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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.

flag	asked Aug 29 2011 at 4:42 Amritanshu Prasad 1,883●6●15

Julian Kuelshammer · Accepted Answer · 2011-08-29 06:42:54Z UTC

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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.

answered Aug 29 2011 at 6:42

Julian Kuelshammer
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Thanks Julian, just the kind of stuff I am looking for. – Amritanshu Prasad Aug 29 2011 at 10:37

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You're welcome. Please mark as answered. – Julian Kuelshammer Aug 29 2011 at 10:43

Done. Sorry about the delay. – Amritanshu Prasad Aug 31 2011 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 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 at 5:45

Are wild problems related to undecidable ones?

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