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  "body": "\n## Summary\n\nA bialgebra is a vector space that is both an [[associative algebra|3CTW1P4HBP]] and a [[coalgebra|A06RA8KB0T]] in a compatible way.\nThe compatibility is that the comultiplication and the counit are algebra homomorphisms.\nGroup algebras, monoid algebras, polynomial rings and universal enveloping algebras of Lie algebras are bialgebras.\nThe compatibility lets one form tensor products of modules, which is the key feature of [[Hopf algebras|7XFYP0N84H]] and quantum groups.\n\n## Explanations\n\n### L1 human\n\nIn a bialgebra one can multiply two elements into one and split one element into pairs.\nThe rule linking the two says that splitting a product is the same as splitting each factor and multiplying the pieces.\nFor a group algebra, each group element $g$ splits as $g \\otimes g$, and indeed $gh$ splits as $gh \\otimes gh = (g \\otimes g)(h \\otimes h)$.\nThis structure is what makes it possible to let a group act on a tensor product of two representations at once.\n",
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