Indecomposable Objects – Part 1

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 Q₀, a set of arrows Q₁, and head and tail maps

h, t : Q₁ → Q₀

We always assume that Q is finite, i.e., the sets Q₀ and Q₁ are finite.

A (complex) representation of a quiver Q consists of complex vector spaces V_i for i ∈ Q₀ and linear maps

φ_a : V_t(a) → V_h(a)

for a ∈ Q₁. A morphism between such representations (V, φ) and (W, ψ) is a collection of linear maps f_i : V_i → W_i for i ∈ Q₀ such that the diagram

commutes ∀ a ∈ Q₁. 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: V₁ → V₂. If W = im(f) is a nonzero proper subspace of V₂, then the splitting V₂ = U ⊕ W, and the corresponding object of Rep(Q) splits as a direct sum of the two representations

V₁ →^ƒ W and 0 → W

Thus if an object f: V₁ → V₂ 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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