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.