If K is a field and Γ is a quiver, then the quiver algebraKΓ 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 and the end vertex and starting vertex of any path are always distinct, then KΓ is a finite-dimensional hereditary algebra over K, i.e. submodules of projective modules over KΓ are projective.
A quiver is a container for arrows, crossbow bolts or darts[?], such as those fired from a bow, crossbow or blowpipe[?].
A quiver may have different forms depending on where it is supposed to be used: Quivers could hang from an archer's belt, from the saddle[?] of a horse or be worn on the back (as most often done in Robin Hood films).
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.
They are commonly used in representation theory: a representation, V, of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a.
If K is a field and Γ is a quiver, then the quiver algebra or path 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 the quiver has finitely many vertices and arrows and the end vertex and starting vertex of any path are always distinct, then KΓ is a finite-dimensional hereditary algebra over K, i.e.
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