The Case of Morphisms of Representation Corresponding to A-Module Holomorphisms. Part 2

Representations of a quiver can be interpreted as modules over a non-commutative algebra A(Q) whose elements are linear combinations of paths in Q.

Let Q be a quiver. A non-trivial path in Q is a sequence of arrows a_m…a₀ such that h_(ai−1) = t_(ai) for i = 1,…, m:

The path is p = a_m…a₀. Writing t(p) = t(a₀) and saying that p starts at t(a₀) and, similarly, writing h(p) = h(a_m) and saying that p finishes at h(a_m). For each vertex i ∈ Q₀, we denote by e_i the trivial path which starts and finishes at i. Two paths p and q are compatible if t(p) = h(q) and, in this case, the composition pq can defined by juxtaposition of p and q. The length l(p) of a path is the number of arrows it contains; in particular, a trivial path has length zero.

The path algebra A(Q) of a quiver Q is the complex vector space with basis consisting of all paths in Q, equipped with the multiplication in which the product pq of paths p and q is defined to be the composition pq if t(p) = h(q), and 0 otherwise. Composition of paths is non-commutative; in most cases, if p and q can be composed one way, then they cannot be composed the other way, and even if they can, usually pq ≠ qp. Hence the path algebra is indeed non-commutative.

Let us define A_l ⊂ A to be the subspace spanned by paths of length l. Then A = ⊕_l≥0A_l is a graded C-algebra. The subring A₀ ⊂ A spanned by the trivial paths e_i is a semisimple ring in which the elements e_i are orthogonal idempotents, in other words e_ie_j = e_i when i = j, and 0 otherwise. The algebra A is finite-dimensional precisely if Q has no directed cycles.

The category of finite-dimensional representations of a quiver Q is isomorphic to the category of finitely generated left A(Q)-modules. Let (V, φ) be a representation of Q. We can then define a left module V over the algebra A = A(Q) as follows: as a vector space it is

V = ⊕_i∈Q0 V_i

and the A-module structure is extended linearly from

e_iv = v, v ∈ M_i

= 0, v ∈ M_j for j ≠ i

for i ∈ Q₀ and

av = φ_a(v_t(a)), v ∈ V_t(a)

= 0, v ∈ V_j for j ≠ t(a)

for a ∈ Q₁. This construction can be inverted as follows: given a left A-module V, we set V_i = e_iV for i ∈ Q₀ and define the map φ_a: V_t(a) → V_h(a) by v ↦ a(v). Morphisms of representations of (Q, V) correspond to A-module homomorphisms.

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