Let $A$ be a Hopf algebra dually paired with a quasi-triangular Hopf algebra $B$. If $x$ is some fixed element of $A$, then we can define a linear map 
$$
P\_x: A \to \mathbb{C}
$$
by setting 
$$
P\_x:a \mapsto \langle R,x \otimes a \rangle.
$$

Let us take the case $A = SL\_q(2)$, $B = U\_q({\mathfrak sl}\_2)$, and let $R$ be the standard universal $R$-matrix for $U\_q({\mathfrak sl}\_2)$, for which
$$
\langle R, u^i\_m \otimes u^j\_n \rangle = R^{ij}\_{mn} = q^{-\frac{1}{2}}.(q^{\delta\_{ij}}\delta\_{im}\delta\_{jn} + (q-q^{-1})\theta (i-j)\delta\_{in}\delta\_{jm}),
$$
where $\theta$ is the Heaviside symbol. If we take $x=u^k\_l$, then 
$$
P\_{u^k\_l}(a) = \langle R, u^k\_l \otimes a \rangle.
$$
Now since $ab = qba$, we should have 
$$
P\_{u^k\_l}(u^1\_1u^1\_2) = q P\_{u^k\_l}(u^1\_2u^1\_1), \qquad \qquad \text{ for all } \quad k,l = 1,2.
$$
However,
$$
P\_{u^2\_1}(u^1\_1u^1\_2) = \langle R, u^2\_1 \otimes u^1\_1u^1\_2 \rangle = \sum\_{z=1}^2 \langle R,u^2\_z \otimes u^1\_1 \rangle \langle R, u^z\_1 \otimes u^1\_2 \rangle = \sum\_{z=1}^2 R^{21}\_{z1}R^{z1}\_{12}.
$$
From the formula for $R^{ij}\_{mn}$, we get that
$$
P(u^1\_1u^1\_2) = \sum\_{z=1}^2 R^{21}\_{z1}R^{z1}\_{12} = R^{21}\_{11}R^{11}\_{12} + R^{21}\_{21}R^{21}\_{12} = 0.0 + q^{-\frac{1}{2}}.1.q^{-\frac{1}{2}}.(q-q^{-1}) = q^{-1}(q-q^{-1}).
$$

On the other hand, we have
$$
qP\_{u^2\_1}(u^1\_2u^1\_1) = \langle R,u^2\_1 \otimes u^1\_2u^1\_1 \rangle = \sum\_{z=1}^2 \langle R,u^2\_z \otimes u^1\_2 \rangle \langle R,u^z\_1 \otimes u^1\_1 \rangle =\sum\_{z=1}^2R^{21}\_{z2}R^{z1}\_{11}.
$$
From the formula for $R^{ij}\_{mn}$, we now get that
$$
qP\_{u^2\_1}(u^1\_2u^1\_1) = q\sum\_{z=1}^2R^{21}\_{z2}R^{z1}\_{11} = qR^{21}\_{12}R^{11}\_{11} + qR^{21}\_{22}R^{21}\_{11} = q.q^{-\frac{1}{2}}.(q-q^{-1}).q^{-\frac{1}{2}}.q + q.0.0 = q(q-q^{-1}).
$$

Thus, the two results are not equal, but instead differ by a factor of $q^2$. A similar problem arises for the action of $P\_{u^2\_1}$ on $bd - qdb$. We get 
$$
P\_{u^2\_1}(u^1\_2u^2\_2) = q^{-1}(q-q^{-1}),
$$
whereas
$$
qP\_{u^2\_1}(u^2\_2u^1\_2) = q(q-q^{-1}).
$$

I've checked and rechecked everything very carefully but can't seem to spot my error. Can anyone see what is going wrong here?