ordinary quiver of an algebra

Let k be a field and A an algebra over k.

Denote by radA the (Jacobson) radicalPlanetmathPlanetmathPlanetmath of A and rad2A=(radA)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={e1,,en}.

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

  1. 1.

    The set of vertices is equal to Q0={1,,n} which is in bijective correspondence with E.

  2. 2.

    If a,bQ0, then the number of arrows from a to b is equal to the dimension of the k-vector spaceMathworldPlanetmath


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