Algebras with irreducible module varieties II: Two vertex case

TL;DR

This paper characterizes finite-dimensional algebras with two simple modules that have irreducible module varieties, revealing their special structure and limitations on their quivers.

Contribution

It provides a classification of geometrically irreducible algebras with two simple modules and constraints on their Gabriel quivers.

Findings

Algebras with two simple modules are of a specific form.

Minimal geometrically irreducible algebras without shortcuts have at most two simple modules.

The structure of such algebras is explicitly described.

Abstract

We call a finite-dimensional K-algebra A geometrically irreducible if for all d all connected components of the affine scheme of d-dimensional A-modules are irreducible. We prove that a geometrically irreducible algebra with exactly two simple modules has to be of a very special form, which we describe. Based on this result we prove that every minimal geometrically irreducible algebra without shortcuts in its Gabriel quiver has at most two simple modules.