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Given two finite dimensional quiver algebras $A$ and $B$ (over a nice field in case that helps, for example a finite field).

Question: Can an there be a finite algorithm that decides whether $A$ and $B$ are derived equivalent?

I am pretty sure that at the moment no such algorithm exists but maybe in 1000 years there is such an algorithm and the above question is rather trivial. On the other hand, there might be some theoretical reason why such an algorithm can not exist.

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  • $\begingroup$ "a nice field...a finite field" I guess the tastes differ, some people only consider fields of characteristic 0 as nice (but I do not know much about quivers, maybe there positive characteristic is the best). $\endgroup$
    – user140765
    Jun 1, 2019 at 14:49
  • $\begingroup$ what is a finite algorithm (or maybe I should ask, what is an infinite algorithm)? An algorithm that halts eventually, or one that can be implemented in finitely many lines of code? $\endgroup$
    – user140765
    Jun 1, 2019 at 14:50
  • $\begingroup$ @kartop_man I would say it has finitely many lines of code but the interpretation can be rather free in case that helps. For representation theory finite fields have some advantages, for example in the GAP package QPA, decomposing modules into indecomposables works only over finite fields as far as I know. $\endgroup$
    – Mare
    Jun 1, 2019 at 14:54

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