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  "body": "\n## Summary\n\nThe smash product $A \\# H$ is an algebra built from a [[Hopf algebra|7XFYP0N84H]] $H$ and an algebra $A$.\nThe Hopf algebra $H$ acts on $A$ compatibly with the multiplication of $A$.\nIt contains $A$ and $H$ as subalgebras, and the action of $H$ on $A$ becomes part of the multiplication.\nFor the group algebra of a group it is the [[skew group ring|PCKK4SFZEK]].\nFor the enveloping algebra of a Lie algebra acting by derivations it is a differential operator ring.\nSo it treats group actions and actions by derivations in one framework.\n\n## Explanations\n\n### L1 human\n\nA group can act on an algebra by symmetries, and a Lie algebra can act on it by derivations.\nA Hopf algebra is a common generalisation of both, and it can also act on an algebra.\nThe smash product puts the algebra and the Hopf algebra side by side, as $A \\otimes H$.\nWhen an element $h$ of $H$ moves past an element $b$ of $A$, it acts on $b$.\nThis works in the same way that a group element moves past a coefficient in a skew group ring.\n",
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