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.