We know that for a path algebra KQ, whether or not KQ is finite dimensional (namely, Q may or may not have oriented cycles), there might be different admissible ideals I and J of KQ for which the corresponded bound quiver algebras are isomorphic. In other words, the cardinal of iso-classes of bound quiver algebras of a path algebra KQ is less or equal to the cardinal of the collection of all admissible ideals of KQ.
Here are my questions:
1) Do we have any machinery which explicitly gives us the collection of admissible ideals of KQ whose bound quiver algebras are isomorphic? i.e, once we have an admissible ideal of KQ, could we find all the other admissible ideals of KQ for which (up to isomorphism) we get the same bound quiver algebra?
2) For a given finite quiver Q, with or without oriented cycles, can we explicitly determine in which cases the cardinal of iso-class of bound quiver algebras of KQ is exactly less than (and on the other hand equal to) the cardinal of the collection of all admissible ideals of KQ?