Tensor Product Question in Kassel's Quantum Groups

Question

In Kassel's book on Quantum Groups, I am stuck on the following computation:

\begin{eqnarray*} [\Delta (E), \Delta (F)] &=& \Delta (E)\Delta (F)-\Delta (F)\Delta (E)\\ &=& (1\otimes E + E \otimes K) (K^{-1} \otimes F + F\otimes 1)\\ &&-(K^{-1}\otimes F+F\otimes 1)(1\otimes E+E\otimes K)\\ &=& K^{-1}\otimes EF + F\otimes E + EK^{-1}\otimes KF + EF\otimes K\\ &&-K^{-1}\otimes FE - K^{-1}E\otimes FK - F\otimes E - FE\otimes K\\ &=& K^{-1} \otimes (EF-FE) + (EF-FE) \otimes K\\ &=& K^{-1}\otimes [E,F] + [E,F]\otimes K\\ &=& \frac{K^{-1}\otimes (K-K^{-1})+(K-K^{-1})\otimes K}{q-q^{-1}}\\ &=& \frac{K\otimes K-K^{-1}\otimes K^{-1}}{q-q^{-1}}\\ &=& \frac{\Delta (K)-\Delta (K^{-1})}{q-q^{-1}} \end{eqnarray*}

The above computation was attempted by me, but I am not sure of one part, namely why is $EK^{-1}\otimes KF = -(K^{-1}E\otimes FK )$.

I understand that $E,F,K$ are variables that generate the quantum group $U_q(\mathfrak{sl}(2))$.

Sincere thanks for any help.

Answer 1

You have a sign error we have $EK^{-1}\otimes KF=K^{-1}E\otimes FK$. This is because of the following two relations: \begin{align*} KEK^{-1}&=q^2E\\ KFK^{-1}&=q^{-2}F \end{align*} Hence we have \begin{align*} EK^{-1}=q^2K^{-1}E\\ KF=q^{-2}FK \end{align*} This gives the above equality (since $q^2q^{-2}=1$ and you can put scalars in front of the tensor product).

Otherwise your computation is correct.