$\tau$-slice algebras of $n$-translation algebras and quasi $n$-Fano algebras
Jin Yun Guo, Cong Xiao

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 -APR tilts of dual -slice algebras of acyclic stable -translation algebras are realized as -mutations. Such dual -slice algebras are quasi -Fano when the -translation algebra is Koszul, and a recursive construction of higher quasi Fano algebras for quasi -Fano algebra obtained in this way is given. The -closure and -closure of such algebras are studied and we show that for an acyclic dual -translation algebras with bound quiver , the Auslander-Reiten quivers of its -closures are truncation of the quiver , and the Auslander-Reiten quiver of its -closure is when it is -representation infinite.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Advanced Topics in Algebra
