Return to Question

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 Aglebras, 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?

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?

Let $A = \mathcal{k}Q/I$ be a gentle algebra ($\mathcal{k}$ is algebraically closed). In the paper Auslander-Reiten Sequences with Few Middle Terms and Application to String Aglebras, 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 $M(b,k,n)$ where $k \in \mathcal{k}^*$ 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?

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 Aglebras, 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?

Let $A = \mathcal{k}Q/I$ be a gentle algebra (let $\mathcal{k}$ be algebraically closed). In the following paper:

https://pub.uni-bielefeld.de/download/1776051/2312059/Ringel_068.pdf

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 $\textit{string module}$ $M(c)$. And for each cyclic string $b$ they produce a family of $\textit{band modules}$ $M(b,k,n)$ where $k \in \mathcal{k}^*$ 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 indecompsible representation $$\mathcal{k}\xrightarrow[1]{\xrightarrow{0}}\mathcal{k}$$

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

Let $A = \mathcal{k}Q/I$ be a gentle algebra ($\mathcal{k}$ is algebraically closed). In the paper Auslander-Reiten Sequences with Few Middle Terms and Application to String Aglebras, 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 $M(b,k,n)$ where $k \in \mathcal{k}^*$ 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?