Some cluster tilting modules for weighted surface algebras

TL;DR

This paper investigates the existence of 3-cluster tilting modules in non-singular weighted surface algebras, establishing conditions under which such modules exist, especially focusing on bipartite Gabriel quivers.

Contribution

It identifies specific conditions for weighted surface algebras to have 3-cluster tilting modules, especially characterizing when these occur in non-singular triangular or spherical cases.

Findings

3-cluster tilting modules exist only in non-singular triangular or spherical weighted surface algebras

Bipartite Gabriel quivers are associated with the existence of certain modules satisfying ext vanishing

Non-singular weighted surface algebras with bipartite quivers do not always admit 3-cluster tilting modules

Abstract

Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext vanishing condition. We show that it is 3-cluster tilting precisely for the non-singular triangular or spherical algebras but not for any other weighted surface algebra with bipartite Gabriel quiver.