I will be appreciated if anyone can give me some clue for the following simple question,

Let $H$,$A$ are both hopf algebras,$\pi :A \rightarrow H$,$\quad f:H\hookrightarrow A$ are both hopf morphism and $\pi f=id_H$.

Let $R= \left\lbrace a\in H| (id\otimes \pi)\triangle_A(a)=a\otimes 1 \right\rbrace$

My question is how to prove $P(R)$(primitive space )$\in {}_H^H\mathcal{YD}$

more concretely $\triangle_R(x):=x_{(1)}f\pi(x_{(2)})\otimes x_{(3)}$,when $x\in P(R)$,I need to prove $x_{(3)}\in P(R)$

I give more information:in fact $R$ is a braided hopf algebra in ${}_H^H\mathcal{YD}$

$P(R)=${ $x\in R|\triangle_R(x)=1\otimes x+x\otimes 1$}

$h\cdot r=h_{(1)}rS_H(h_{(2)}),h\in H,r\in R$

$\triangle_l(r)=r_{(-1)}\otimes r_{(0)}=\pi(r_{(1)})\otimes r_{(2)}$