Cluster algebras were introduced by Fomin and Zelevinsky in a seminal paper published at the beginning of this century (Journal of the AMS 15, (2002), 497–529) as commutative rings with unit defined by a distinguished set of generators (the so-called, cluster variables), grouped into overlapping sets (clusters) of the same finite cardinality (its rank). Usually, an algebra is defined by listing a complete set of its generators and relations. What distinguishes a cluster algebra from the very beginning is that instead of giving all its generators, a distinguished finite set of them is given together with an iterative process to produce the rest of them.
The initial data, the seed, includes the cluster variables, and the iterative process or mutation is codified in an exchange matrix. For the sake of simplicity one may assume that a cluster algebra of rank n is a subring of a given field of rational functions in n variables generated by all cluster variables. This very definition emphasizes the geometric and combinatorial nature of a cluster algebra, which is not surprising if one remembers that cluster algebras were defined in the context of Lie theory. Thus, examples of cluster algebras include coordinate rings of Grassmannians, Lagrangian-Grassmannians, and several other flag varieties important in the representation theory of algebraic groups.
There are several connections of cluster algebras to other subjects in mathematics. Perhaps the most fruitful of these is the connection to the representation theory of finite dimensional algebras, as first noticed in the influential paper “Generalized associahedra via quiver representations” (Trans. A.M.S. 355 (2003) 4171-4186) by the author of the monograph under review in joint work with M. Reineke and A. Zelevinsky. An exposition based on this approach is given in the recent paper “Cluster algebras: an introduction” by L. Williams (Bull. A.M.S. 51, 2014, 1-26), based on her talk on the Current Events Session of the AMS in January 2013 in San Diego. By a natural approach, going from categories of modules over hereditary algebras, cluster categories were introduced to categorify the essential components in the definition of a cluster algebra in some important cases, basically replacing some components of the definition by a corresponding categorical concept which captures the essence of the component.
I must also mention the intimate connection of cluster algebras and Teichmüller theory, in particular with Riemann surfaces with a set of marked points, and their connection with Poisson geometry, already exposed in the recent monograph Cluster algebras and Poisson geometry by M. Gekhtman and M. Shapiro (AMS, 2010).
This roll-call is only intended to show the variety of diverse subjects where cluster algebras play a fundamental role. Thus, the task of presenting an introduction to this relatively new field in mathematics involves making several choices. In the book under review, the author’s main goals are to give an accessible and motivated introduction to the theory of cluster algebras, within the reach of graduate students or researchers working in related fields, and to give a glimpse on how cluster algebras interact with other areas of mathematics.
The basic notions and properties of cluster algebras are the content of the first three chapters. Here the emphasis is on motivation, and many explicit examples are discussed at length. Proofs are given of many of the statements, but the reader is assumed to be willing to fill in some details or go the detailed references to find a proof of a resulted being quoted. The next three chapters treat the finite type classification theory in terms of the ubiquitous Dynkin diagrams and a simplicial complex associated to a cluster algebra of finite type.
The last three chapters are devoted to illustrating how cluster algebras arise in some other areas of mathematics. From the Zamolodchikov periodicity conjecture in mathematical physics to quivers associated to marked Riemann surfaces, and to the cluster algebra structure of the coordinate ring of the Grassmannian variety.
As may be gathered from the above remarks, this is a beautiful subject, simple from its definition, and yet so basic that its interactions with many diverse fields of mathematics seem very natural.
The book under review succeeds in all its goals: It is timely introduction to a fast growing field of mathematics that was discovered in the first years of this century; it also shows some of the beautiful and deep connections with other important areas of mathematics. All of these in just 100 pages!
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.