Quiver representation theory MOC

Reducibility of quiver representations

Let be a quiver representation. We classify the reducibility of according to its reducibility as a -module: So is

• simple or irreducibile iff it has no nonzero subrepresentation;
• decomposable iff it is the direct sum of two nonzero subrepresentations;
• indecomposable iff it is the direct sum of two nonzero subrepresentations.

Decomposition of a quiver representation states that every finite-dimensional quiver representation is isomorphic to a direct sum of indecomposable representations, unique up to isomorphism and permutation of factors.

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