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  "body": "\n## Definition\n\nLet $k$ be a field and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $k$ with coproduct $\\Delta$.\nLet $\\tau \\colon H \\otimes H \\to H \\otimes H$ be the flip $a \\otimes b \\mapsto b \\otimes a$, and let $\\Delta^{\\mathrm{op}} = \\tau \\circ \\Delta$.\nFor $R = \\sum_i a_i \\otimes b_i$ in the [[tensor product|0BF2DE2SYK]] $H \\otimes H$, put $R_{12} = \\sum_i a_i \\otimes b_i \\otimes 1$, $R_{13} = \\sum_i a_i \\otimes 1 \\otimes b_i$ and $R_{23} = \\sum_i 1 \\otimes a_i \\otimes b_i$.\nThe pair $(H, R)$ is a *quasitriangular Hopf algebra* if $R$ is invertible in $H \\otimes H$ and:\n\n1. $R \\, \\Delta(x) \\, R^{-1} = \\Delta^{\\mathrm{op}}(x)$ for all $x \\in H$.\n2. $(\\Delta \\otimes \\id)(R) = R_{13} R_{23}$.\n3. $(\\id \\otimes \\Delta)(R) = R_{13} R_{12}$.\n\nThe element $R$ is the *universal R-matrix*.\n\n## Remarks\n\nExample: let $H = k[G]$ be the group algebra of a group $G$, with $\\Delta(g) = g \\otimes g$, and let $R = 1 \\otimes 1$.\nThen $\\Delta^{\\mathrm{op}} = \\Delta$, so clause 1 holds, and both sides of clauses 2 and 3 equal $1 \\otimes 1 \\otimes 1$.\nNon-example: let $G$ be a finite non-abelian group and let $H = k^G$ be the dual Hopf algebra of functions on $G$.\nSince $k^G$ is commutative, $R \\, \\Delta(x) \\, R^{-1} = \\Delta(x)$ for every invertible $R$, so clause 1 would force $\\Delta = \\Delta^{\\mathrm{op}}$.\nThis fails because $k^G$ is not cocommutative, so no $R$ makes $k^G$ quasitriangular.\nThe axioms imply $(\\varepsilon \\otimes \\id)(R) = 1 = (\\id \\otimes \\varepsilon)(R)$ for the counit $\\varepsilon$.\nThey also imply the Yang–Baxter equation $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$.\nA *triangular* Hopf algebra is a quasitriangular one with $R_{21} R = 1$, where $R_{21} = \\tau(R)$.\n",
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