admissible ideals,, bound quiver and its algebra
Assume, that is a quiver and is a field. Let be the associated path algebra. Denote by the two-sided ideal in generated by all paths of length , i.e. all arrows. This ideal is known as the arrow ideal.
It is easy to see, that for any we have that is a two-sided ideal generated by all paths of length . Note, that we have the following chain of ideals:
Definition. A two-sided ideal in is said to be admissible if there exists such that
If is an admissible ideal in , then the pair is said to be a bound quiver and the quotient algebra is called bound quiver algebra.
The idea behind this is to treat some paths in a quiver as equivalent. For example consider the following quiver
Then the ideal generated by is not admissible () but an ideal generated by is. We can see that this means that ,,walking” from to directly and through 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 . In particular, for any we have .
More generally, it can be easily checked, that if is a finite quiver without oriented cycles, then there exists such that
|Title||admissible ideals,, bound quiver and its algebra|
|Date of creation||2013-03-22 19:16:42|
|Last modified on||2013-03-22 19:16:42|
|Last modified by||joking (16130)|