Quasitriangular Sweedler bialgebra

Question

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?

Answer 1

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.