Extensions of simple modules over Leavitt path algebras

TL;DR

This paper constructs explicit projective resolutions for Chen simple modules over Leavitt path algebras associated with infinite paths in directed graphs, and describes extension groups, revealing the existence of indecomposable modules of any finite length.

Contribution

It provides explicit projective resolutions for Chen simple modules over Leavitt path algebras and characterizes their extension groups, advancing understanding of module structure.

Findings

Explicit projective resolutions for modules from rational and irrational paths.

Description of Ext^1 groups between simple modules.

Existence of indecomposable modules of arbitrary finite length.

Abstract

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.