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) p2 i = 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|p h 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 = xh 1 . . . xh m : Vh m → Vh 1 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.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.