20 2. Basic notions of representation theory

Definition 2.8.4. The path algebra PQ of a quiver Q is the algebra

whose basis is formed by oriented paths in Q, including the trivial

paths pi, i ∈ I, corresponding to the vertices of Q, and multiplication

is the concatenation of paths: ab is the path obtained by first tracing

b and then a. If two paths cannot be concatenated, the product is

defined to be zero.

Remark 2.8.5. It is easy to see that for a finite quiver

∑

i∈I

pi = 1, so

PQ is an algebra with unit.

Problem 2.8.6. Show that the algebra PQ is generated by pi for

i ∈ I and ah for h ∈ E with the following defining relations:

(1) pi 2 = pi, pipj = 0 for i = j,

(2) ahph = ah, ahpj = 0 for j = h ,

(3) ph ah = ah, piah = 0 for i = h .

We now justify our statement that a representation of a quiver is

the same thing as a representation of the path algebra of a quiver.

Let V be a representation of the path algebra PQ. From this

representation, we can construct a representation of Q as follows: let

Vi = piV, and for any edge h, let xh = ah|ph

V

: ph V −→ ph V be

the operator corresponding to the one-edge path h.

Similarly, let (Vi,xh) be a representation of a quiver Q. From this

representation, we can construct a representation of the path algebra

PQ: let V =

i

Vi, let pi : V → Vi → V be the projection onto Vi,

and for any path p = h1

. . . hm let ap = xh1 . . . xhm : Vhm → Vh1 be

the composition of the operators corresponding to the edges occurring

in p (and the action of this operator on the other Vi is zero).

It is clear that the above assignments V → (piV) and (Vi) →

i

Vi are inverses of each other. Thus, we have a bijection between

isomorphism classes of representations of the algebra PQ and of the

quiver Q.

Remark 2.8.7. In practice, it is generally easier to consider a rep-

resentation of a quiver as in Definition 2.8.3.

We lastly define several previous concepts in the context of quiver

representations.