Quivers with relations

A relation is a special property that can be imposed on a quiver. A path in a quiver is a succession of arrows such that the next arrow in the succession starts at the point where the previous arrow ended. Consider the quiver;

We have two paths cb and a from T to R. The length of a path in a quiver is determined by the number or arrows in the path. A trivial path is a path associated to a point and has no arrows, while an arrow is a path of length 1. If the starting and ending points of a path coincide, the path is called a cyclic path. Our example has three trivial paths and three paths of length 1. The total number of paths in the quiver is 7.

A path algebra over a field K is an algebra with the basis consisting of all the distinct paths a quiver. If the paths are finite, the path algebra is finite dimensional. A path algebra is finite dimensional if and only if the underlying quiver has no loops. A relation is an ideal of the path algebra. If we quotient the relations from the path algebra, the quotient algebra will be finite dimensional.

Consider the loop, Paths of this quiver are, e, x, xx, xxx, - - - , x---x,- - - There are infinitely many paths in this quiver, which implies that its path algebra is not finite dimensional.

If we let

 

then the path algebra of the quiver will be finite dimensional. the path e in the quiver is like 1 in the path algebra.

 

The element

that we set to 0 in the path algebra is what we call a relation on the quiver. The path algebra of the loop, with this relation is finite dimensional. It is similar to the algebra of complex numbers.