General solution of the Yung–Baxter equation with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$

The problem of constructing the $\mathrm R$-matrix is considered in the case of an integrable spin chain with symmetry group $\mathrm{SL}(\mathrm n,\mathbb C)$. A fairly complete study of general $\mathrm R$-matrices acting in the tensor product of two continuous series representations of $\mathrm{SL}(\mathrm n,\mathbb C)$ is presented. On this basis, $\mathrm R$-matrices are constructed that act in the tensor product of Verma modules (which are infinite-dimensional representations of the Lie algebra $\mathrm{sl}(n)$), and also $\mathrm R$-matrices acting in the tensor product of finite-dimensional representations of the Lie algebra $\mathrm{sl}(n)$.