$n$-cluster tilting subcategories for radical square zero algebras

TL;DR

This paper characterizes radical square zero bound quiver algebras that admit $n$-cluster tilting subcategories, revealing a lattice structure related to the quiver's type and divisors of an integer.

Contribution

It provides a characterization of $n$-cluster tilting subcategories for radical square zero algebras based on the quiver's structure and describes the lattice of such subcategories.

Findings

Characterization of algebras admitting $n$-cluster tilting subcategories.

Lattice structure of $n$-cluster tilting subcategories for non-cyclically oriented extended Dynkin quivers.

Isomorphism of the lattice to the opposite of divisors of an integer.

Abstract

We give a characterization of radical square zero bound quiver algebras $\mathbf{k} Q/\mathcal{J}^2$ that admit $n$-cluster tilting subcategories and $n\mathbb{Z}$-cluster tilting subcategories in terms of $Q$. We also show that if $Q$ is not of cyclically oriented extended Dynkin type $\tilde{A}$, then the poset of $n$-cluster tilting subcategories of $\mathbf{k} Q/\mathcal{J}^2$ with relation given by inclusion forms a lattice isomorphic to the opposite of the lattice of divisors of an integer which depends on $Q$.