## 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: $${\cal R} \sim q^{Q(H\otimes H)}\prod_i \exp \Big\{(q-q^{-1})\cdot e_i\otimes f_i\Big\}$$ 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:hep-th/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 $$\varkappa_{_R} = \nu_R - \nu_{\tilde R} = \sum_{(i,j)\in R} (i-j)$$ is the eigenvalue of the cut-and-join 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 block-diagonal components, $${\cal R} = q^{-2\varkappa_{_{R_1}}-2\varkappa_{_{R_2}}}\,A^{-|R_1|-|R_2|} \oplus_Q \ W_Q\otimes q^{\varkappa_Q}\cdot Id$$