Cluster algebras and derived categories

By B. Keller. 2012

160 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9d861b30ba9f225337ca08b9e2fae9778b2a80e9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

Lattice Properties of Oriented Exchange Graphs and Torsion Classes (2015, methodology): “Maximal directed paths of finite length in the oriented exchange graph, known as maximal green sequences [3], may be used to compute the refined Donaldson-Thomas invariant of Q (see [22]) introduced by Kontsevich and Soibelman in [23].”
Cluster Algebras of Grassmannians are Locally Acyclic (2014, methodology): “The specific form of cluster algebras we use in this note is that of skew-symmetric cluster algebras of geometric type, and we define them using ice quivers ; a more thorough introduction to this approach can be found in [Kel12].”