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Let $A = \mathcal{k}Q/I$ be a gentle algebra (where $\mathcal{k}$ is algebraically closed). In the paper Auslander-Reiten Sequences with Few Middle Terms and Application to String Algebras, Butler and Ringel show that string and band modules classify the indecomposable modules of $A$ (pages 157–161). To flesh out the details a little more, for each string $c$ of $Q$ they produce a string module $M(c)$. And for each cyclic string $b$ they produce a family of band modules $M(b,x,n)$ where $x \in \mathcal{k}^*$ and $n \geq 1$.

I am trying to compare this to the classification of indecomposable representations of the $2$-Kronecker quiver. But as an example I don't see where the indecomposable representation

$$\mathcal{k}\overset{0}{\underset{1}{\rightrightarrows}} \mathcal{k}$$

appears in Butler and Ringel's classification. What am I missing?

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If the $2$-Kronecker quiver is

$$ \overset 1\circ\overset{\alpha}{\underset{\beta}{\rightrightarrows}} \overset 2\circ ,$$

then the representation

$$\mathcal{k}\overset{0}{\underset{1}{\rightrightarrows}} \mathcal{k}$$

corresponds to the string module $M(\beta)$.

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