Sweedler's 4-dimensional Hopf algebra admits a one-parameter family of triangular structures given by \begin{equation} R_{\lambda}:=1\otimes1+1\otimes g+g\otimes 1-g\otimes g-\frac{\lambda}{2}(x\otimes x-gx\otimes x+x\otimes gx+gx\otimes gx), \lambda\in \mathbb{C}. \end{equation} Is there a value for $\lambda$ such that the resulting triangular Hopf algebra is factorizable?
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The answer is no. The easiest to see this is the following: Consider the element $$X:=(R_{\lambda})_{21}R_{\lambda}\in H\otimes H.$$ The Hopf algebra is factorizable if and only if $X$ has maximal length in the tensor product. But if this would have been the case, $X$ would have also had a maximal length in the quotient $H/(x)\otimes H/(x)$. But as can easily be seen, the image of $X$ in that space is simply $1\otimes 1$. Alternatively we can use the following argument: the one-dimensional non-trivial representation $k_{-}$ (upon which $g$ acts as $-1$ and $x$ as zero) is a transparent object in the braided monoidal category $R_{\lambda}$ defines, which implies that $R_{\lambda}$ cannot be factorizable. (Transparent objects are the objects $V$ such that $c_{U,V}c_{V,U}=Id_{V\otimes U}$ for every object $U$).
