Given a fixed connected quiver $Q$. Are there only finitely many quiver algebra $KQ/I$ (I an admissible ideal) up to isomorphism which have finite representation type and finite global dimension?
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1$\begingroup$ If you take $Q$ to be a loop and $K=\mathbb{Q}$ then $KQ=\mathbb{Q}[x]$. Among its quotients you find all finite field extensions of $\mathbb{Q}$. There are infinitely many of those and all of them have finite representation type and global dimension $0$. $\endgroup$– Fernando MuroCommented Dec 14, 2019 at 18:21
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$\begingroup$ @FernandoMuro For me in the definition of quiver algebra, I is an admissible ideal. $\endgroup$– MareCommented Dec 14, 2019 at 18:50
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1$\begingroup$ Mare, for me the quiver algebra is actually the path algebra. Now that you added a condition on the ideal $I$ my previous answer doesn't work. $\endgroup$– Fernando MuroCommented Dec 14, 2019 at 18:58
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