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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 $2$.

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