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  • What are tame and wild hereditary algebras?

  • Are they related to hereditary rings? (Those are rings for which every left (resp. right) ideal is projective, equivalently, for which every left (resp. right) submodule of a projective module is again projective).

  • Googling them I can see they seem related to the "tame representation type", but this concept is also new to me.

  • I would also like to know what is their relation to path algebras, since sometimes they appear mentioned together.

Do you know any good (newbie) references for all this? (Or can you elaborate in any of the questions?)

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up vote 15 down vote accepted

An $k$-algebra $A$ is tame (or, equivalently, it has tame representation type) if, for every dimension $d\geq0$, you can parametrize all isoclasses of indecomposable $A$-modules of dimension $d$, apart from a finite number of them, by a finite number of $1$-parameter families. On the other hand, a finite dimensional $k$-algebra $A$ is wild (or, equivalently, it has wild representation type) if in the category $\mathrm{mod}_A$ of finite dimensional modules contains a copy of the category $\mathrm{mod}_{k\langle x,y\rangle}$ of modules over the free $k$-algebra on two generators. It is an amazing theorem of Drozd that a finite dimensional algebra is either tame or wild; this is the so called dichotomy theory. One of the reasons that make this theorem so amazing is that one can show that if $A$ is wild then $\mathrm{mod}_A$ contains copies of the module categories of all finite dimensional algebras; in other words, wild algebras are really wild...

In particular, this concepts of tame and wild apply to hereditary algebras, which are those of global dimension $1$.

Now, a finite dimensional hereditary algebra is Morita equivalent to the path algebra $kQ$ on a quiver without oriented cycles. A well-known theorem of Gabriel and others tells us that such a path algebra $kQ$ is tame iff the quiver $Q$ is, when you forget the orientation of the arrows, an Dynkin or an extended-Dynkin diagram. In all other cases the parth algebra is wild.

Two great references on all of this are the (first volume of the) book by Assem, Skowroński and Simson, or the book by Auslander, Reiten and Smalø.

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Great and fast. Thank you very much! I'm working in Leavitt path algebras, which arise from path algebras modulo some relations, and which happen to be hereditary, so this info is fully appreciated! :-) –  Jose Brox Nov 18 '09 at 4:34
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The mentioned books are: * Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory (London Mathematical Society Student Texts), by Ibrahim Assem, Andrzej Skowronski, and Daniel Simson. Cambridge University Press. * Representation Theory of Artin Algebras, by Maurice Auslander, Idun Reiten and Sverre O. Smalo. Cambridge Studies in Advanced Mathematics, Vol. 36. –  Jose Brox Nov 18 '09 at 13:22
    
@JoseBrox, I guess this is a bit late, but anyways: the relations that define Leavitt algebras are not admissible in the sense of Gabriel, so the connection between them and the theory mentioned in my answer above should be done with care. –  Mariano Suárez-Alvarez Apr 7 at 4:23
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