Cluster algebra

Definition

This is the coefficient-free, skew-symmetrizable version. Let $n \ge 1$ and let $\mathcal{F} = \Q(u_1, \dots, u_n)$ be the field of rational functions in $n$ indeterminates. An integer $n \times n$ matrix $B = (b_{ij})$ is skew-symmetrizable if there are positive integers $d_1, \dots, d_n$ with $d_i b_{ij} = -d_j b_{ji}$ for all $i, j$. A seed is a pair $(\mathbf{x}, B)$, where $\mathbf{x} = (x_1, \dots, x_n)$ is a free generating set of $\mathcal{F}$ over $\Q$ and $B$ is skew-symmetrizable. For $k \in \set{1, \dots, n}$, the mutation of $(\mathbf{x}, B)$ at $k$ is the seed $(\mathbf{x}', B')$ given as follows, where $[a]_+ = \max(a, 0)$:

1. $x'_i = x_i$ for $i \ne k$, and $x'_k x_k = \prod_{i} x_i^{[b_{ik}]_+} + \prod_{i} x_i^{[-b_{ik}]_+}$.
2. $b'_{ij} = -b_{ij}$ if $i = k$ or $j = k$.
3. $b'_{ij} = b_{ij} + \tfrac{1}{2}\left(\abs{b_{ik}} b_{kj} + b_{ik} \abs{b_{kj}}\right)$ otherwise.

Fix an initial seed $(\mathbf{x}, B)$. The cluster variables are the entries of all seeds reached from it by finite sequences of mutations, and each such $\mathbf{x}'$ is a cluster. The cluster algebra $\mathcal{A}(B)$ is the subring of $\mathcal{F}$ generated by all cluster variables.