Reducibility of quiver representations

Let $𝑉 : Γ ―― \to 𝖵 𝖾 𝖼 𝗍 𝕂$ be a quiver representation. We classify the reducibility of $𝑉$ according to its reducibility as a [[path algebra| $𝕂 [Γ ――]$ ]]-module: So $𝑉$ 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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Reducibility of quiver representations

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