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  "body": "\n## Definition\n\nLet $K$ be a field.\nA *Hopf algebra* over $K$ is a [[bialgebra|Q9HHVN606G]] $(H, m, \\eta, \\Delta, \\varepsilon)$ over $K$ together with a linear map $S \\colon H \\to H$ such that\n$$m \\circ (S \\otimes \\id_H) \\circ \\Delta = \\eta \\circ \\varepsilon = m \\circ (\\id_H \\otimes S) \\circ \\Delta.$$\nThe map $S$ is the [[antipode|PG3ECG5D0G]] of $H$.\n\n## Remarks\n\nExample: the [[group algebra|MNP2HBWB5Q]] $KG$ of a group $G$, with $\\Delta(g) = g \\otimes g$, $\\varepsilon(g) = 1$ and $S(g) = g^{-1}$.\nThe check on a basis element $g$: $m(S \\otimes \\id)\\Delta(g) = g^{-1} g = 1 = \\varepsilon(g) 1$, and likewise $g g^{-1} = 1$.\nExample: the universal enveloping algebra of a Lie algebra $\\mathfrak{g}$, with $\\Delta(x) = x \\otimes 1 + 1 \\otimes x$, $\\varepsilon(x) = 0$ and $S(x) = -x$ for $x \\in \\mathfrak{g}$.\nNon-example: the monoid algebra $K[x]$ of $(\\N, +)$, with $\\Delta(x) = x \\otimes x$ and $\\varepsilon(x) = 1$, is a bialgebra but not a Hopf algebra.\nAn antipode would need $S(x) x = \\varepsilon(x) 1 = 1$, but $x$ has no inverse in $K[x]$.\nThe antipode of a Hopf algebra is unique, because it is the inverse of $\\id_H$ in the convolution algebra $\\End_K(H)$.\n",
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