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  "body": "\n## Summary\n\nThe Yang–Baxter equation is the identity $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ for a linear map $R$ of $V \\otimes V$.\nThe indices say on which two of three tensor factors $R$ acts.\nIt first appeared in exactly solvable models of statistical mechanics and in quantum scattering in one dimension.\nComposing with the flip turns a solution into a solution of the braid relation.\nSo an invertible solution gives representations of the [[braid group|BNR4H5GCTP]].\nThe universal R-matrix of a [[quasitriangular Hopf algebra|NCRPWS9W5R]] satisfies it, and [[quantum groups|AP2ZDXANKC]] are a rich source of solutions.\n\n## Explanations\n\n### L1 human\n\nPicture three particles on a line that pass each other, and let $R$ describe how two of them interact when they cross.\nThe three particles can be brought from one order to the reverse order in two different sequences of pairwise crossings.\nThe Yang–Baxter equation says that both sequences give the same result.\nIn pictures of braids, it says that a strand can slide across the crossing of two other strands.\nThe plain swap of tensor factors is the simplest solution.\n",
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