2025/04/26 by Sengupta, Annoy, Kuber, Amit · 1 citation
#16G20 #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2504.18996
For a zero-relation algebra over a field \mathcal K, Crawley-Boevey introduced the concept of a tree module and provided a combinatorial description of a basis for the space of homomorphisms between two tree modules--the basis elements are called graph maps. The indecomposability of tree modules is essentially due to Gabriel. We relax a condition in the definition of a tree module to define generalised tree modules and when char(\mathcal K)≠2, under a certain condition, provide a combinatorial description of a finite generating set for the space of homomorphisms between two such modules--we call the generators generalised graph maps. As an application, we provide a sufficient condition for the (in)decomposability of certain generalised tree modules. We also show that all indecomposable modules over a Dynkin quiver of type \mathbf D are isomorphic to generalised tree modules--this result also follows from a theorem of Ringel which states that all exceptional modules over the path algebra \mathcal KQ of a finite quiver Q are generalised tree modules.