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I think the only issue here is a harmless error in your calculation and that there is a normalization of the $R$-matrix for $U_q(sl_N)$ by a factor of $q^{1/2}$ which you have omitted (See 8.4.2 of Klymik Schmudgen).

First, I get $(R^{21}_{12})^{-1} (R^{-1})^{21}_{12} = -q^{-1} R^{21}_{12}$, because $\langle(S\otimes id)(R),a^2_1\otimes a^1_2\rangle = \langle R,S(a^2_1)\otimes a^1_2 \rangle = -q^{-1} \langle R,a^2_1\otimes a^1_2\rangle$, using that $S(b)=-q^{-1} b$ from Proposition 4.1.2.3 of Klymik Schmudgen. So the actual matrix you should get should just be $q^{-1}$ times what you had expected to get.

Now the factor of q^{-1} here is because you had multiplied the actual universal R matrix by $q^{1/2}$ and so $(\lambda A)^{-1} = \lambda^{-1} A^{-1}$, so there's a factor of $\lambda^2$ as a discrepancy.

I hope this helps!

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I think the only issue here is a harmless error in your calculation and that there is a normalization of the $R$-matrix for $U_q(sl_N)$ by a factor of $q^{1/2}$ which you have omitted (See 8.4.2 of Klymik Schmudgen).

First, I get $(R^{21}_{12})^{-1} = -q^{-1} R^{21}_{12}$, because $\langle(S\otimes id)(R),a^2_1\otimes a^1_2\rangle = \langle R,S(a^2_1)\otimes a^1_2 \rangle = -q^{-1} \langle R,a^2_1\otimes a^1_2\rangle$, using that $S(b)=-q^{-1} b$ from Proposition 4.1.2.3 of Klymik Schmudgen. So the actual matrix you should get should just be $q^{-1}$ times what you had expected to get.

Now the factor of q^{-1} here is because you had multiplied the actual universal R matrix by $q^{-1/2}$ q^{1/2}$and so$(\lambda A)^{-1} = \lambda^{-1} A^{-1}$, so there's a factor of$\lambda^2$as a discrepancy. I hope this helps! 2 added 48 characters in body I think the only issue here is a harmless error in your calculation and that there is a normalization of the$R$-matrix for$U_q(sl_N)$by a factor of$q^{1/2}$which you have omitted (See 8.4.2 of Klymik Schmudgen). First, I get $(R^{21}{12})^{-1} (R^{21}_{12})^{-1} = -q^{-1} R^{21}{12}$, R^{21}_{12}$, because $<(S\ot \\langle(S\otimes id)(R),a^2_1\otimes a^1_2> a^1_2\rangle = \langle R,S(a^2_1)\otimes a^1_2 \rangle = -q^{-1} \langle R,a^2_1\otimes a^1_2\rangle$, using that $S(b)=-q^{-1} b$ from Proposition 4.1.2.3 of Klymik Schmudgen. So the actual matrix you should get should just be $q^{-1}$ times what you had expected to get.

Now the factor of q^{-1} here is because you had multiplied the actual universal R matrix by $q^{-1/2}$ and so (\lambda $(\lambda A)^{-1} = \lambda^{-1} A^{-1}A^{-1}$, so there's a factor of $\lambda^2$ as a discrepancy.

I hope this helps!

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