# Are wild problems related to undecidable ones?

In representation theory, there is a well-known notion of a wild classification problem (such problems have been discussed often on this forum, for example, here). In logic, there is a notion of an undecidable problem.

Is there a theorem which says that there is something undecidable about a wild classification problem?

A reference where such issues are discussed would be very helpful.

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## 1 Answer

Yes, there is a connection, but I think it is conjectural in its full generality. The mosst general reference could be, where it is proven, that for a subclass of wild algebras, the representation theory is undecidable:

Mike Prest: Wild representation type and undecidability, Comm. Alg. 19 (3), 1991.

It is also well-known (it is stated with references for example in Benson), that the representation theory of the algebra used to define wildness (i.e. $k\langle X,Y\rangle$) is undecidable.

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Thanks Julian, just the kind of stuff I am looking for. –  Amritanshu Prasad Aug 29 '11 at 10:37
You're welcome. Please mark as answered. –  Julian Kuelshammer Aug 29 '11 at 10:43
Done. Sorry about the delay. –  Amritanshu Prasad Aug 31 '11 at 16:00
Note that all articles I could find in the literature show that the infinite dimensional representation theory is undecidable for wild quivers. I have a proof for the finite dimensional representation theory and will post one day on ArXiv. It is not difficult. –  Benjamin Steinberg Sep 17 '12 at 15:06
@Benjamin: would Prest's proof fail for the theory of finite dimensional representations? In that case it does not really address the matrix pair problem, which is really about finite dimensional representations. –  Amritanshu Prasad Sep 20 '12 at 5:45