Cluster algebras: an introduction (Q2864920)

The purpose of this expository paper is to introduce the notion of cluster algebras, which are commutative rings with a set of distinguished generators having a remarkable combinatorial structure, and give several important examples from various domains where the framework of cluster algebra apply.NEWLINENEWLINECluster algebras were introduced by \textit{S. Fomin} and \textit{A. Zelevinsky} [J. Am. Math. Soc. 15, No. 2, 497--529 (2002; Zbl 1021.16017)] as a tool for studying total positivity and dual canonical bases in Lie theory. A cluster algebra of rank \(n\) is a subring of an ambient field of rational functions \(\mathcal{F}\), which is constructed from an initial data, a \textit{seed}, which consists of a set of \(n\) generators together with an exchange matrix. Through an iterative process called \textit{mutation} new elements of \(\mathcal{F}\) called cluster variables, are constructed and the algebra is the subring generated by all these cluster variables.NEWLINENEWLINEThe author first gives a very simple definition of cluster algebras using quivers as initial seed and uses the example of the type A cluster algebras to illustrate the definition. Then a more general definition is given where coefficient variables are introduced and various fundamental properties and conjecture are presented, such as the Laurent phenomenon and the positivity conjecture.NEWLINENEWLINEThe third chapter explains how the framework of cluster algebras appears in Teichmüller theory, through the notions of decorated Teichmüller space and lambda lengths, introduced by \textit{R. C. Penner} [Commun. Math. Phys. 113, 299--339 (1987; Zbl 0642.32012)]. In this setting, the lambda-lengths of ideal arcs in a triangulation of an hyperbolic marked surface play the role of the cluster variables, and the mutation process corresponds to the notion of flip inside a triangulation, as it is explored by \textit{S. Fomin} et al. [Acta Math. 201, No. 1, 83--146 (2008; Zbl 1263.13023)]. Other ideas relating Thurston shear coordinates to coefficients and space of laminations to a tropical version of cluster algebras are sketched.NEWLINENEWLINEThe last section concerns the periodicity conjecture of Zamolodchikov on the thermodynamic Bethe ansatz. Namely the conjecture states that the solutions to the so-called Y-systems related to the Bethe Ansatz equations for diagonal scattering theories, are periodic. The notion of Y-system associated to a Dynkin diagram or a pair of Dynkin diagrams is explained here, and the author introduces the general techniques from the theory of cluster algebras that were used to prove these conjectures.