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

Abstract

Jasso-Külshammer 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 $n>2$. 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.

Figures (2)

• Figure 1: quiver $Q$
• Figure 2: quiver $\dot{Q}$

Theorems & Definitions (108)

• Definition 2.1: Iya11IY08IJ17
• Proposition 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• Definition 2.6
• Remark 2.7
• Proposition 2.8
• Proposition 2.9
• Proposition 2.10