Representations of free products of semisimple algebras via quivers

TL;DR

This paper establishes a correspondence between modules over free products of semisimple algebras and quiver representations, providing criteria for simplicity and semisimplicity, and applies these results to free products of finite groups.

Contribution

It introduces a quiver-based framework to study modules over free products of semisimple algebras, linking module properties to quiver stability conditions.

Findings

Category of modules over free product is equivalent to a subcategory of quiver representations.

Simple modules correspond to stable quiver representations under certain conditions.

Modules in general position are always semisimple.

Abstract

Let $\mathbb{K}$ denote an algebraically closed field and $A$ a free product of finitely many semisimple associative $\mathbb{K}$-algebras. We associate to $A$ a finite acyclic quiver $\Gamma$ and show that the category of finite dimensional $A$-modules is equivalent to a full subcategory of the category ${\rm rep}(\Gamma)$ of finite dimensional representations of $\Gamma$. Under this equivalence, the simple $A$-modules correspond exactly to the $\theta$-stable representations of $\Gamma$ for some stability parameter $\theta$. This gives us necessary conditions for an $A$-module to be simple, conditions which are also sufficient if the module is in general position. Even though there are indecomposable modules that are not simple, we prove that a module in general position is always semisimple. We also discuss the construction of arbitrary finite dimensional modules using nilpotent representations of quivers. Finally, we apply our results to the case of a free product of finite groups when $\mathbb{K}$ has characteristic zero.