One thing that comes in my mind -which is similar and in some sense generalises the first construction discussed in the OP- is the smash product algebra (some call it crossed product algebra), between a Hopf algebra $H$ and an algebra $A$, which also a $H$-module algebra:
Definition: Let $Η$ be a $k$-Hopf algebra and $Α$ a $k$-algebra, which is also a $H$-module algebra. The smash product of $Α$ and $Η$, denoted by $Α \sharp Η$ will be the algebraic structure determined by the following data:
- As $k$-vector spaces $Α \sharp Η = Α \otimes_{k} Η$. We will denote the element $a \otimes h$ as $a \sharp h$.
- For any $a, b \in A$ and for any $h, g \in H$ define a bilinear multiplication by:
$$
(a \sharp h)(b \sharp g) =
\sum a(h_{(1)} \cdot b) \sharp h_{(2)}g
$$
where $\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ the comultiplication in $H$.
It is relatively easy to show that the $k$-v.s. $Α \sharp Η$ equipped with the above defined multiplication becomes an associative $k$-algebra with unity. This algebra is called the smash product algebra or crossed product algebra between the hopf algebra $H$ and the $H$-module algebra $A$.
It can furtherore be shown that -under suitable circumstances- $Α \sharp Η$ also becomes a $k$-Hopf algebra called the crossed product hopf algebra.
(in the above $k$ is the field).
Now what might be particularly interesting, is that the above smash product algebra, generalizes some similar but well known constructions. So let us study closer some specific cases of smash product algebras:
- Example 1: The skew group algebra
Let $Α$ be a (left) $kG$-module algebra. Using the definition above, we get that for all $g \in G$ we have: $$ g\cdot (ab) = (g \cdot a)(g \cdot b) $$ i.e.: the elements of $G$ act as automorphisms of $Α$. We thus get a group homomorphism: $$ \sigma: G \longrightarrow Aut_{k}A $$ and conversely: given any automorphism $\sigma: G \rightarrow Aut_{k}A$, $Α$ becomes a (left) $kG$-module algebra. By the definition of the multiplication in $A \sharp kG$ : $$ (a \sharp g)(b \sharp h) = a(g \cdot b) \sharp gh = a \sigma(g)(b) \sharp gh $$ which proves that, in this case the smash product algebra is isomorphic to the skew group algebra: $$A \sharp kG \cong A \star G$$ as $k$-algebras.
The next example, may be seen as a special case of the above:
- Example 2: The semidirect product of groups
Let $Κ$ and $Η$ two groups and let us assume that the elements of $K$ act as automorphisms of $H$. (We thus have a group homomorphism: $\phi: Κ \rightarrow Aut(H)$). Thus, the elements of $Κ$ act as automorphisms of the group algebra $kH$, which becomes a $kK$-module algebra. From the definition of the multiplication in $kH \sharp kK$ we get that for all $h, h' \in H$ και $g, g' \in K$: $$ (h \sharp g)(h' \sharp g') = h(g \cdot h') \sharp gg' = h \phi(g)(h') \sharp gg' $$ i.e., in this case: $$ kH \star K \cong kH \sharp kK \cong k(H \rtimes_{\phi} K) $$ where $H \rtimes_{\phi} K$ stands for the semidirect product of the groups $Η$, $Κ$ (which is a group itself) and $k(H \rtimes_{\phi} K)$ is its group algebra.
A remarkable consequence, is that, if the action $\phi$ is trivial, in the sense that, for all $g \in K$ and for all $h \in H$: $$ g \cdot h = \phi(g)(h) = h $$ then the semidirect product is identified with the direct product (as groups) and the smash product is identified with the tensor product (as algebras). In that case, the former relation reproduces the well known (from group representation theory): $$ kH \otimes kK \cong k(H \times K) \equiv k(H \oplus K) $$
P.S.: On the other hand, the smash product algebra described above, plays a much more important role in the structure theory of hopf algebras than simply generalizing or mimicking other known similar constructions: For example, it is well known that any cocommutative hopf algebra $H$ over an algebraically closed field of characteristic zero is isomorphic to the smash product hopf algebra between the group hopf algebra of its grouplikes $G(H)$ and the UEA of the Lie algebra of its primitives $U(P(H))$: $$ H\cong U(P(H))\sharp kG(H) $$ This is commonly refered to as the Cartier-Kostant-Milnor-Moore theorem.