Description of modules over self-injective algebras of finite representation type

Question

Is there any description of indecomposable modules and irreducible morphisms over self-injective algebras of finite representation type? I am interested mainly in such a description for nonstandard algebras of tree type $D_{3n}$. Modulo stable equivalence, such an algebra can be represented by:

• a quiver $Q$ whose vertices are $Q_0 = \{0, \ldots, n-1\}$ (considered modulo $n$), and whose arrows are $b\colon 0\to 0$ and $a_i\colon i\to i+1$ for $(0 \leq i \leq n-1)$,

• and an ideal $I$ generated by the elements $a_i\cdots a_0 a_{n-1}\cdots a_i$, $b^2 + a_{n-1}\cdots a_0$ and $a_0 a_{n-1} + a_0 b a_{n-1}$.

The characteristic of the field equals $2$.

Answer 1

Such algebras were named "penny-farthing algebras" by Gabriel. Note first that this algebra is socle equivalent to the "standard" penny-farthing algebras, so the representation theory of the standard and non-standard ones is more or less the same. The article https://eudml.org/doc/152308 summarises the representaiton-theory of this algebra, but without giving too much details and its in German. Chapter 4 of the recent textbook by Skowronski and Yamagata discusse in general how to obtain the Auslander-Reiten quiver of representation-finite selfinjective algebras. It is an exercise there to calculate the Auslander-Reiten quiver for penny farthing algebras with 6 simples modules. Id be interested in a quick solution of this exercise. Full detail solution should take hours to write down with all proofs.