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Suppose $Q$ and $Q'$ are two quivers. I am curious as to what relation $\mathbb{C}Q$ bears to $\mathbb{C}Q'$ when $Q$ and $Q'$ share the same underlying graph and only differ by direction.

Since quivers and path algebras are in correspondence, surely something may be said about this situation, yet it seems the algebras may be wildly different. Consider for instance $C_4$, the cycle quiver over it has a very different path algebra from the diamond-shaped quiver.

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    $\begingroup$ Are you familiar with reflection functors? For acyclic graphs this is well understood see math.berkeley.edu/~serganov/math252/notes11.pdf. Experts can probably tell you what happens in general. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 2, 2022 at 18:59
  • $\begingroup$ @BenjaminSteinberg This seems to be if $Q$ and $Q'$ share an underlying graph => there exists these adjoint reflection functors, but it doesn't look like the converse doesn't seem to hold right? $\endgroup$
    – tox123
    Commented Dec 3, 2022 at 4:53
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    $\begingroup$ I don't think there is a converse. My understanding, but I'm not an expert, is there reflection functors led to tilting theory. And obviously it's a different game if your graph is not a tree and you compare an acyclic orientation with one with a cycle $\endgroup$
    – Benjamin Steinberg
    Commented Dec 3, 2022 at 11:58

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