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 finite-dimensional, 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 finite-dimensional 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 well-defined, 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	2013-03-22 19:17:41
Last modified on	2013-03-22 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