properties of the ordinary quiver
Let be a field and be a finite-dimensional algebra over . Denote by the ordinary quiver (http://planetmath.org/OrdinaryQuiverOfAnAlgebra) of .
Theorem. The following statements hold:
- 1.
If is basic and connected, then is a connected quiver.
- 2.
If is a finite quiver and is an admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) in and , then and are isomorphic
.
- 3.
If is basic and connected, then is isomorphic to for some (not necessarily unique) admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) .
For proofs please see [1, Chapter II.3].
References
- 1 I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras, vol 1., Cambridge University Press 2006, 2007