Quasitriangular Sweedler bialgebra Say we have Sweedler Hopf defined like here:
https://en.wikipedia.org/wiki/Sweedler%27s_Hopf_algebra. We define universal R-matrix:
$$R_\lambda=\frac{1}{2}(1\otimes1+1\otimes g-g\otimes1+g\otimes g)+\frac{\lambda}{2}(x\otimes x+x\otimes gx+gx\otimes gx-gx\otimes x)$$
In order to check if it's quasitrinagular we need to verify:
\begin{equation}
(\Delta\otimes id)R=R_{13}R_{23}
\end{equation}
Left hand side can be done easily using bialgebra compatibility condition and properties of g and x:
$$\Delta(ab)=\Delta(a)\Delta(b)$$
$$\Delta(g)=g\otimes g\quad\quad \Delta{x}=1\otimes x+x\otimes g$$
It goes as follow:
\begin{eqnarray}
(\Delta\otimes id)R&=&\frac{1}{2}(1\otimes1\otimes1+1\otimes1\otimes g+g\otimes g\otimes1-g\otimes g\otimes g)\\
&+&\frac{\lambda}{2}((1\otimes x+x\otimes g)\otimes x+(1\otimes x+x\otimes g)\otimes gx\\
&+&(g\otimes g)(1\otimes x+x\otimes g)\otimes gx-(g\otimes g)(1\otimes x+x\otimes g)\otimes x)
\end{eqnarray}
But the problem is I can't decipher right hand side and do explicit check. In case of simple R like $R=a\otimes b$ we have:
\begin{eqnarray}
R_{23}&=&1\otimes a\otimes b\\
R_{13}&=&a\otimes 1\otimes b\\
R_{13}R_{23}&=&=a\otimes a\otimes bb
\end{eqnarray}
But how to understand it in case of multicomponent R?
 A: You have maps $\phi_{ij}:A\otimes A\to A\otimes A\otimes A$ for $i<j$ and $i,j\in\{1,2,3\}$. The subscript tells you where the terms of your $2$-tensor are mapped to in the $3$-tensor, in the sense that: $\phi_{12}(f\otimes g)=f\otimes g\otimes 1$. Where as you write at the end of your question:
$$R_{23}=\phi_{23}(R)$$
$$R_{13}=\phi_{13}(R)$$
$$R_{13}R_{23}=\phi_{13}(R)\phi_{23}(R)$$
Hence, for example
$$R_{13}=\frac{1}{2}(1\otimes1\otimes 1+1\otimes1\otimes g-g\otimes 1\otimes1+g\otimes 1\otimes g)+\frac{\lambda}{2}(x\otimes1\otimes x+x\otimes1\otimes gx+gx\otimes1\otimes gx-gx\otimes1\otimes x)$$
$$R_{23}=\frac{1}{2}(1\otimes1\otimes1+1\otimes1\otimes g-1\otimes g\otimes1+1\otimes g\otimes g)+\frac{\lambda}{2}(1\otimes x\otimes x+1\otimes x\otimes gx+1\otimes gx\otimes gx-1\otimes gx\otimes x)$$
Where hence
$$R_{13}R_{23}=\frac14(1\otimes1\otimes1+1\otimes 1\otimes g-g\otimes 1\otimes 1+g\otimes 1\otimes g+1\otimes 1\otimes g+1\otimes 1\otimes g^2-1\otimes g\otimes g +1\otimes g\otimes g^2-g\otimes 1\otimes 1-g\otimes1\otimes g+g\otimes g\otimes 1-g\otimes g\otimes g+g\otimes 1\otimes g+g\otimes 1\otimes g^2-g\otimes 1\otimes g^2+g\otimes g\otimes g^2)+\cdots$$
where I've only bothered to multiply the first term of each, and haven't reduced the terms by the relations.
