Let $f:G\rightarrow H$ be a polynomial map, between algebraic groups $G,H$ (ie defined by polynomial equations), such that the comorpism $f^*:K(H)\rightarrow K(G),\ g\mapsto g\circ f$ is bijective.
Is $f$ bijective and $f^{-1}$ a polynomial map?
If $f$ is an isomorphism of algebraic groups, then it is straightforeward that $f^*$ is bijective, but what about the converse?
This seems to follow from Tauvel and Yu's Lie Algebra and Algebraic groups 11.4.4, but I don't understand the proof. On the other hand I see that $f^*$ surjective implies $f$ injective, that $f^*$ injective implies $f$ dominant hence surjective, so $f$ is bijective.