From groups to clusters

TL;DR

This paper introduces a new class of symmetric tame algebras linked to cluster tilting objects in 2-Calabi-Yau categories, connecting surface triangulations, group algebra blocks, and cluster algebras.

Contribution

It constructs a novel class of symmetric tame algebras using triangulation quivers, bridging modular representation theory and cluster algebra theory.

Findings

Includes algebras of quaternion type

Contains Jacobian algebras from surface triangulations

Modules are $ ext{Ω}$-periodic of period dividing 4

Abstract

We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are $\Omega$-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin-Shapiro-Thurston and Labardini-Fragoso to triangulations of closed surfaces with punctures, hence our construction may serve as a bridge between the modular representation theory of finite groups and the theory of cluster algebras.