Quantum ${\cal R}$matrices

Quantum ${\cal R}$matrix is known for arbitrary pair of representations $R_1\otimes R_2$ of arbitrary simple Lie algebra:
$$
{\cal R} \sim q^{Q(H\otimes H)}\prod_i \exp \Big\{(qq^{1})\cdot e_i\otimes f_i\Big\}
$$
where product is over all simple roots.
The first factor contains a bilinear form of Cartan elements, while
$e_i$ and $f_i$ are Chevalley generators in the given representations $R_1$ and $R_2$ respectively.
For associated formulas for group elements see arXiv:hepth/9409093
and arXiv:1403.1834 for their mutation transforms.
 For algebras $SL(N)$ irreducible representations are labeled by Young diagrams $R$.
Then $R$ is the size (number of boxes) and
$$\varkappa_{_R} = \nu_R  \nu_{\tilde R} = \sum_{(i,j)\in R} (ij)$$
is the eigenvalue of the cutandjoin operator.

This universal ${\cal R}$matrix is in the "vertical" framing, it is converted to topological framing
by additional factor $q^{\varkappa_{_{R_1}}+\varkappa_{_{R_2}}}\, A^{R_1+R_2}$.

If the product of representations is decomposed into irreducible $Q$'s,
$R_1\otimes R_2 = \oplus_Q W_Q\otimes Q$, then ${\cal R}$ is made of blockdiagonal components,
$${\cal R} = q^{2\varkappa_{_{R_1}}2\varkappa_{_{R_2}}}\,A^{R_1R_2} \oplus_Q \ W_Q\otimes q^{\varkappa_Q}\cdot Id $$