$nd\mathbb{Z}$-cluster tilting subcategories of $d$-Nakayama algebras

TL;DR

This paper classifies $ndbZ$-cluster tilting subcategories in $d$-Nakayama algebras, identifying conditions for their existence and providing explicit examples, extending prior results on classical Nakayama algebras.

Contribution

It offers a complete classification of $ndbZ$-cluster tilting subcategories in non-self-injective $d$-Nakayama algebras, including explicit constructions.

Findings

Existence of at most one $ndbZ$-cluster tilting subcategory for each algebra.

Such subcategories exist only if $n$ divides $m$ and $n$ divides $l-2$ in self-injective cases.

Explicit example constructed when $n=l-2$.

Abstract

Jasso-K\"{u}lshammer introduced the class of $d$-Nakayama algebras as a higher dimensional analogue of Nakayama algebras. In particular, they are endowed with a distinguished $d\mathbb{Z}$-cluster tilting subcategory. In this paper, we investigate which $d$-Nakayama algebras admit an $nd\mathbb{Z}$-cluster tilting subcategory for $n>1$. The radical square zero case is already covered by results on classical Nakayama algebras due to Herschend-Kvamme-Vaso. For each remaining non-self-injective $d$-Nakayama algebra, we provide a complete classification of its $nd\mathbb{Z}$-cluster tilting subcategories. In fact, there exists at most one for a suitable integer $n$. A self-injective $d$-Nakayama algebra is determined by two positive integers $m$ and $l$. We show that an $nd\mathbb{Z}$-cluster tilting subcategory is only possible if $n|m$ and $n|(l-2)$. In case $n=l-2$, we show that such subcategory does indeed exist by constructing an explicit example.