Quantum Serre relations and braided commutator. I am reading the lecture notes. On page 21, it is said that when $a_{ij}=-1$, we have
\begin{align}
ad_c(x_i)^{1-a_{ij}}(x_j)=x_i^2x_j - (q+q^{-1})x_ix_jx_i+x_jx_i^2. \quad (1)
\end{align}
Here $ad_c(x_i)(x_j)=x_ix_j - q^{a_{ij}}x_jx_x$.
I am trying to verify $(1)$. We have
\begin{align}
& ad_c(x_i)^{1-a_{ij}}(x_j)\\
& =ad_c(x_i)^{2}(x_j) \\
& =ad_c(x_i)(x_ix_j - q^{-1}x_jx_i) \\
& = x_i^2 x_j -q^{-1}x_ix_jx_i -q^{-1}( x_ix_j - q^{-1}x_jx_i )x_i \\
& = x_i^2 x_j -2q^{-1}x_ix_jx_i + q^{-2}x_jx_i^2.
\end{align}
But I didn't get $x_i^2x_j - (q+q^{-1})x_ix_jx_i+x_jx_i^2$. I don't know where I made a mistake. Thank you very much.
 A: I think you are making a mistake in the fourth line of your computation: you compute $-q^{-1}ad_c(x_i)\big(x_j\big)x_i \ $, instead of the correct:
$\ -q^{-1}ad_c(x_i)\big(x_jx_i\big)$. However, it seems that it does not affect the final result. Here's what I get:
\begin{align}
& ad_c(x_i)^{1-a_{ij}}(x_j)=\\
& =ad_c(x_i)^{2}(x_j)= \\
& =ad_c(x_i)(x_ix_j - q^{-1}x_jx_i)= \\
& = x_i^2 x_j -q^{-1}x_ix_jx_i -q^{-1}ad_c(x_i)\big(x_jx_i\big)= \\
& = x_i^2 x_j -q^{-1}x_ix_jx_i-q^{-1}\big(x_ix_jx_i-q^{-1}x_jx_i^2\big)= \\
& = x_i^2 x_j -q^{-1}x_ix_jx_i-q^{-1}x_ix_jx_i+q^{-2}x_jx_i^2 = \\
& = x_i^2 x_j -2q^{-1}x_ix_jx_i+q^{-2}x_jx_i^2
\end{align}
but it's still different from the result of the paper. Maybe there's some typo in the paper.  
It is interesting to note that the two expressions 
$$
x_i^2 x_j -2q^{-1}x_ix_jx_i+q^{-2}x_jx_i^2
$$
and
$$
x_i^2x_j - (q+q^{-1})x_ix_jx_i+x_jx_i^2
$$
coincide if $q=q^{-1}$. (However, in such cases, $q$ is customarily considered not to be a root of unity).
Hope that helps a bit.
A: When $a_{ij}=-1$, 
\begin{align}
& ad_\Psi(x_i)^{1-a_{ij}}(x_j) \\
& = ad_c(x_i)^2(x_j) \\
& = ad_c(x_i)(x_i \otimes x_j - K_i.(x_j) \otimes x_i) \\
& = ad_c(x_i)(x_i \otimes x_j - q^{-1} x_j \otimes x_i) \\
& =  x_i \otimes x_i \otimes x_j - K_i.(x_i \otimes x_j) \otimes x_i - q^{-1} ( x_i \otimes x_j \otimes x_i - K_i.(x_j \otimes x_i) \otimes x_i) \\
& =  x_i \otimes x_i \otimes x_j - K_i.x_i \otimes K_i.x_j \otimes x_i - q^{-1} ( x_i \otimes x_j \otimes x_i - K_i.x_j \otimes K_i.x_i \otimes x_i) \\
& =  x_i \otimes x_i \otimes x_j - q^2 x_i \otimes q^{-1} x_j \otimes x_i - q^{-1} ( x_i \otimes x_j \otimes x_i - q^{-1} x_j \otimes q^2 x_i \otimes x_i) \\
& =  x_i \otimes x_i \otimes x_j - q x_i \otimes x_j \otimes x_i - q^{-1}  x_i \otimes x_j \otimes x_i +  x_j \otimes x_i \otimes x_i \\
& =  x_i \otimes x_i \otimes x_j - (q+q^{-1}) x_i \otimes x_j \otimes x_i +  x_j \otimes x_i \otimes x_i.
\end{align}
