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.
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