Derived equivalence classification of the cluster-tilted algebras of Dynkin type E

TL;DR

This paper provides a complete classification of derived equivalence classes of cluster-tilted algebras of Dynkin types E6, E7, and E8, using computational methods and explicit standard forms.

Contribution

It introduces an algorithm for classifying these algebras into derived equivalence classes and explicitly describes each class with standard forms and lists of contained algebras.

Findings

67, 416, 1574 algebras classified into 6, 14, 15 classes

Derived equivalence corresponds to equivalent bilinear forms of Cartan matrices

Classification aligns with sequences of 'good' mutations connecting algebras

Abstract

We obtain a complete derived equivalence classification of the cluster-tilted algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and E8 which turn out to fall into 6, 14, 15 derived equivalence classes, respectively. This classification can be achieved computationally and we outline an algorithm which has been implemented to carry out this task. We also make the classification explicit by giving standard forms for each derived equivalence class as well as complete lists of the algebras contained in each class; as these lists are quite long they are provided as supplementary material to this paper. From a structural point of view the remarkable outcome of our classification is that two cluster-tilted algebras of Dynkin type E are derived equivalent if and only if their Cartan matrices represent equivalent bilinear forms over the integers which in turn happens if and only if the two algebras are connected by a sequence of "good" mutations. This is reminiscent of the derived equivalence classification of cluster-tilted algebras of Dynkin type A, but quite different from the situation in Dynkin type D where a far-reaching classification has been obtained using similar methods as in the present paper but some very subtle questions are still open.