Mutating loops and 2-cycles in 2-CY triangulated categories

TL;DR

This paper develops an algorithm for mutating quivers with loops and 2-cycles in 2-Calabi-Yau categories and classifies related algebras arising from finite 2-CY triangulated categories.

Contribution

It introduces a mutation algorithm for complex quivers with loops and 2-cycles and classifies associated 2-CY tilted algebras from finite categories.

Findings

Algorithm for mutating quivers with loops and 2-cycles

Classification of 2-CY tilted algebras from finite categories

Conditions under which the mutation algorithm applies

Abstract

We derive an algorithm for mutating quivers of 2-CY tilted algebras that have loops and 2-cycles, under certain specific conditions. Further, we give the classification of the 2-CY tilted algebras coming from standard algebraic 2-CY triangulated categories with a finite number of indecomposables. These form a class of algebras that satisfy the setup for our mutation algorithm.