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  "body": "\n## Definition\n\nLet $K$ be a field and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ with counit $\\varepsilon$.\nA *left integral* in $H$ is an element $\\Lambda \\in H$ such that $h \\Lambda = \\varepsilon(h) \\Lambda$ for all $h \\in H$.\nA *right integral* in $H$ is an element $\\Lambda \\in H$ such that $\\Lambda h = \\varepsilon(h) \\Lambda$ for all $h \\in H$.\n\n## Remarks\n\nThe left integrals form a subspace of $H$, often written $\\int_H^l$.\nExample: for a finite group $G$, the element $\\Lambda = \\sum_{g \\in G} g$ of the [[group algebra|MNP2HBWB5Q]] $KG$ is a left and right integral.\nThe check: for $h \\in G$, $h \\Lambda = \\sum_{g} hg = \\Lambda = \\varepsilon(h) \\Lambda$, since $g \\mapsto hg$ is a bijection of $G$; linearity gives all $h \\in KG$.\nNon-example: in $KG$ with $G \\ne 1$, the element $1$ is not an integral, because $g \\cdot 1 = g \\ne 1 = \\varepsilon(g) 1$ for $g \\ne 1$.\nIn $K[x]$ with $\\Delta(x) = x \\otimes 1 + 1 \\otimes x$ and $\\varepsilon(x) = 0$, a left integral must satisfy $x \\Lambda = 0$, so $0$ is the only integral.\nSome authors call these elements integrals *in* $H$, and call integrals *on* $H$ the linear forms $\\lambda \\colon H \\to K$ with $\\sum h_{(1)} \\lambda(h_{(2)}) = \\lambda(h) 1$.\n",
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