$\tau$-slice algebras of $n$-translation algebras and quasi $n$-Fano algebras

TL;DR

This paper explores the structure and properties of $ au$-slice algebras related to $n$-translation algebras, including their tilts, mutations, and classifications as quasi $n$-Fano, with implications for their Auslander-Reiten quivers.

Contribution

It introduces a recursive construction of higher quasi Fano algebras and studies the $ au_n$-closure and $ u_n$-closure of these algebras, linking their AR quivers to specific truncations.

Findings

Dual $ au$-slice algebras are quasi $(n-1)$-Fano when the algebra is Koszul.

The $ au_n$-closure's AR quiver is a truncation of $ extbf{Z}|_n Q^{ot}$.

The $ u_n$-closure's AR quiver is $ extbf{Z}|_n Q^{ot}$ for $n$-representation infinite cases.

Abstract

In this paper, we show that the $n$-APR tilts of dual $\tau$-slice algebras of acyclic stable $n$-translation algebras are realized as $\tau$-mutations. Such dual $\tau$-slice algebras are quasi $(n-1)$-Fano when the $n$-translation algebra is Koszul, and a recursive construction of higher quasi Fano algebras for quasi $n$-Fano algebra obtained in this way is given. The $\tau_n$-closure and $\nu_n$-closure of such algebras are studied and we show that for an acyclic dual $n$-translation algebras with bound quiver $Q^{\perp}$, the Auslander-Reiten quivers of its $\tau_n$-closures are truncation of the quiver $\mathbb{Z}|_n Q^{\perp}$, and the Auslander-Reiten quiver of its $\nu_n$-closure is $\mathbb{Z}|_n Q^{\perp}$ when it is $n$-representation infinite.