Formation of Cluster Algebras
In a continuation of the example we left off on, you may be wondering at this point how any of this has something to do with “rings.” If we take a look at the diagram above and imagine every possible sequence of mutations that could take place, we will notice two things. First, that in this particular example, the mutations will never go “out of control,” as they will end up generating repeats of previous seeds --- cluster algebras with this property are known as being of “finite-type” since the graph relating seeds to one another via mutation (as shown in the most recent diagram) is finite. Second, that certain expressions pop up in seeds in terms of x1 and x2 --- in general, this is always the case (the rational expressions that appear will be in terms of the initial set of variables in the initial seed used), and these expressions are referred to as “cluster variables.”
The set of these cluster variables, X, which is simply the union of all clusters (vertex sets of quivers) generated through this process, forms the cluster algebra. In particular, they form a ring R[X] that is a subset of the field of rational functions F of which the initial set (in this example, {x1, x2}) are members. The other cluster variables that are in terms of these initial variables are rational expressions of them, and are thus also members of F.
Note: I am the author of the image above.
