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.
