Grothendieck Group and Generalized Mutation Rule for 2-Calabi--Yau Triangulated Categories

TL;DR

This paper explores the Grothendieck group of specific 2-Calabi--Yau triangulated categories and generalizes the mutation rule in cluster algebras, linking quiver representations to algebraic structures.

Contribution

It computes the Grothendieck group for certain 2-Calabi--Yau categories and extends the mutation rule in cluster algebra theory.

Findings

Computed Grothendieck group for 2-Calabi--Yau categories

Generalized Fomin--Zelevinsky mutation rule

Established connections between quiver representations and cluster algebras

Abstract

We compute the Grothendieck group of certain 2-Calabi--Yau triangulated categories appearing naturally in the study of the link between quiver representations and Fomin--Zelevinsky's cluster algebras. In this setup, we also prove a generalization of Fomin--Zelevinsky's mutation rule.