On higher spin $U_q(\operatorname{sl}_2)$-invariant $R$-matrices

The spectral decomposition of regular $U_q(\operatorname{sl}_2)$-invariant solutions of the Yang-Baxter equation is studied. An algorithm for the search of all possible spin $s$ solutions is developed, also allowing reconstruction of the $R$-matrix by a given nearest neighbor spin chain Hamiltonian. The algorithm is based on reduction of the Yang-Baxter equation to certain subspaces. As an application, a complete list of nonequivalent regular $U_q(\operatorname{sl}_2)$-invariant $R$-matrices is obtained for generic $q$ and spins $s\leq\frac{3}{2}$. Some further results about spectral decompositions for higher spins are also proved. In particular, it is shown that certain types of regular $\operatorname{sl}_2$-invariant $R$-matrices have no $U_q(\operatorname{sl}_2)$-invariant counterparts.