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Extensions of simple modules over Leavitt path algebras

Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. For each rational infinite path $c^\infty$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[c^\infty]}$. Further, when $E$ is row-finite, for each irrational infinite path $p$ of $E$ we explicitly construct a projective resolution of the corresponding Chen simple left $L_K(E)$-module $V_{[p]}$. For Chen simple modules $S,T$ we describe ${\rm Ext}_{L_K(E)}^1(S,T)$ by presenting an explicit $K$-basis. For any graph $E$ containing at least one cycle, this description guarantees the existence of indecomposable left $L_K(E)$-modules of any prescribed finite length.