Representations and corepresentations of $p$-equipped posets

TL;DR

This paper constructs two classes of right-peak algebras from $p$-equipped posets and explores their module categories, establishing a correspondence between their Auslander-Reiten components despite differences in almost split sequences.

Contribution

It introduces two new right-peak algebras associated with $p$-equipped posets and describes the relationship between their module categories and Auslander-Reiten components.

Findings

Established bijective correspondence between Auslander-Reiten components

Described the shape differences of almost split sequences

Constructed new algebras from $p$-equipped posets

Abstract

For $p$ a prime number and $\mathscr{P}$ a $p$-equipped finite partially ordered set we construct two different right-peak algebras (in the sense of \cite{KS}) $\Lambda^{(r)}$ and $\Lambda^{(c)}$. We consider the category $\mathcal{U}^{(r)}$ $\left(\mathcal{U}^{(c)}\right)$ consisting of the finitely generated right $\Lambda^{(r)}$-modules ($\Lambda^{(c)}$-modules) which are socle-projective. The categories $\mathcal{U}^{(r)}$ and $\mathcal{U}^{(c)}$ have almost split sequences. We describe he Auslander-Reiten components $\mathcal{C}_{\mathcal{U}}^{(r)}$ and $\mathcal{C}_{\mathcal{U}}^{(c)}$ of the corresponding simple projective modules in $\mathcal{U}^{(r)}$ and $\mathcal{U}^{(c)}$. Then we prove that there is a bijective correspondence between $\mathcal{C}_{\mathcal{U}}^{(r)}$ and $\mathcal{C}_{\mathcal{U}}^{(c)}$, although the corresponding almost split sequences have different shapes.