Let $(H,R)$ be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements (I actually need it for $H$ fin. dim. pointed with $G(H)$ abelian). Let $u\in H$ be the Drinfeld't element and $\gamma=uS(u)^{-1} \in G(H)$ the associated group-like element. Does there always exist another grouplike $g \in G(H)$, s.t. $g^2=\gamma$?

Let $(H,R)$ be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements (I actually need it for $H$ fin. dim. pointed with $G(H)$ abelian). Let $u\in H$ be the Drinfeld't element and $\gamma=uS(u)^{-1} \in G(H)$ the associated group-like element. Does there always exist another grouplike $g \in G(H)$, s.t. $g^2=\gamma$?

EDIT: I just saw that if $(H,R)$ admits a ribbon element $\nu$, then $g:=\nu^{-1}u$ is group-like and $S(\nu)=\nu$ implies $g^2=uS(u)^{-1}=\gamma$.

Let $(H,R)$ be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements ($G(H)$ abelian would also be fine). Let $u\in H$ be the Drinfeld't element and $\gamma=uS(u)^{-1} \in G(H)$ the associated group-like element. Does there always exist another grouplike $g \in G(H)$, s.t. $g^2=\gamma$?

Let $(H,R)$ be a finite-dimensional quasi-triangular Hopf algebra, lets say generated by group-like and skew-primitive elements (I actually need it for $H$ fin. dim. pointed with $G(H)$ abelian). Let $u\in H$ be the Drinfeld't element and $\gamma=uS(u)^{-1} \in G(H)$ the associated group-like element. Does there always exist another grouplike $g \in G(H)$, s.t. $g^2=\gamma$?