Categorification of Acyclic Cluster Algebras: An Introduction

Abstract

Summary. This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category. MSC 2010 classification: 18E30, 16S99

Cluster Multiplication in Regular Components via Generalized Chebyshev Polynomials

“…For a general overview of these notions, we refer, for instance, to the survey [18] and references therein.…”

“…Since then, it appeared that triangulated 2-Calabi-Yau categories provide a fruitful way for categorifying a wide class of cluster algebras (see [18] and references therein). Given such a categorification of a cluster algebra A, one has a notion of cluster character in the sense of Palu, which allows to realize explicitly cluster variables in A from objects in the corresponding category [13,20].…”

Calabi–Yau triangulated categories

“…, n} is the one associated with the matrix B whose (i, j)-coefficient equals the number of arrows from i to j minus the number of arrows from j to i. In [32], the reader can find a translation of Fomin-Zelevinsky's construction of A Q into the quiver language.…”

Introduction to Cluster Algebras

“…These notes are far from exhaustive and the references above only touch on the vast literature. Other overviews of cluster algebras can be found in the works [32,52,21] which will also provide additional references.…”