On representations of algebras with radical square zero

TL;DR

The paper addresses representations of algebras with radical square zero and introduces a new correspondence with species representations. It constructs a functorial bridge between the stable category $A\text{-}{\underline{Mod}}$ and representations of a double/bipartite species $\Gamma$, via minimal projective/injective presentations, enabling the transfer of Auslander-Reiten theory to the species setting. A key result is that the Auslander-Reiten quiver of $A$ can be reconstructed from the AR-quiver of $\Gamma$, providing a practical method to study modules through the (simpler) species framework. The work includes explicit descriptions and examples illustrating how AR-components transform under this correspondence and how to recover $\mathrm{AR}(A)$ from $\mathrm{AR}(\Gamma)$, with dual statements for the related $\Gamma^*$ construction.

Abstract

We consider a new correspondence between representations of algebras with radical square zero and representations of species. We show that the stable category of representations of such algebra embeds into the representation category of the corresponding species and show how one reconstruct the Auslander-Reiten quiver of the algebra from the Auslander-Reiten quiver of the species.

Theorems & Definitions (26)

• Definition 1.1
• Definition 1.2
• Proposition 1.3
• proof : The proof is evident
• Remark
• Theorem 2.1
• proof
• Theorem 2.1a
• Lemma 3.1
• proof