quantum groups - Map constructed from the coquasitriangular structure of SLq(2) which appears not to respect the standard commutation relations

First of all I believe a factor $q^{-1/2}$ is missing in your definition of the coefficients of the universal R-matrix.

Then, as far as I remember, the commutation relations of $mathcal O_q(SL_2)$ are
$ba = qab$, $db = qbd$, $ca = qac$, $dc = qcd$, $bc = cb$, $da-ad=(q-q^{-1})bc$, and $ad-q^{-1}bc=1$. So we do NOT have the relation $ab=qba$.

Finally, according to your convention it seems that you have $a=u_1^1$, $b=u_2^1$ $c=u_1^2$ $d=u_2^2$ (it seems that Kassel has a different convention for indices, but his R-matrix coefficients are also organized in a different way, so...). So let me compute $P_c(ab)$ and $P_c(ba)$ following your notation.

$P_c(ab) = R^{21}_{11} R^{11}_{12} + R^{21}_{21} R^{21}_{12} = 0$

and

$P_c(ba) = R^{21}_{12} R^{11}_{11} + R^{21}_{22} R^{21}_{11} = q(q-q^{-1})$

Then I believe the definition of the coefficients you gave is wrong (also I can't really follow your computations: there are a few typos, and also errors - or it might be that I did not understand what is going on).

Now if I compute following Kassel's definition of R-matrix coefficients I find :

$P_c(ab) = R^{21}_{11} R^{11}_{12} + R^{21}_{21} R^{21}_{12} = 0$

and

$P_c(ba) = R^{21}_{12} R^{11}_{11} + R^{21}_{22} R^{21}_{11} = 0$

By the way, even following uniquely your definitions I can't see how you get (on line 16) the following:

$P(u^1_1u^1_2) = quad sum_z R^{21}_{z1}R^{zi}_{1z} quad = quad R^{21}_{12}R^{21}_{12} quad = quad q^{-1}(q-q^{-1})$

First of all there is a typo, the second term should be $sum_z R^{21}_{z1}R^{z1}_{12}$. Then there seems to be two errors:

1. how can you find $R^{21}_{12}R^{21}_{12}$ ?


2. I can't see how $R^{21}_{12}R^{21}_{12}=q(q-q^{-1})$.