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

Question

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?

Answer 1

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.