6
$\begingroup$

I would like to know Auslander-Reiten quivers of which types of algebras have been found.

For quantum groups, there is a paper http://arxiv.org/abs/1202.1714 by Julian Külshammer. I search the Internet, but was not able to find other papers about Auslander-Reiten theory for quantum groups.

We know that the irreducible representations of quantum groups are parameterized by Drinfeld polynomials. I think that the the vertices of Auslander-Reiten quivers can be given by Drinfeld polynomials (or correspond to Drinfeld polynomials). Are there some papers describe Auslander-Reiten quivers for quantum groups whose vertices correspond to Drinfeld polynomials explicitly?

For group algebras, there are some papers by Karin Erdmann.

There are also many papers about Auslander-Reiten quivers for projective algebras by Christof Geiss, Bernard Leclerc, and Jan Schroer.

Are there some other papers about Auslander-Reiten quivers of finite dimensional algebras? Thank you very much.

$\endgroup$
2
  • $\begingroup$ A remark concerning my preprint: I do not describe the AR quiver completely but only the components containing graded modules. You might also be interested in a similar question on MO I asked a while ago about known Auslander-Reiten quivers for wild algebras: mathoverflow.net/questions/73967/…. $\endgroup$ May 23, 2013 at 17:18
  • $\begingroup$ @Julian, thank you very much. $\endgroup$
    – LJR
    May 25, 2013 at 11:20

1 Answer 1

5
$\begingroup$

This is not an answer, but the comments are getting too long, so I just write it here:

This question is difficult to answer, in the way that there are so many classes of algebras where people have worked on AR-quiver, and I dare not to say I have the complete list. But I can try to list some, and the list would be even more messy if you want AR-quiver of the related categories (derived/homotopy/functorially finite subcat).

  1. Finite/tame-type hereditary algebras - well-known and in many literature.
  2. Finite-type self-injective algebras - work of (mainly) Riedtmann and others.
  3. Tame-type group algebras - work of (mainly) Erdmann, as you said.
  4. Many classes of tame-type self-injective algebras, done by too many people and I couldn't keep track, but probably most are related to works of Skowronski. He also has a survey on tame-type self-injective algebras, which should contain some material on this.
  5. I think there is also a theorem stating that AR component of some of the wild-type self-inejctive algebras (such as group algebras) with infinitely many indecomposable modules must be of type $\mathbb{Z}A_\infty$ or $\mathbb{Z}A_\infty/\langle \tau^m\rangle$. And there are probably more work done in studying the types of AR components which are tubes.
  6. Skorwonski's series of books (Elements of Rep. Thy and Assoc. Algebras) also has quite detail treatmeant to AR-quiver components of tame/wild-type algebras, arising from tilting (tilted algebras) and such like (quasi-tilted algebras). This kind of extends the list from (1).
  7. Preprojective algebras, as you said there. (you missed the "pre-" prefix there, which is quite important, because it is originally about preprojective components of AR-quiver).
  8. Certain quasi-hereditary algebras arising from Lie theory/type A representation theory. The AR-quviers are known to some people, such as Ringel, Reiten, Miemietz, Madsen, and probably Koenig and Xi, and some other Chinese scholars influenced by the work of Xi whose name I forgot. But I doubt if any of these are written down explicitly anywhere.
  9. I think Julian Külshammer also has some works on the AR-quiver of Frobenius kernel of Lie algebra, but not entirely sure (see comments).

I should also point out that the term 'quantum group' is misleading, I think you mean small quantum group whenever you said quantum group in your question? There is probably some hope in working in this direction, because small quantum groups are finite dimensional algebras. Apart from Kulshammer, Mazorchuk and H.H.Anderson probably also know something on this issue. In the case of 'other quantum group', I think studying AR-quiver would be incredibly difficult. I may be wrong though. However, there is something done in this direction by concerning the BGG category $\mathcal{O}$ instead, which is what I mentioned in point 8 above.

$\endgroup$
2
  • 1
    $\begingroup$ I think your type 5. result is not correct. It should be $\mathbb{Z}A_\infty$ or $\mathbb{Z}A_\infty/(\tau^m)$ for some $m$. But this is not known for many self-injective algebras, just some (such as group algebras) or for particular components for some algebras (such as restricted enveloping algebras (by work of Farnsteiner)). Concerning 9. there are some results in arxiv.org/abs/1201.5303, but again not all the components are described just some. $\endgroup$ May 23, 2013 at 17:25
  • $\begingroup$ Thanks Julian! I have corrected it. $\endgroup$
    – Aaron
    May 24, 2013 at 10:46

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .