Use this tag for questions about constructively defined commutative rings equipped with a distinguished set of generators grouped into overlapping subsets of the same finite cardinality.
Suppose that $F$ is an integral domain such as the field $\mathbb Q(x_1, \dots ,x_n)$ of rational functions in $n$ variables over the rational numbers $\mathbb Q.$
A cluster of rank $n$ consists of a set of $n$ elements {$x, y, \ldots$} of $F,$ usually assumed to be an algebraically independent set of generators of a field extension $F.$
A seed consists of a cluster {$x, y, \dots$} 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 as the generating set by drawing $b_{x,y}$ arrows from $x$ to $y$ if that number is positive. If $b_{x,y}$ 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 $b_{x,y}b_{y,z} + b_{x,z}.$ If $b_{x,y} < 0$ and $b_{y,z} < 0,$ then replace $b_{x,z}$ by $-b_{x,y}b_{y,z} + b_{x,z}.$ If $b_{x,y}b_{y,z} \le 0,$ do not change $b_{x,z}.$ Finally, replace $y$ by a new generator $w$ where $$wy=\prod_{t,b_{t,y}>0}t^{b_{t,y}} + \prod_{t,b_{t,y}<0}t^{-b_{t,y}}$$ where the products run through the elements $t$ in the cluster of the seed such that $b_{t,y}$ is positive or negative respectively. The inverse of a mutation is also a mutation, i.e., if $A$ is a mutation of $B,$ then $B$ is a mutation of $A.$
A cluster algebra is constructed from a seed as follows. If we repeatedly mutate the seed in all possible ways, we get a finite or infinite graph of seeds where two seeds are joined if one can be obtained by mutating the other. The underlying algebra of the cluster algebra is the algebra generated by all the clusters of all the seeds in this graph. The cluster algebra also comes with the extra structure of the seeds of this graph.
A cluster algebra is said to be of finite type if it has a finite number of seeds only.
See Wikipedia article "Cluster algebra" or "What is a Cluster Algebra?" by one of its inventors.
