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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 $g^2=uS(u)^{-1}=\gamma$.

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  • $\begingroup$ Any reason that $G(H)$ can't be an elementary abelian 2-group? $\endgroup$
    – zibadawa timmy
    Commented Apr 5, 2017 at 19:08
  • $\begingroup$ It can be, but why should this contradict the statement $\endgroup$
    – Bipolar Minds
    Commented Apr 5, 2017 at 19:26
  • $\begingroup$ It immediately contradicts it if $\gamma$ is non-trivial. I just can't recall if there's anything that forces $\gamma=1$ when $H$ is fin. dim. pointed with $G(H)$ an elementary abelian 2-group. $\endgroup$
    – zibadawa timmy
    Commented Apr 5, 2017 at 19:37
  • $\begingroup$ I don't know either but that's definetely a good starting point $\endgroup$
    – Bipolar Minds
    Commented Apr 5, 2017 at 20:09

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