Rooted tree modules

Abstract

A rooted tree module (RTM) $M:=M(T,F)$ over a zero-relation algebra $Λ:=\mathcal KQ/\langleρ\rangle$ over a field $\mathcal K$ is given by the data of a quiver morphism $F:T\to Q$ from a rooted tree $T$ (either with a source or a sink) taking paths in $T$ to paths in $Q$ not lying in $\langleρ\rangle$. When $\mathrm{char}(\mathcal K)\neq2$, we provide a checkable combinatorial characterization of the indecomposability of the RTM $M$ in terms of non-existence of idempotent quiver morphisms $ι:T\to T$ satisfying $F\circι=F$ and $ι\neq 1_T$. Further, we provide an iterative method to decompose an RTM into indecomposable RTMs as well as a method to recursively construct indecomposable RTMs.