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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?

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The set of ideals containing a fixed power of the ideal generated by the arrows is an algebraic variety (a closed subset of a Grassmanian) on which the group of automorphisms of the path algebra which preserve the vertices acts. The orbits correspond to isoclasses of álgebras, and each point in the orbit to an ideal. Everything can be made explicit. –  Mariano Suárez-Alvarez Mar 1 at 19:15
    
For your second question: I would guesd that there are more ideals than algebras exactly when there is a path parallel to an arrow which starts at a non-source vertex or which ends at a non-sink vertex. –  Mariano Suárez-Alvarez Mar 1 at 19:17

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