The dual actions, equivariant autoequivalences and stable tilting objects

TL;DR

This paper investigates the relationship between dual actions of finite abelian groups on linear categories and their equivariant autoequivalences, establishing a bijection in stable tilting objects and applying results to weighted projective lines.

Contribution

It proves an isomorphism between groups of equivariant autoequivalences under dual actions and classifies stable tilting objects in this context, with applications to weighted projective lines.

Findings

Groups of equivariant autoequivalences are isomorphic under dual actions.

Stable tilting objects are in bijection between categories with dual actions.

Applications to stable tilting complexes on weighted projective lines of tubular type.

Abstract

For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant autoequivalences on these two categories are isomorphic. In the triangulated situation, this isomorphism implies that the classifications of stable tilting objects for these two categories are in a natural bijection. We apply these results to stable tilting complexes on weighted projective lines of tubular type.