Auslander-Reiten theory of wild algebras known in examples? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:01:41Z http://mathoverflow.net/feeds/question/73967 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73967/auslander-reiten-theory-of-wild-algebras-known-in-examples Auslander-Reiten theory of wild algebras known in examples? Julian Kuelshammer 2011-08-29T12:54:20Z 2011-10-31T21:17:03Z <p>In my PhD thesis I am studying the Auslander-Reiten theory of a particular class of wild algebras. My question is:</p> <p>Are there instances of algebras/categories, where the Auslander-Reiten theory of a wild algebra is understood (in the sence that one knows which shapes of components occur) beside the following:</p> <ul> <li>Hereditary algebras (Ringel: Finite dimensional hereditary algebras of wild representation type)</li> <li>Canonical algebras and coherent sheaves on a weighted projective line (Lenzing, de la Peña: Wild canonical algebras)</li> <li>Group algebras (Erdmann: On Auslander-Reiten components for group algebras)</li> <li>Local restricted enveloping algebras (Erdmann: The Auslander-Reiten quiver of restricted enveloping algebras)</li> <li>Quantum complete intersections (Bergh, Erdmann: The stable Auslander-Reiten quiver of a quantum complete intersection)</li> </ul> http://mathoverflow.net/questions/73967/auslander-reiten-theory-of-wild-algebras-known-in-examples/79653#79653 Answer by Dag Oskar Madsen for Auslander-Reiten theory of wild algebras known in examples? Dag Oskar Madsen 2011-10-31T21:07:31Z 2011-10-31T21:17:03Z <p>For self-injective Koszul algebras of Loewy length greater than three such that the Yoneda algebra is Noetherian, all (stable) components of the Auslander-Reiten quiver for <em>graded</em> modules are of the form <code>$\mathbb Z A_{\infty}$</code> (Martinez-Villa, Zacharia: Approximation with modules having linear resolution).</p> <p>It shouldn't be too hard to find examples where all modules are gradeable, it is more difficult to keep track of graded shifts within each component. The easy way out is to say that each (ungraded) component is of the form <code>$\mathbb Z A_{\infty}/G$</code> for some group of automorphisms $G$. (Edit: Also known as a tube when $G$ is non-trivial.)</p>