Cluster algebra

Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002, 2003, 2007). A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions.

Suppose that F is an integral domain, such as the field Q(x₁,...,x_n) of rational functions in n variables over the rational numbers Q.

A cluster of rank n consists of a set of n elements {x, y, ...} of F, usually assumed to be an algebraically independent set of generators of a field extension F.

A seed consists of a cluster {x,y,...} of F, together with an exchange matrix B with integer entries b_x,y indexed by pairs of elements x, y of the cluster. The matrix is sometimes assumed to be skew symmetric, so that b_x,y = –b_y,x. More generally the matrix might be skew symmetrizable, meaning there are positive integers d_x associated with the elements of the cluster such that d_xb_x,y = –d_yb_y,x. It is common to picture a seed as a quiver with vertices the generating set, by drawing b_x,y arrows from x to y if this number is positive. When b_x,y is skew symmetrizable the quiver has no loops or 2-cycles.

A mutation of a seed, depending on a choice of vertex y of the cluster, is a new seed given by a generalization of tilting as follows. Exchange the values of b_x,y and b_y,x for all x in the cluster. If b_x,y > 0 and b_y,z > 0 then replace b_x,z by b_x,yb_y,z + b_x,z. If b_x,y < 0 and b_y,z < 0 then replace b_x,z by -b_x,yb_y,z + b_x,z. If b_x,yb_y,z <= 0 do not change b_x,z. Finally replace y by a new generator w, where

...
Wikipedia