Cluster algebras were introduced by Fomin and Zelevinsky in a series of papers, the first of which appeared in 2002. A cluster algebra is a commutative algebra, generated by a (typically infinite) set of elements, called "cluster variables", which are grouped into overlapping sets, called "clusters", of a fixed size, which is called the "rank" of the cluster algebra.

There is a fairly wide list of topics related to cluster algebras, including:

  • total positivity in algebraic groups
  • canonical bases in homogeneous co-ordinate rings of homogeneous varieties
  • Poisson geometry
  • Teichmüller theory
  • representation theory of finite-dimensional algebras
  • combinatorics of Coxeter groups
  • Donaldson-Thomas theory
  • the Thermodynamic Bethe Ansatz

There are also non-commutative versions of cluster algebras, called quantum cluster algebras.

A number of excellent surveys on cluster algebras have been written. Here are links to the ICM 2010 talks by Sergey Fomin and Bernard Leclerc.

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