Let $k$ be an algebraically closed field of positive characteristic $p > 0$, and let $X$ be an intedeterminate over $k$. I am interested in the additive group scheme $\mathbb{G}_a$, that is, the affine algebraic group scheme having coordinate ring $k[X]$. We can identify $\mathbb{G}_a$ with the unipotent radical of a Borel subgroup $B$ for the group scheme $SL_2$, e.g., if we identify $B$ with the subgroup of upper triangular matrices in $SL_2$, then we can identify $\mathbb{G}_a$ with the subgroup $U$ of upper triangular unipotent matrices in $SL_2$.

I am specifically interested in rational modules for $\mathbb{G}_a$, or equivalently, $k[X]$-comodules. Given a rational $B$-module $M$, I can of course restrict $M$ to $U = \mathbb{G}_a$ and obtain a rational $\mathbb{G}_a$-module. But what I'd really like are some explicit examples of rational $\mathbb{G}_a$-modules that do not arise via restriction from a rational $B$-module, and I haven't had any luck so far finding explicit examples in the literature.

Can anyone point me to explicit examples in the literature of (finite-dimensional) rational $\mathbb{G}_a$-modules that do not arise as the restriction of rational $B$-modules?

Here's the (single) example I've been able to cook up from scratch for the case $p=2$. Let $V$ be the $k$-vector space with basis vectors $v_0$ and $v_1$. I define a linear map $\Delta: V \rightarrow V \otimes k[X]$ by

$\Delta(v_0) = v_0 \otimes 1 + v_1 \otimes (X+X^2)$,

and $\Delta(v_1) = v_1 \otimes 1$. Then it's straightforward to check that this defines a $k[X]$-comodule structure on $V$ (equivalently, a rational $\mathbb{G}_a$-module structure on $V$), which cannot come from the restriction of a rational $B$-module structure on $V$ because of the non-homogeneous term $X+X^2$.