---
tag: TKK0TV2MYC
kind: concept
name: Cluster algebra
msc: [13F60, 16G20]
specializes: [0310REJJY8]
sources:
  - url: https://en.wikipedia.org/wiki/Cluster_algebra
same_as:
  wikidata: Q5136690
  wikipedia: Cluster algebra
---

## Summary

A cluster algebra is a [[commutative ring|0310REJJY8]] inside a field of rational functions, generated by special elements called cluster variables.
The cluster variables are grouped into overlapping sets of the same size, the clusters.
One passes from one cluster to the next by a rule called mutation.
Cluster algebras were introduced by Fomin and Zelevinsky.
They link the [[representation theory of quivers|TXVWFWP6VQ]], total positivity, Teichmüller theory and combinatorics.

## Explanations

### L1 human

Start with a few variables and a table of integers that says how the variables interact.
Mutation replaces one variable by a new rational function of the others, built from the table, and also updates the table.
Repeating mutations in all directions produces a family of variables, possibly infinite, and the cluster algebra is the ring they generate.
A surprising fact is that every new variable is a Laurent polynomial in the starting variables, so the divisions always cancel.
For two variables with the simplest table, mutation gives $x_1$, $x_2$, $(x_2 + 1)/x_1$, $(x_1 + x_2 + 1)/(x_1 x_2)$, $(x_1 + 1)/x_2$, and then the list repeats.