$n$-APR tilting and $\tau$-mutations

Abstract

APR tilts for path algebra $kQ$ can be realized as the mutation of the quiver $Q$ in $\mathbb{Z}Q$ with respect to the translation. In this paper, we show that we have similar results for the quadratic dual of truncations of $n$-translation algebras, that is, under certain condition, the $n$-APR tilts of such algebras are realized as $\tau$-mutations.For the dual $\tau$-slice algebras with bound quiver $Q^{\perp }$, we show that their iterated $n$-APR tilts are realized by the iterated $\tau$-mutations in $\mathbb{Z}|n-1Q^{\perp }$.

This paper is a extended and generalized version of the results concerning $n$-APR tilts in '$\tau$-slice algebras of $n$-translation algebras and quasi $n$-Fano algebras, arXiv:1707.01393, which is discontinued