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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $k$ with comultiplication $\\Delta$ and counit $\\varepsilon$.\nWrite $\\Delta(h) = \\sum h_{(1)} \\otimes h_{(2)}$ in [[Sweedler notation|NC26CE0Y8B]].\nA *left $H$-module algebra* is an [[associative algebra|3CTW1P4HBP]] $A$ over $k$ with identity that is also a left $H$-[[module|0NZ8JA25WK]], written $h \\cdot a$, such that:\n\n1. $h \\cdot (ab) = \\sum (h_{(1)} \\cdot a)(h_{(2)} \\cdot b)$ for all $h \\in H$ and $a, b \\in A$.\n2. $h \\cdot 1_A = \\varepsilon(h) 1_A$ for all $h \\in H$.\n\nThe *smash product* $A \\# H$ is the vector space $A \\otimes_k H$, with $a \\otimes h$ written $a \\# h$, and with the multiplication\n$$(a \\# h)(b \\# g) = \\sum a \\, (h_{(1)} \\cdot b) \\# h_{(2)} g.$$\n\n## Remarks\n\nWith this product $A \\# H$ is an associative algebra with identity $1 \\# 1$, and $a \\mapsto a \\# 1$ and $h \\mapsto 1 \\# h$ embed $A$ and $H$ as subalgebras.\nExample: let $G$ be a group acting on $A$ by algebra automorphisms and let $H = kG$, with $\\Delta(g) = g \\otimes g$ and $\\varepsilon(g) = 1$.\nThen clause 1 reads $g \\cdot (ab) = (g \\cdot a)(g \\cdot b)$, clause 2 reads $g \\cdot 1 = 1$, and the product is $(a \\# g)(b \\# h) = a (g \\cdot b) \\# gh$, so $A \\# kG$ is the [[skew group ring|PCKK4SFZEK]] $A * G$.\nExample: if $H = U(\\mathfrak g)$ for a Lie algebra $\\mathfrak g$, with $\\Delta(x) = x \\otimes 1 + 1 \\otimes x$ for $x \\in \\mathfrak g$, clause 1 says that each $x$ acts by a derivation of $A$.\nNon-example: let $G = \\set{e, g}$ have order $2$, $A = k[x]$, and let $g$ act linearly by $g \\cdot 1 = 1$ and $g \\cdot x^n = -x^n$ for $n \\ge 1$.\nThis is a $kG$-module with clause 2, but $g \\cdot (x \\cdot x) = -x^2$ while $(g \\cdot x)(g \\cdot x) = x^2$, so clause 1 fails when the characteristic of $k$ is not $2$.\n",
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