Quiver Representations

Gabriel's theorem (1972): A connected quiver Q has finitely many indecomposable representations (over algebraically closed field) iff the underlying graph is ADE Dynkin: Aₙ, Dₙ, E₆, E₇, E₈.

The indecomposables correspond bijectively to positive roots of the ADE root system.

Each indecomposable M has a dimension vector d(M) ∈ ℤⁿ₀, and d(M) is always a positive root of the associated root system.

For A₃: 6 positive roots → 6 indecomposables.
For D₄: 12 positive roots → 12 indecomposables.
For Ã₃: infinitely many indecomposables (infinite type).