Cluster algebras and derived categories

TL;DR

This survey explores the combinatorial and categorical aspects of cluster algebras, highlighting their structure, examples, and the role of derived categories and Ginzburg algebras in their categorification.

Contribution

It provides an overview of cluster algebra theory and introduces a categorification framework using derived categories of Ginzburg algebras, connecting combinatorics with categorical proofs.

Findings

Cluster variables, coefficients, c-vectors, and g-vectors are interconnected.

Categorification leads to proofs of Fomin-Zelevinsky conjectures.

Quantum cluster algebras relate to the quantum dilogarithm.

Abstract

This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coefficients, c-vectors and g-vectors. We show how c-vectors appear in the study of quantum cluster algebras and their links to the quantum dilogarithm. We then present the framework of additive categorification of cluster algebras based on the notion of quiver with potential and on the derived category of the associated Ginzburg algebra. We show how the combinatorics introduced previously lift to the categorical level and how this leads to proofs, for cluster algebras associated with quivers, of some of Fomin-Zelevinsky's fundamental conjectures.