element $x \in
G$ are $x \cdot f =\sum f_1f_2(x)$; $f \cdot x =\sum f_1(x)f_2$.

A –not necessarily finite dimensional– rational (left) $G$–module $M$ can be defined in terms of a (right) $\Bbbk[G]$–comodule structure $\chi_M: M \to M \otimes \Bbbk[G]$, and this structure map is written à la Sweedler as $\chi(m)=\sum m_0 \otimes m_1 \in M \otimes
\Bbbk[G]$. It is related with the action of $G$ on $M$ by the formula ($x \in G\,,\, m \in M$): $x \cdot m=\sum m_0m_1(x)$. The category of rational $G$–modules is denoted as ${{}_{G}\mathcal M}$, and by definition it coincides with the category of $\Bbbk[G]$–comodules. If $N \in
{{}_{G}\mathcal M}$, we denote as ${}^GN:=\{n \in N: x\cdot n=n \text{ for all } x
\in G\}$ and it is clear that ${}^GN=\{n \in N: \chi(n)=n \otimes 1\}$ with $\chi$ the $\Bbbk[G]$–comodule structure on $N$. If $M$ is a finite dimensional rational $G$–module and $m \in M\,,\,\alpha \in M^*$ we call $\alpha|m \in \Bbbk[G]$ the polynomial $\alpha|m=\sum \alpha(m_0)m_1$ or in explicit terms: $(\alpha|m)(x)=\alpha(x\cdot m)$ for $x \in G$. It is clear that $x\cdot (\alpha|m)=\alpha|(x \cdot m)$ for all $x \in G$. Also, in the case of a closed inclusion $H \subseteq G$ of affine algebraic groups, 