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.

  • $\begingroup$ The cluster variety is not necessarily the spectrum of the cluster algebra. Rather, it is defined by glueing together toric charts, using the mutation formulas as glueing maps. $\endgroup$ – Nick Jul 31 at 3:34

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