Bernstein, Gelfand and Ponomarev gave a proof with reflection functors. A reflection functor at a sink or source changes the orientation at one vertex, and acts on dimension vectors by the simple reflection of the root system. Applying them repeatedly brings every indecomposable representation to a simple one, whose dimension vector is a simple root. Example: for $1 \to 2$, of type $A_2$, the positive roots are $(1, 0)$, $(0, 1)$ and $(1, 1)$, matching the indecomposables $k \to 0$, $0 \to k$ and $k \xrightarrow{\id} k$. The Dynkin hypothesis is necessary: for the Kronecker quiver, with two arrows $a, b \colon 1 \to 2$, the representations $V_1 = V_2 = k$, $V_a = \id$, $V_b = \lambda \id$ are indecomposable and pairwise non-isomorphic for $\lambda \in k$. The proof by reflection functors works over every field.

Going deeper

For a student

The quadratic form $q$ is the Tits form of the quiver; it does not depend on the orientation, and it is positive definite exactly for the Dynkin graphs. This links representation theory of quivers with the classification of simple Lie algebras, where the same diagrams appear. Kac's theorem extends the description of dimension vectors to all quivers without loops, through the roots of Kac–Moody algebras. For path algebras of Dynkin quivers, the Auslander–Reiten quiver can be built from the projective modules by the Auslander–Reiten translation.