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Cluster algebras, quiver representations and triangulated categories

Bernhard Keller

Abstract

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.

Cluster algebras, quiver representations and triangulated categories

Abstract

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.

Paper Structure

This paper contains 52 sections, 40 theorems, 198 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Contents
  4. An informal introduction to cluster-finite cluster algebras
  5. The classification theorem
  6. The knitting algorithm
  7. Symmetric cluster algebras without coefficients
  8. Quivers
  9. Seeds and mutations
  10. Examples of seed and quiver mutations
  11. Definition of cluster algebras
  12. The example $A_2$
  13. The example $A_3$
  14. Cluster algebras with finitely many cluster variables
  15. Cluster algebras with coefficients
  16. ...and 37 more sections

Key Result

Theorem 3.1

Let $Q$ be a finite connected quiver without loops or $2$-cycles with vertex set $\{1, \ldots, n\}$. Let ${\mathcal{A}}_Q$ be the associated cluster algebra.

Figures (5)

  • Figure 1: The repetition of type $A_n$
  • Figure 2: Some dimension vectors of indecomposables in ${\mathcal{D}}_{\vec{A}_5}$
  • Figure 3: A cluster-tilting set in $A_5$
  • Figure 4: The quiver $\vec{A}_4\otimes\vec{D}_5$
  • Figure 5: The quiver $\vec{A}_4 \square \vec{D}_5$

Theorems & Definitions (41)

  • Theorem 3.1: Fomin-Zelevinsky FominZelevinsky03
  • Proposition 4.1: FominZelevinsky03
  • Theorem 4.2: GekhtmanShapiroVainshtein03
  • Theorem 4.3: GekhtmanShapiroVainshtein03
  • Theorem 4.4: GekhtmanShapiroVainshtein03
  • Theorem 4.5: BerensteinFominZelevinsky05
  • Theorem 5.3: Gabriel Gabriel72
  • Theorem 5.4: Donovan-Freislich DonovanFreislich73, Nazarova Nazarova73
  • Theorem 5.5
  • Corollary 5.6
  • ...and 31 more