Smash product

Remarks

With 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. Example: let $G$ be a group acting on $A$ by algebra automorphisms and let $H = kG$, with $\Delta(g) = g \otimes g$ and $\varepsilon(g) = 1$. Then 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 $A * G$. Example: 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 $A$. Non-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 $n \ge 1$. This 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 $2$.