I'm trying to check some details from this paper. Let $U_q(\mathfrak{sl}_3)$ be the usual Drinfeld-Jimbo quantum group, with generators $E_1,E_2,F_1,F_2,K_1^{\pm},K_2^{\pm}$ and well known relations, for example $$K_1E_1K_1^{-1} = q^2E_1$$ $$E_1F_1 – F_1E_1 = \frac{K_1 – K_1^{-1}}{q-q^{-1}}$$
etc… For notational ease let us also define $$[K_i;a] := \frac{q^aK_i – q^{-a}K_i^{-1}}{q-q^{-1}}$$
Define operators $T_1, T_2$ as follows : $T_i(F_j) = – K_i^{-1}E_i$ if $i=j$ and $F_iF_j – qF_jF_i$ else, $T_i(E_j) = -F_iK_i$ if $i=j$, and
$E_iE_j – q^{-1}E_jE_i$ else. Finally $T_i(K_j) = K_i^{-1}$ if $i=j$ and $K_iK_j$ else.
Question : how to check the equalities $T_1T_2(F_1) = F_2$ ?
Details of my computation :
\begin{equation} T_1T_2F_1= T_1(F_2F_1 – qF_1F_2) = \\ (F_2F_1 – qF_1F_2)(-K_1^{-1}E_1) -q(-K_1^{-1}E_1)(F_2F_1-qF_1F_2) = \\ -q^{-1}K_1^{-1}(F_2F_1-qF_1F_2)E_1 +q(K_1^{-1}E_1)(F_1F_2 – qF_1F_2) \end{equation}
Where we used $K_1F_2 = q F_2K_1$ so $F_2K_1^{-1} = qK_1^{-1}F_2$ and $F_1K_1^{-1} = q^{-2}K_1^{-1}F_1$.
Now since $E_1F_2 = F_2E_1$ and $F_1E_1 = E_1F_1 – [K_1;0]$ we get $$(F_2F_1 – qF_1F_2)E_1 = (F_2F_1 – qF_1F_2) +q[K_1;0]F_2 – F_2[K_1;0]$$
Using this we obtain
$\begin{equation} -q^{-1}K_1^{-1}(F_2F_1-qF_1F_2)E_1 -q(K_1^{-1}E_1)(F_1F_2 – qF_1F_2) = \\ -q^{-1}K_1^{-1}E_1(F_2F_1 – qF_1F_2) – q^{-1}K_1^{-1}(q[K_1;0]F_2-F_2[K;0]) + q K_1^{-1}E_1(F_2F_1 – q F_1F_2) = \\ (q-q^{-1})K_1^{-1}E_1(F_2F_1-qF_1F_2) – q^{-1}K_1^{-1}F_2(q[K_1;1] + [K_1;0]) \end{equation}$
Edit : the first version asked why $T_1T_2T_1 = T_2T_1T_2$ but it is pretty easy once you know that $T_iT_j(F_i) = F_i$ and similarly with $E_i$. Indeed we get $T_2T_1T_2(F_1) = T_2(F_2) = -K_2^{-1}E_2$ and $T_1T_2T_1(F_1) = T_1T_2(-K_1^{-1}E_1) = -T_1(K_1^{-1}K_2^{-1}T_2(E_1))) = -K_2^{-1}T_1T_2(E_1) = -K_2^{-1}E_2$.
Best Answer
Since you didn't give any details, I can't tell what went wrong with your calculation, but I will note that one definition you have is incorrect. You should define $T_i(F_i)=-K_i^{-1}E_i$ (as opposed to $-K_i^{+1}E_i$, which you have written multiple times in your post).
I double checked and if you apply commutation relations correctly the result follows quite easily.