Group actions on cluster algebras and cluster categories

TL;DR

This paper explores how group actions on cluster algebras and categories produce orbit spaces with new generalized cluster structures, extending previous models and classifying low-rank cases.

Contribution

It introduces admissible group actions on cluster structures, describes the resulting generalized cluster algebras, and classifies low-rank cases, expanding the understanding of symmetries in cluster theory.

Findings

Orbit spaces have generalized cluster algebra structures different from previous models.

Complete classification of exchange polynomials for surface-based cluster algebras.

Classification of rank 1 and 2 algebras under group actions.

Abstract

We introduce admissible group actions on cluster algebras, cluster categories and quivers with potential and study the resulting orbit spaces. The orbit space of the cluster algebra has the structure of a generalized cluster algebra. This generalized cluster structure is different from those introduced by Chekhov-Shapiro and Lam-Pylyavskyy. For group actions on cluster algebras from surfaces, we describe the generalized cluster structure of the orbit space in terms of a triangulated orbifold. In this case, we give a complete list of exchange polynomials, and we classify the algebras of rank 1 and 2. We also show that every admissible group action on a cluster category induces a precovering from the cluster category to the cluster category of orbits. Moreover this precovering is dense if the categories are of finite type.