Rooted tree modules

Abstract

A rooted tree module (RTM) $M:=M(T,F)$ over a zero-relation algebra $\hat{I}›:=\mathcal{K}Q/⟨\ddot{I}⟩$ 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 $⟨\ddot{I}⟩$. When $\mathrm{char}(\mathcal{K})\ne 2$, we provide a checkable combinatorial characterization of the indecomposability of the RTM $M$ in terms of non-existence of idempotent quiver morphisms ${\hat{I}}^1:T\to T$ satisfying $F\circ {\hat{I}}^1=F$ and ${\hat{I}}^1\ne 1_T$. Further, we provide an iterative method to decompose an RTM into indecomposable RTMs as well as a method to recursively construct indecomposable RTMs.

15 pages