Step 1: Dimension vectors and the Euler form

For representations $V, W$, define the Euler form: $\langle \underline{\dim}(V), \underline{\dim}(W) \rangle = \sum_{i \in Q_0} \dim V_i \dim W_i - \sum_{\alpha \in Q_1} \dim V_{s(\alpha)} \dim W_{t(\alpha)}$

This is a bilinear form on $\mathbb{Z}^{Q_0}$ determined by the quiver structure. The associated quadratic form is: $q(\mathbf{d}) = \langle \mathbf{d}, \mathbf{d} \rangle = \sum_{i} d_i^2 - \sum_{\alpha: i \to j} d_i d_j$

Step 2: Positive definiteness criterion

Key Lemma: $Q$ has finite representation type if and only if the quadratic form $q$ is positive definite.

Proof idea ($\Leftarrow$): If $q$ is positive definite, there are finitely many dimension vectors $\mathbf{d}$ with $q(\mathbf{d}) = 1$ (finitely many integral points in a bounded region). For each such $\mathbf{d}$, there are finitely many indecomposables with that dimension vector, giving finite representation type overall.

Proof idea ($\Rightarrow$): If $q$ is not positive definite, construct infinitely many indecomposables using preprojective representations or other techniques, contradicting finite representation type.

Step 3: Classification of positive definite forms

The quadratic forms corresponding to connected graphs are positive definite precisely for the Dynkin diagrams $A_n, D_n, E_6, E_7, E_8$.

This is a classical result from the classification of root systems and Cartan matrices. The Cartan matrix $C$ of the quiver satisfies: $q(\mathbf{d}) = \mathbf{d}^T C \mathbf{d}$

Positive definiteness of $C$ characterizes finite Dynkin type.

Step 4: Construct bijection with roots

For a Dynkin quiver, define a map: $\Phi: \{\text{indecomposables}\} \to \Phi^+ \text{ (positive roots)}$ by sending an indecomposable $V$ to its dimension vector (viewed as a root).

Injectivity: Two indecomposables with the same dimension vector have $\text{Hom}(V, W) \neq 0$ or $\text{Ext}^1(V, W) \neq 0$, and for Dynkin quivers, this forces $V \cong W$ (by analyzing the Euler form).

Surjectivity: For each positive root $\alpha$, construct an indecomposable with dimension vector $\alpha$ using reflection functors or explicit construction.

Conclusion: This establishes the bijection $\{\text{indecomposables}\} \leftrightarrow \Phi^+$.