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?
