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Let $Q$ be a finite connected quiver. An admissible quotient algebra is an algebra of the form $KQ/I$ with an admissible ideal $I$.

Question 1: Is there a nice closed formula for the number of admissible quotient algebras of a tree $Q$?

For example when $Q$ is of linear orineted Dynkin type $A_n$ we get the Catalan numbers and for linear oriented Dynkin type $D_n$ we get a difference between two neighboring Catalan numbers.

Question 2: For which acyclic $Q$ are there only a finite number of admissible quotient algebras up to isomorphism?

For example when $Q$ is the Hasse quiver of the Boolean lattice of a 2-set we get a finite number, while for $Q$ the quiver of a canonical algebra of type [2,2,2,2] for example we get infinitely many admissible quotient algebras.

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  • $\begingroup$ Is $K$ a finite field or an arbitrary field? $\endgroup$
    – Benjamin Steinberg
    Commented Mar 1, 2023 at 16:30
  • $\begingroup$ @BenjaminSteinberg An arbitrary field. $\endgroup$
    – Mare
    Commented Mar 1, 2023 at 16:48

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