---
tag: HM9HBVSM54
kind: definition
name: Cluster algebra
msc: [13F60, 16G20]
defines: TKK0TV2MYC
uses: [0310REJJY8, 0XDK9CGYKJ, 0VD9ZSWWJG]
sources:
  - url: https://en.wikipedia.org/wiki/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|0XDK9CGYKJ]] in $n$ indeterminates.
An integer $n \times n$ [[matrix|0VD9ZSWWJG]] $B = (b_{ij})$ 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.

## Remarks

Example: $n = 2$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, which is skew-symmetric.
Mutation at $1$ gives $x'_1 = (1 + x_2)/x_1$, since $b_{21} = -1$ and $b_{11} = 0$.
Alternating mutations give the cluster variables $x_1$, $x_2$, $(x_2 + 1)/x_1$, $(x_1 + x_2 + 1)/(x_1 x_2)$, $(x_1 + 1)/x_2$, five in all.
This is the cluster algebra of type $A_2$.
Non-example: $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is not skew-symmetrizable, because $d_1 b_{12} = -d_2 b_{21}$ would give $d_1 = -d_2$ with $d_1, d_2 > 0$.
Mutation is an involution: mutating twice at the same $k$ returns the original seed.
By the Laurent phenomenon of Fomin and Zelevinsky, every cluster variable is a Laurent polynomial with integer coefficients in the initial cluster.
The general definition also allows coefficients from a semifield.