What do we know about group-like elements in coquasitriangular bialgebras?

In particular, we know that a commutative bialgebra in which all group-like elements are invertible is a Hopf algebra (see this question); does this result generalize to coquasitriangular bialgebras?

More precisely : Let $(B, r)$ be a coquasitriangular $k$-bialgebra, i.e., a $k$-bialgebra with a $k$-linear form $r : B \otimes B \rightarrow k$ satisfying :

  • $r$ has a convolution inverse $\bar{r}$, i.e., $r \ast \bar{r} = \bar{r} \ast r = \varepsilon$,
  • $H$ is quasi-commutative, i.e., $\mu^{op} = r \ast \mu \ast \bar{r}$
  • $r (\mu \otimes \text{id}) = r_{13} \ast r_{23}$ and $r(\text{id} \otimes \mu) = r_{13} \ast r_{12}$, with $r_{12} = r \otimes \varepsilon$, $r_{23} = \varepsilon \otimes r$ and $r_{13} = (\varepsilon \otimes r)(\tau_{H, H} \otimes \text{id})$,

and let $GLE$ = { $g \in B ~|~ g \neq 0, \Delta (g) = g \otimes g$ } is the set of group-like elements. Then if every element in $GLE$ is invertible, does $B$ have an antipode?


Take a look at [T. HAYASHI , Quantum groups and quantum determinants, J. Algebra 152 (1992), pp 146–165.] I'm not sure if he answer completely your question but for sure he discuss it in the context of CQT bialgebras. He construct families of (CQT) Hofp algebras by inverting all group-like elements.


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