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  "body": "\n## Definition\n\nLet $K$ be a field.\nA *bialgebra* over $K$ is a vector space $B$ over $K$ with linear maps $m \\colon B \\otimes B \\to B$, $\\eta \\colon K \\to B$, $\\Delta \\colon B \\to B \\otimes B$ and $\\varepsilon \\colon B \\to K$ such that:\n\n1. $(B, m, \\eta)$ is an [[associative algebra|3CTW1P4HBP]] with multiplication $m$ and unit $\\eta(1)$.\n2. $(B, \\Delta, \\varepsilon)$ is a [[coalgebra|A06RA8KB0T]].\n3. $\\Delta$ and $\\varepsilon$ are algebra homomorphisms, where $B \\otimes B$ has the product $(a \\otimes b)(c \\otimes d) = ac \\otimes bd$.\n\n## Remarks\n\nClause 3 means $\\Delta(ab) = \\Delta(a)\\Delta(b)$, $\\Delta(1) = 1 \\otimes 1$, $\\varepsilon(ab) = \\varepsilon(a)\\varepsilon(b)$ and $\\varepsilon(1) = 1$.\nIt is equivalent to asking that $m$ and $\\eta$ be coalgebra homomorphisms.\nExample: the [[group algebra|MNP2HBWB5Q]] $KG$ of a group $G$, with $\\Delta(g) = g \\otimes g$ and $\\varepsilon(g) = 1$ for $g \\in G$.\nThe check of clause 3 on the basis: $\\Delta(gh) = gh \\otimes gh = (g \\otimes g)(h \\otimes h) = \\Delta(g)\\Delta(h)$ and $\\varepsilon(gh) = 1 = \\varepsilon(g)\\varepsilon(h)$.\nNon-example: on $K[x]$, the coalgebra with $\\Delta(x^n) = \\sum_{k=0}^{n} x^k \\otimes x^{n-k}$ and $\\varepsilon(x^n) = \\delta_{n,0}$ fails clause 3.\nIndeed $\\Delta(x)^2 = (x \\otimes 1 + 1 \\otimes x)^2 = x^2 \\otimes 1 + 2\\, x \\otimes x + 1 \\otimes x^2$, while $\\Delta(x^2) = x^2 \\otimes 1 + x \\otimes x + 1 \\otimes x^2$, and $2 \\ne 1$ in every field.\nWith $\\Delta(x^n) = \\sum_{k} \\binom{n}{k} x^k \\otimes x^{n-k}$ instead, $K[x]$ is a bialgebra.\n",
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