the rational functions $x_k$ defined recursively by $$x_{k+1}x_{k-1} = 1+ x_k^a,\qquad (k\in \Z).$$ The clusters of $\A_\Si$ are the subsets $\{x_k,x_{k+1}\}$, and the cluster monomials are the special elements of the form $$x_k^l x_{k+1}^m,\qquad (k\in\Z,\ l,m\in \N).$$ It turns out that when $a=1$, there are only five different clusters and cluster variables, namely $$x_{5k+1}=x_1,\ \
 x_{5k+2}=x_2,\ \
 x_{5k+3}=\frac{1+x_2}{x_1},\ \
 x_{5k+4}=\frac{1+x_1+x_2}{x_1x_2},\ \
 x_{5k}=\frac{1+x_1}{x_2}.$$ For $a\ge 2$ though, the sequence $(x_k)$ is no longer periodic and $\A_\Si$ has infinitely many cluster variables.

The next theorem summarizes the first deep results of this theory obtained by Fomin and Zelevinsky.

\[thFZ\]

-   Every cluster variable of $\A_\Si$ is a Laurent polynomial with coefficients in $\Z$ in the cluster variables of any single fixed cluster.

-   $\A_\Si$ has finitely many clusters if and only if the mutation class $\CC(\Si)$ contains a seed whose quiver is an orientation of a Dynkin diagram of type $A$, $D$, $E$.

An example from Lie theory {#sect2.2}
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We illustrate Theorem \[thFZ\] (ii) with a prototypical example. Let $G=SL_4$ and denote by $N$ the subgroup of upper unitriangular matrices. In [@BFZ §2.6] explicit initial seeds for a cluster algebra structure in the coordinate ring of the big cell of the 