ordinary quiver of an algebra
Let $k$ be a field and $A$ an algebra over $k$.
Denote by $\mathrm{rad}A$ the (Jacobson) radical^{} of $A$ and ${\mathrm{rad}}^{2}A={(\mathrm{rad}A)}^{2}$ a square of radical.
Since $A$ is finitedimensional, then we have a complete set of primitive orthogonal idempotents (http://planetmath.org/CompleteSetOfPrimitiveOrthogonalIdempotents) $E=\{{e}_{1},\mathrm{\dots},{e}_{n}\}$.
Definition. The ordinary quiver of a finitedimensional algebra $A$ is defined as follows:

1.
The set of vertices is equal to ${Q}_{0}=\{1,\mathrm{\dots},n\}$ which is in bijective correspondence with $E$.

2.
If $a,b\in {Q}_{0}$, then the number of arrows from $a$ to $b$ is equal to the dimension of the $k$vector space^{}
$${e}_{a}\left(\mathrm{rad}A/{\mathrm{rad}}^{2}A\right){e}_{b}.$$
It can be shown that the ordinary quiver is welldefined, i.e. it is independent on the choice of a complete set of primitve orthogonal idempotents. Also finite dimension of $A$ implies, then the ordinary quiver is finite.
Title  ordinary quiver of an algebra 

Canonical name  OrdinaryQuiverOfAnAlgebra 
Date of creation  20130322 19:17:41 
Last modified on  20130322 19:17:41 
Owner  joking (16130) 
Last modified by  joking (16130) 
Numerical id  4 
Author  joking (16130) 
Entry type  Definition 
Classification  msc 16S99 
Classification  msc 20C99 
Classification  msc 13B99 