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}…a_{0} such that h_{(ai−1)} = t_{(ai)} for i = 1,…, m:

The path is p = a_{m}…a_{0}. Writing t(p) = t(a_{0}) and saying that p starts at t(a_{0}) and, similarly, writing h(p) = h(a_{m}) and saying that p finishes at h(a_{m}). For each vertex i ∈ Q_{0}, 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≥0}A_{l} is a graded C-algebra. The subring A_{0} ⊂ 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_{i}e_{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_{i}v = v, v ∈ M_{i}

= 0, v ∈ M_{j} for j ≠ i

for i ∈ Q_{0 }and

av = φ_{a}(v_{t(a)}), v ∈ V_{t(a)}

= 0, v ∈ V_{j} for j ≠ t(a)

for a ∈ Q_{1}. This construction can be inverted as follows: given a left A-module V, we set V_{i} = e_{i}V for i ∈ Q_{0} 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.

[…] a quiver with relations (Q, R) is a quiver Q together with a set R = {ri} of elements of its path algebra, where each ri is contained in the subspace A(Q)aibi of A(Q) spanned by all paths p starting at […]