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  "body": "\n## Definition\n\nLet $V$ be a [[vector space|05EGHEQZAW]] over a field $k$ and let $R \\colon V \\otimes V \\to V \\otimes V$ be a linear map, where $\\otimes$ is the [[tensor product|0BF2DE2SYK]] over $k$.\nDefine linear maps of $V \\otimes V \\otimes V$ by $R_{12} = R \\otimes \\id_V$ and $R_{23} = \\id_V \\otimes R$.\nLet $R_{13}$ act as $R$ on the first and third factors and as the identity on the second.\nIf $\\tau_{23}$ swaps the second and third factors, then $R_{13} = \\tau_{23} R_{12} \\tau_{23}$.\nThe map $R$ satisfies the *(quantum) Yang–Baxter equation* if\n$$R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}.$$\n\n## Remarks\n\nLet $\\tau \\colon V \\otimes V \\to V \\otimes V$ be the flip $u \\otimes v \\mapsto v \\otimes u$ and $c = \\tau \\circ R$.\nThen $R$ satisfies the Yang–Baxter equation if and only if $c$ satisfies the *braid relation* $c_{12} c_{23} c_{12} = c_{23} c_{12} c_{23}$, with $c_{12} = c \\otimes \\id_V$ and $c_{23} = \\id_V \\otimes c$.\nExample: $R = \\tau$.\nOn $u \\otimes v \\otimes w$, the left side sends it through $u \\otimes w \\otimes v$ and $v \\otimes w \\otimes u$ to $w \\otimes v \\otimes u$.\nThe right side sends it through $v \\otimes u \\otimes w$ and $w \\otimes u \\otimes v$ to $w \\otimes v \\otimes u$, so the two sides agree.\nExample and non-example: let $R = A \\otimes B$ for linear maps $A, B \\colon V \\to V$.\nThen $R_{12} R_{13} R_{23} = A^2 \\otimes BA \\otimes B^2$ and $R_{23} R_{13} R_{12} = A^2 \\otimes AB \\otimes B^2$.\nIf $AB = BA$ the equation holds.\nIf $A, B$ are invertible and $AB \\ne BA$, it fails, because $A^2 \\otimes (BA - AB) \\otimes B^2 \\ne 0$.\nVersions with a spectral parameter, $R(u)$ depending on $u \\in \\C$, are also called Yang–Baxter equations.\n",
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