admissible ideals,, bound quiver and its algebra

Assume, that Q is a quiver and k is a field. Let kQ be the associated path algebra. Denote by RQ the two-sided idealMathworldPlanetmath in kQ generated by all paths of length 1, i.e. all arrows. This ideal is known as the arrow ideal.

It is easy to see, that for any m1 we have that RQm is a two-sided ideal generated by all paths of length m. Note, that we have the following chain of ideals:


Definition. A two-sided ideal I in kQ is said to be admissible if there exists m2 such that


If I is an admissible ideal in kQ, then the pair (Q,I) is said to be a bound quiver and the quotient algebra kQ/I is called bound quiver algebra.

The idea behind this is to treat some paths in a quiver as equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath. For example consider the following quiver


Then the ideal generated by ab-c is not admissible (ab-cRQ2) but an ideal generated by ab-ef is. We can see that this means that ,,walking” from 1 to 3 directly and through 2 is not the same, but walking in the same number of steps is.

Note, that in our case there is no path of length greater then 2. In particular, for any m>2 we have RQm=0.

More generally, it can be easily checked, that if Q is a finite quiver without oriented cycles, then there exists m such that RQm=0

Title admissible ideals,, bound quiver and its algebra
Canonical name AdmissibleIdealsBoundQuiverAndItsAlgebra
Date of creation 2013-03-22 19:16:42
Last modified on 2013-03-22 19:16:42
Owner joking (16130)
Last modified by joking (16130)
Numerical id 6
Author joking (16130)
Entry type Definition
Classification msc 14L24