Reference requests : Presentation of the braided dual of $U_q(\frak{sl_2})$ I am interested in the braided dual of the quantum group $U_q(\frak{sl_2})$. This is the algebra generated by the matrix coefficients but where the multiplication is twisted by an action of the $R$-matrix. I have seen (for example in https://arxiv.org/pdf/1908.05233.pdf example 1.23) that it is isomorphic to the algebra generated by elements $a^1_1, a^1_2, a^2_1$ and $a^2_2$ together with the relations :
\begin{align*}
a^1_2 a^1_1 &= a^1_1 a^1_2 + ( 1-q^{-2})a^1_2a^2_2\\
a^2_1 a^1_1 &= a^1_1 a^2_1 - ( 1-q^{-2})a^2_2a^2_1\\
a^2_1 a^1_2 &= a^1_2 a^2_1 + ( 1-q^{-2})(a^1_1a^2_2 -a^2_2a^2_2)\\
a^2_2a^1_1 &= a^1_1a^2_2\\
a^2_2a^1_2 &=  q^2 a^1_2a^2_2 \\
a^2_2a^2_1 &= q^{-2} a^2_1a^2_2\\
a^1_1a^2_2 &= 1 -q^{-2}a^1_2a^2_1
\end{align*}
If $V$ is the standard representation of $U_q(\frak{sl_2})$ and we set $a^i_j := v^i \otimes v_j$, I can see that those elements indeed generate the whole algebra, but  I don't know if there are more relations needed. According to the literature this is enough, but I cannot find a proof of this.
 A: There are a few different ways to see that these relations are sufficient.

*

*One can appeal to the fact that the braided dual is a flat PBW deformation of the algebra O(SL_2), so that a basis is given by ordered monomials in the generators $a^i_j$, as has been proved in many places (and in which one may assume $a^1_2a^2_1$ does not appear, using the q-determinant relation.  One can then confirm that these expressions satisfy the criteria of Bergman's diamond lemma, so that ordered expressions in the $a^i_j$'s form a basis of the algebra thusly presented, hence if there were any additional relations, it would break the flatness.  This kind of computation is done in Juliet Cooke's paper (in more complicated examples) https://arxiv.org/abs/1811.09293, though of course this particular result you're asking about is much much older, appearing in papers in the 90's which I won't try and dig up.

*This is essentially a variation on the above, or one way of proving the PBW claim being made there.  Since Repq(SL_2) is semisimple, its braided dual has a Peter-Weyl type decomposition as the direct sum of $C(\lambda) = V_\lambda^* \otimes V_\lambda$, and one can see that the degree $\leq k$ elements in the filtration on the algebra above map isomorphically onto the subspace $C(0) + C(1) + ... + C(k)$.  Here's a filtration because the q-determinant relation is not homogeneous.  One can then see that if there were more relations than those listed, it would not define an injective map.

