Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations:
$$(\Delta \otimes \mathrm{id}) (R) = R\_{13} R\_{23}$$
$$(\mathrm{id} \otimes \Delta) (R) = R\_{13} R\_{12}$$
It appears that these two equations are not equivalent, although I don't know a single example for an $R$ that satisfies one of the equations and not the other.


**Are there known *additional* conditions on the Hopf algebra that imply the equivalence of the two braid relations?**


Or otherwise:


**Is there a Hopf algebra $H$ and an element $R \in H \otimes H$ that satisfies e.g. the first braid relation all further axioms, but *not* the second braid relation?**


Motivation: I think I have discovered a case where they are equivalent, which is the case of a *quantum group with half-twist*, as described in [this article by Noah Snyder and Peter Tingley](http://arxiv.org/abs/0810.0084). It deals with Hopf algebras where $R = \left(t^{-1} \otimes t^{-1}\right) \Delta(t)$ for a suitable element $t$ in the Hopf algebra which is called the half-twist. I'm currently [writing this particular instance up](https://www.manuelbaerenz.de/article/half-ribbon-categories), but I'm very curious whether there are other conditions that make the braid relations equivalent, or counterexamples. It's nice if they are equivalent, since it saves you a lot of work when searching for new examples.



When are the braid relations in a quasitriangular Hopf algebra equivalent?
The condition $R\_{21}\,R=I$ (to be triangular) implies the equivalence of the two equations.


For the second question, let $A$ be a finite abelian group, and $H=k^A$ the Hopf algebra of function on $A$. Functions $f:A\times A\to \mathbb{C}^\*$ (or equivalently maps from $A$ to $\operatorname{Maps}(A,\mathbb{C}^\*$)) are in correspondence with invertible elements in $H^{\otimes 2}$. The first equation is equivalent to $f(a,b+c)=f(a,b)f(a,c)$ and similarly for the second one. Then a map $R: A\to \operatorname{Hom}(A,\mathbb C^\*)$ that is not a group homomorphism and $R(0)=0$, works as an example.


