I saw this post recently: Tannaka–Krein duality I have this question please: in the following which iI report here:
"The problem is with surjectivity: let us denote G':=G(R(G)) and ^:R(G)→R(G') is an isomorphism from the previous theorem. One shows that δ∗:F↦F∘δ is right inverse of ^ and therefore is also an isomorphism. Since R(G) and R(G') are dense in C(G) and C(G') resp. it follows that δ∗ extends to the isomomorhism between C(G) and C(G') and thus δ is surjective."
The problem is with surjectivity: let us denote $\mathcal{G}:=\mathcal{G}(\mathcal{R}(G))$ and $\hat{}:\mathcal{R}(G) \to \mathcal{R}(\mathcal{G})$ is an isomorphism from the previous theorem. One shows that $\delta^*:F \mapsto F \circ \delta$ is right inverse of $\hat{}$ and therefore is also an isomorphism. Since $\mathcal{R}(G)$ and $\mathcal{R}(\mathcal{G})$ are dense in $C(G)$ and $C(\mathcal{G})$ resp. it follows that $\delta^*$ extends to the isomomorhism between $C(G)$ and $C(\mathcal{G})$ and thus $\delta$ is surjective.
Let me plsplease ask the couple of questions below:
- why ^Why is $\hat{}$ an isomorphism of Hopf algebras!?
- Why if δ∗$\delta^*$ is right inverse of ^$\hat{}$ then it is also an isomorphism? Have left invertibilty can be proved? And can we use only algebraic arguments on G$G$?
