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  "body": "\n## Summary\n\nA cluster algebra is a [[commutative ring|0310REJJY8]] inside a field of rational functions, generated by special elements called cluster variables.\nThe cluster variables are grouped into overlapping sets of the same size, the clusters.\nOne passes from one cluster to the next by a rule called mutation.\nCluster algebras were introduced by Fomin and Zelevinsky.\nThey link the [[representation theory of quivers|TXVWFWP6VQ]], total positivity, Teichmüller theory and combinatorics.\n\n## Explanations\n\n### L1 human\n\nStart with a few variables and a table of integers that says how the variables interact.\nMutation replaces one variable by a new rational function of the others, built from the table, and also updates the table.\nRepeating mutations in all directions produces a family of variables, possibly infinite, and the cluster algebra is the ring they generate.\nA surprising fact is that every new variable is a Laurent polynomial in the starting variables, so the divisions always cancel.\nFor 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.\n",
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