Example of a cluster variety

Question

This question is basically just me asking for something to be either verified or rebutted.

So I'm trying to work with cluster varieties, and no matter how much I look around, I simply am not fortunate enough to come across an example that is simple enough for me to feel I have understood things properly. Thence cometh my question.

A cluster variety is defined as the spectrum of maximal ideals of the corresponding cluster algebra.

The simplest cluster algebra is simply the Laurent polynomial ring in one variable, $\mathbb{F}[x,x^{-1}]$, and since this is isomorphic to $\mathbb{F}[x,y]/(xy-1)$, the cluster variety should simply be $$\{(x,y) \in \mathbb{F}^2 | xy = 1\} \cong \mathbb{F}^{*} .$$ Similarly, in the case of the cluster algebra associated with the quiver of two nodes joined by a single arrow, since we have cluster variables $$x_1 , x_2 , \frac{1+x_1}{x_2} , \frac{1+x_2}{x_1} , \frac{1+x_1 + x_2}{x_1 x_2},$$ the cluster algebra is $\mathbb{F} [x_1 , x_2 , \frac{1+x_1}{x_2} , \frac{1+x_2}{x_1} , \frac{1+x_1 + x_2}{x_1 x_2}] \subset \mathbb{F}[x_1^{\pm 1}, x_2^{\pm 1}]$ which is isomorphic to $\mathbb{F} [x,y,z,v,w] / (xz-y-1,yv-x-1,xyw-x-y-1)$, and so the cluster variety is $$\{ (x,y,z,v,w) \in \mathbb{F}^5 | xz-y= 1, yv-x=1, xyw-x-y=1 \} .$$ Are these two examples accurate, or is there some silly mistake in there somewhere? Look forward to your comments.

Answer 1

There are easy examples of cluster varieties for cluster algebras associated with acyclic quivers, for example trees. In this case, the article "Cluster Algebras III" by Berenstein, Fomin and Zelevinsky contains a theorem to the effect that the cluster algebra has the following finite presentation : it is generated by the cluster variables in the union of one acyclic cluster and the $n$ neighbor clusters, and the relations are the $n$ exchange relations of the fixed initial cluster. In your second example, you would only need 4 variables and 2 relations.

Usually, the cluster variety is strictly larger than the union of the glued tori associated with clusters. This happens for example in the case of the quiver

o --> o --> o ,

where there is an additional singular point.