properties of the ordinary quiver

Let $k$ be a field and $A$ be a finite-dimensional algebra over $k$ . Denote by $Q_{A}$ the ordinary quiver (http://planetmath.org/OrdinaryQuiverOfAnAlgebra) of $A$ .

Theorem. The following statements hold:

1.

If $A$ is basic and connected, then $Q_{A}$ is a connected quiver.
2.

If $Q$ is a finite quiver and $I$ is an admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) in $k Q$ and $A=kQ/I$ , then $Q_{A}$ and $Q$ are isomorphic.
3.

If $A$ is basic and connected, then $A$ is isomorphic to $kQ_{A}/I$ for some (not necessarily unique) admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) $I$ .

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

Title	properties of the ordinary quiver
Canonical name	PropertiesOfTheOrdinaryQuiver
Date of creation	2013-03-22 19:17:44
Last modified on	2013-03-22 19:17:44
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	4
Author	joking (16130)
Entry type	Theorem
Classification	msc 16S99
Classification	msc 20C99
Classification	msc 13B99