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  "body": "\n## Definition\n\nLet $(H, R)$ be a [[quasitriangular Hopf algebra|NCRPWS9W5R]] over a field $k$, with coproduct $\\Delta$, counit $\\varepsilon$ and antipode $S$.\nWrite $R = \\sum_i a_i \\otimes b_i$, let $R_{21} = \\sum_i b_i \\otimes a_i$, and let $u = \\sum_i S(b_i)\\, a_i$ be the Drinfeld element.\nA *ribbon element* of $H$ is a central invertible element $v \\in H$ such that:\n\n1. $v^2 = u\\, S(u)$.\n2. $S(v) = v$.\n3. $\\varepsilon(v) = 1$.\n4. $\\Delta(v) = (R_{21} R)^{-1} (v \\otimes v)$.\n\nA *ribbon Hopf algebra* is a quasitriangular Hopf algebra together with a ribbon element.\n\n## Remarks\n\nExample: let $H = k[G]$ be the group algebra of a group $G$ with $R = 1 \\otimes 1$.\nThen $u = S(1) \\cdot 1 = 1$, and $v = 1$ is central and invertible.\nClauses 1 to 3 read $1 = 1$, and clause 4 reads $1 \\otimes 1 = (1 \\otimes 1)^{-1}(1 \\otimes 1)$.\nNon-example: in the same algebra with $\\operatorname{char} k \\ne 2$, the element $v = -1$ satisfies clauses 1 and 2 but fails clause 3, since $\\varepsilon(-1) = -1$.\nThe Drinfeld element $u$ is invertible and satisfies $S^2(x) = u x u^{-1}$ for all $x \\in H$.\nSome authors use $v^{-1}$ in place of $v$, which changes clause 4 to $\\Delta(v) = (v \\otimes v) R_{21} R$.\n",
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