What are tame and wild hereditary algebras? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:59:03Z http://mathoverflow.net/feeds/question/5895 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5895/what-are-tame-and-wild-hereditary-algebras What are tame and wild hereditary algebras? Jose Brox 2009-11-18T01:43:26Z 2009-11-18T04:38:08Z <ul> <li><p>What are tame and wild hereditary algebras?</p></li> <li><p>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).</p></li> <li><p>Googling them I can see they seem related to the "tame representation type", but this concept is also new to me.</p></li> <li><p>I would also like to know what is their relation to path algebras, since sometimes they appear mentioned together.</p></li> </ul> <p>Do you know any good (newbie) references for all this? (Or can you elaborate in any of the questions?)</p> http://mathoverflow.net/questions/5895/what-are-tame-and-wild-hereditary-algebras/5906#5906 Answer by Mariano Suárez-Alvarez for What are tame and wild hereditary algebras? Mariano Suárez-Alvarez 2009-11-18T02:35:30Z 2009-11-18T04:38:08Z <p>An $k$-algebra $A$ is <em>tame</em> (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 <em>wild</em> (or, equivalently, it has wild representation type) if in the category <code>$\mathrm{mod}_A$</code> of finite dimensional modules contains a copy of the category <code>$\mathrm{mod}_{k\langle x,y\rangle}$</code> 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 <code>$\mathrm{mod}_A$</code> contains copies of the module categories of <em>all</em> finite dimensional algebras; in other words, wild algebras are really wild...</p> <p>In particular, this concepts of tame and wild apply to hereditary algebras, which are those of global dimension $1$. </p> <p>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.</p> <p>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.</p>