[[Quiver representation theory MOC]]
# Reducibility of quiver representations

Let $V: \underline{\Gamma} \to \Vect_{\mathbb{K}}$ be a [[quiver representation]].
We classify the reducibility of $V$ according to its [[Reducibility of modules|reducibility]] as a [[path algebra|$\mathbb K[\underline \Gamma]$]]-module:
So $V$ is

- **simple** or **irreducibile** iff it has no nonzero [[Quiver subrepresentation|subrepresentation]]; ^simple
- **decomposable** iff it is the direct sum of two nonzero [[Quiver subrepresentation|subrepresentations]]; ^decomposable
- **indecomposable** iff it is the direct sum of two nonzero [[Quiver subrepresentation|subrepresentations]]. ^indecomposable

[[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.

#
---
#state/tidy | #lang/en | #SemBr