(Note: I had this posted on MSE for a while but didn't get much of a response... so I'm posting it here now.)
Let $G$ be an affine algebraic group over an algebraically closed field $k$ of characteristic zero. Then the coordinate ring $k[G]$ of $G$ is a Hopf algebra, and it comes with a map $\epsilon:k[G]\to k$ given by evaluation at the identity. The augmentation ideal $I$ of $G$ is the kernel of this map, and thus is a maximal ideal.
Now $G$ acts on itself by conjugation, and this action stabilizes $I$. Thus $G$ acts on $I/I^2$, which is just $I/I^2=T_e^*G=\mathfrak{g}^*$, where $\mathfrak{g}=T_eG$ is the Lie algebra of $G$. This action of $G$ on $\mathfrak{g}^*$ is just the usual coadjoint action.
My question is: does the $G$-module homomorphism $I\to I/I^2=\mathfrak{g}^*$ split as a $G$-module? Or if it doesn't always are there conditions (something weaker than the group being reductive) that implies we get a splitting?
