Motivations of cross product

Question

Let $H$ be a bialgebra and $B$ an $H$-module. The cross product $B \rtimes H$ of $B$ and $H$ is $B \otimes H$ as a vector space and the multiplication in $B \rtimes H$ is defined as follows: $(a \otimes g)(b \otimes h) = a(g_{(1)}.b) \otimes (g_{(2)}h)$, where $a,b \in B$, $g, h \in H$, $\Delta(g) = g_{(1)} \otimes g_{(2)}$. What are the motivations of the definition of cross product? Thank you very much.

Answer 1

It's the Hopf-algebraic version of the semidirect product of groups. To see this, just consider the case you have groups $G, H, G\rtimes H$. The group rings (over, say, a field $k$) are Hopf algebras (with $\Delta(g)=g\otimes g$); compare $kG\rtimes kH$ and $k(G\rtimes H)$ and you will see they have the same multiplication.

Answer 2

Another motivation I personally like is the following.

Assumptions: Let $B$ be an $k$-algebra ($k$ a field) such that multiplication and unit are morphisms of $H$-modules for a $k$-Hopf algebra $H$ (this is a more precise set of assumptions needed in the question). One says that $A$ is a $H$-module algebra.

Consider the category $B-Mod_H$ of left modules over $B$ which are also left modules over $H$ such that the $B$-action is a morphism of $H$-modules. This is quite a common phenomenon:

• For algebras that are $\mathbb Z_2$-graded (or $\mathbb Z$-graded, or graded by any group $G$) it is natural to consider modules that are graded in the same way. This gives for $\mathbb Z_2$ super-algebras and their super-modules, for $\mathbb Z$ graded algebras and modules. In these examples $H=k[G]$, the function ring on these finite groups.
• For $H=k[d]/d^2$ one recovers DG-algebras $B$ and their DG-modules. Strictly speaking, this is a bit interated: $k[d]/d^2$ is a graded super-Hopf algebra (i.e. $\mathbf Z$-graded) and DG-algebras are $H$-module algebras that are also $k[\mathbf Z]$-module algebras. So DG-modules are $(B-Mod(H-Mod))(k[\mathbf Z]-Mod)$, where we use the symmetric monoidal structure on $k[\mathbf Z]-Mod$ give by $v\otimes w \mapsto (-1)^{|v||w|}w\otimes v$.
• The positive parts $U_q(\mathfrak n_+)$ of the quantum groups are $H$-comodule algebras for $H=k\mathbf Z^n$. The category $U_q(\mathfrak n_+)-Mod_H$ consist of modules with weight space decomposition. (Note that this uses a comodule version)

A motivation for the cross product $B\rtimes H$ is now that the category $B-Mod_H$ is equivalent to the category $B\rtimes H-Mod$. The latter is simply a category of $k$-modules which sometimes offers advantages.