Clusters and Weaves

Cluster algebras were defined by Sergey Fomin and Andrei Zelevinsky [9] in 2001, with the goal of providing a combinatorial framework for problems related to total positivity and canonical bases in Lie theory.Since then, various connections have been developed between the theory of cluster algebras and many areas of mathematics, including representation theory and categorification, integrable systems, mathematical physics, symplectic and algebraic geometry, and higher Teichmller theory, to name just a few.The goal of this article is to introduce weaves as a technique to construct cluster algebra structures on the coordinate rings of certain algebraic varieties that appear naturally in the study of Legendrian links and in Lie theory, and which generalize many of the motivating examples of cluster algebras.1.1.Cluster algebras.Cluster algebras are a special type of commutative algebras that enjoy nice algebraic and combinatorial properties.In contrast with the commutative algebras we encounter in a first algebra course, or even in an algebraic geometry class, cluster algebras are typically not explicitly given by generators and relations.Rather, cluster algebras are defined to be subalgebras of a field of rational functions ( 1 , , ) specified by a set of generators , known as the set of cluster variables.While the set may be infinite, it can be expressed as a union of distinguished finite subsets of the same cardinality , which is the transcendence degree of the field .Each of these finite subsets is called a cluster.A key property of cluster algebras is that, in order to know all of the cluster variables, it is enough to know a single cluster and a certain combinatorial datum together with a rule, called mutation, for how to get all the other such pairs from it.