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  "body": "\n## Definition\n\nLet $K$ be a field and let $(H, m, \\eta, \\Delta, \\varepsilon)$ be a [[bialgebra|Q9HHVN606G]] over $K$.\nAn *antipode* for $H$ is a [[linear map|0NRYHJPQG2]] $S \\colon H \\to H$ such that\n$$m \\circ (S \\otimes \\id_H) \\circ \\Delta = \\eta \\circ \\varepsilon = m \\circ (\\id_H \\otimes S) \\circ \\Delta.$$\n\n## Remarks\n\nThe *convolution* of linear maps $f, g \\colon H \\to H$ is $f * g = m \\circ (f \\otimes g) \\circ \\Delta$.\nIt makes $\\End_K(H)$ an associative algebra with identity $\\eta \\circ \\varepsilon$, and the condition says $S * \\id_H = \\eta \\circ \\varepsilon = \\id_H * S$.\nSo $S$ is a two-sided inverse of $\\id_H$, and an antipode is unique when it exists.\nExample: $H = K[x]$ with $\\Delta(x) = x \\otimes 1 + 1 \\otimes x$ and $\\varepsilon(x) = 0$ has the antipode $S(x^n) = (-x)^n$.\nOn $x$ the check is $S(x) \\cdot 1 + S(1) \\cdot x = -x + x = 0 = \\varepsilon(x) 1$.\nNon-example: the identity map of $K[x]$ is not an antipode, since $m(\\id \\otimes \\id)\\Delta(x) = 2x \\ne 0$ when the characteristic of $K$ is not $2$.\nAn antipode is an algebra anti-homomorphism, $S(ab) = S(b)S(a)$, and a coalgebra anti-homomorphism.\nIf $H$ is commutative or cocommutative, then $S \\circ S = \\id_H$.\n",
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