Quiver (mathematics)

In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed. They are commonly used in representation theory: a representation of a quiver assigns a vector space to each vertex of the quiver and a linear map to each arrow.

If K is a field and Γ is a quiver, then the quiver algebra KΓ is defined as follows: it is the vector space having all the paths in the quiver as basis; multiplication is given by composition of paths. If two paths cannot be composed because the end vertex of the first is not equal to the starting vertex of the second, their product is defined to be zero. This defines an associative algebra over K. This algebra has a unit element if and only if the quiver has only finitely many vertices. In this case, the modules over KΓ are naturally identified with the representations of Γ.

If the quiver has finitely many vertices and arrows, then KΓ is a finite-dimensional hereditary algebra over K, i.e. submodules of projective modules over KΓ are projective.