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  "body": "\n## Statement\n\nLet $K$ be a field and let $H$ be a finite-dimensional [[Hopf algebra|7XFYP0N84H]] over $K$.\nThen:\n\n1. The space of left [[integrals|RJA5026RGX]] in $H$ is one-dimensional over $K$.\n2. $H$ is a [[Frobenius algebra|8EVP1K2CZH]].\n\n## Remarks\n\nThe proof applies the fundamental theorem of [[Hopf modules|N8X7XJBKT0]] to the dual space $H^*$, which is a Hopf module over $H$.\nThis gives an isomorphism $H^* \\cong (H^*)^{\\mathrm{co}H} \\otimes H$, and the coinvariants of $H^*$ are the left integrals on $H$.\nComparing dimensions shows that this space is one-dimensional, and the isomorphism $H^* \\cong H$ of $H$-modules is the Frobenius property; applying this to $H^*$ gives the statement about integrals in $H$.\nExample: for a finite group $G$, the left integrals in $KG$ are the multiples of $\\sum_{g \\in G} g$.\nIndeed, if $\\Lambda = \\sum_g a_g g$ satisfies $h \\Lambda = \\Lambda$ for all $h \\in G$, then all $a_g$ are equal.\nWithout finite dimension the conclusion fails: in $K[x]$ with $x$ primitive the only integral is $0$.\nThe same work shows that the antipode of a finite-dimensional Hopf algebra is bijective.\n\n## Explanations\n\n### L1 human\n\nAn integral is a Hopf algebra version of the sum of all elements of a finite group.\nThe theorem says that, in a finite-dimensional Hopf algebra, such an element always exists and is unique up to a scalar.\nIt also says that the algebra is Frobenius, a strong symmetry property: as a module over itself it looks like its own dual.\n\n### L2 human\n\nThe theorem is the starting point for the structure theory of finite-dimensional Hopf algebras.\nTogether with it, Larson and Sweedler proved the Hopf algebra form of Maschke's theorem: $H$ is semisimple if and only if $\\varepsilon(\\Lambda) \\ne 0$ for a nonzero left integral $\\Lambda$.\nFor $H = KG$ this gives $\\varepsilon(\\sum_g g) = \\abs{G}$, and so recovers the classical condition that the characteristic of $K$ does not divide $\\abs{G}$.\nIntegrals also define the distinguished group-like elements of $H$ and $H^*$, which appear in Radford's formula for $S^4$.\n",
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