# How do I view $k$ as $kG$-module?

How could I view $k$ as $kG$-module? Does $kG$ acts on $k$ by $rg\cdot r'=rr'$, just ignore $g$? Could we do that?

Yes, Hopf algebras generally act on the field via the counit which in the case of group algebras can be defined as the linear extension of the map $$\epsilon:\mathcal{G}(kG)\longrightarrow k$$ $$x \mapsto 1_k$$ where $\mathcal{G}(kG)$ denotes the set of all group-like elements, that is, elements $x\in kG$ such that $\Delta (x)= x\otimes x$, where $\Delta$ denotes the comultiplication. Note that this set can canonically be identified with the set of elements of the group $G$ itself.