Universal \(\mathcal R\)-matrix for quantum affine algebras \(U_q(A_2^{(2)})\) and \(U_q(\widehat{\text{osp}}(1|2))\) with Drinfeld co\-mul\-ti\-plication.
In two previous articles [Transform. Groups 5, No. 1, 35-59 (1999; Zbl 1018.17010); Lett. Math. Phys. 53, No. 2, 121--141 (2000; Zbl 1066.17501), the latter with \textit{S. Pakuliak}], the authors described the universal \(R\)-matrix for quantum nontwisted affine algebras with Drinfeld comultiplication. This was presented up to a standard factor as a series of contour integrals of certain canonical tensors over the system of factorizable cycles in deformed configuration spaces. They did this first for \(U_q(\widehat {\text{sl}}_2)\), and then extended the construction to the general nontwisted case using a current braid group action constructed in the first reference above. In the paper under review, they handle the two twisted quantum affine algebras \(U_q(A^{(2)}_2)\) and its super partner \(U_q(\widehat{\text{osp}} (1|2))\). This turns out to be technically more difficult than the \(U_q(\widehat {\text{sl}}_2)\) untwisted case, since the poles and zeroes of the current operators, together with the vanishing Serre conditions, form a complicated structure of the deformed configuration space. The final description of the universal \(R\)-matrices is as a series of integrals of canonical tensors over the system of factorizable cycles using the abstract theory of the second reference above. The description involves integrating over a product of unit circles. The authors plan to study other twisted quantum affine algebras, which will require the construction of current group actions.
scientific article; zbMATH DE number 2136508
scientific article