Representations and corepresentations of $p$-equipped posets

Abstract

For $p$ a prime number and $\mathcal{P}$ 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.