# 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) $E=\{e_{1},\ldots,e_{n}\}$.

Definition. The ordinary quiver of a finite-dimensional algebra $A$ is defined as follows:

1. 1.

The set of vertices is equal to $Q_{0}=\{1,\ldots,n\}$ which is in bijective correspondence with $E$.

2. 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

 $e_{a}\big{(}\mathrm{rad}A/\mathrm{rad}^{2}A\big{)}e_{b}.$

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 OrdinaryQuiverOfAnAlgebra 2013-03-22 19:17:41 2013-03-22 19:17:41 joking (16130) joking (16130) 4 joking (16130) Definition msc 16S99 msc 20C99 msc 13B99