Quiver
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). Sometimes, especially in quivers made for hanging on a saddle, the quiver had a compartment for holding the bow itself. Many quivers had lids to protect the arrows from rain, which could potentially ruin them (for example by affecting the glue holding the fletching[?] in place).Quivers have seen use in all cultures where bow and arrow have been used. One certain archaeological source of evidence is Oetzi, the Bronze Age man found in the Alps.
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 isn't 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.
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