An object X in a category C with an initial object is called indecomposable if X is not the initial object and X is not isomorphic to a coproduct of two noninitial objects. A group G is called indecomposable if it cannot be expressed as the internal direct product of two proper normal subgroups of G. This is equivalent to saying that G is not isomorphic to the direct product of two nontrivial groups.

A quiver Q is a directed graph, specified by a set of vertices Q_{0}, a set of arrows Q_{1}, and head and tail maps

h, t : Q_{1} → Q_{0}

We always assume that Q is finite, i.e., the sets Q_{0} and Q_{1} are finite.

A (complex) representation of a quiver Q consists of complex vector spaces V_{i} for i ∈ Q_{0 }and linear maps

φ_{a} : V_{t(a)} → V_{h(a)}

for a ∈ Q_{1}. A morphism between such representations (V, φ) and (W, ψ) is a collection of linear maps f_{i} : V_{i} → W_{i} for i ∈ Q_{0} such that the diagram

commutes ∀ a ∈ Q_{1}. A representation of Q is finite-dimensional if each vector space V_{i} is. The dimension vector of such a representation is just the tuple of non-negative integers (dim V_{i})_{i∈Q0}.

Rep(Q) is the category of finite-dimensional representations of Q. This category is additive; we can add morphisms by adding the corresponding linear maps f_{i}, the trivial representation in which each V_{i} = 0 is a zero object, and the direct sum of two representations is obtained by taking the direct sums of the vector spaces associated to each vertex. If Q is the one-arrow quiver, • → •, then the classification of indecomposable objects of Rep(Q), yields the objects E ∈ Rep(Q) which do not have a non-trivial direct sum decomposition E = A ⊕ B. An object of Rep(Q) is just a linear map of finite-dimensional vector spaces f: V_{1} → V_{2}. If W = im(f) is a nonzero proper subspace of V_{2}, then the splitting V_{2} = U ⊕ W, and the corresponding object of Rep(Q) splits as a direct sum of the two representations

V_{1} →^{ƒ} W and 0 → W

Thus if an object f: V_{1} → V_{2} of Rep(Q) is indecomposable, the map f must be surjective. Similarly, if ƒ is nonzero, then it must also be injective. Continuing in this way, one sees that Rep(Q) has exactly three indecomposable objects up to isomorphism:

C → 0, 0 → C, C →^{id} C

Every other object of Rep(Q) is a direct sum of copies of these basic representations.

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