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.
type: Item
instance of: scholarly article
title: 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.
zbMATH Open document ID: 1070.17005
author: Jintai Ding
author: Sergey Khoroshkin
published in: Advances in Mathematics
publication date: 2005-02-22T00:00:00Z
review text: 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.
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 17B37
zbMATH DE Number: 2136508
zbMATH Keywords: quantum affine algebras
zbMATH Keywords: universal \(R\)-matrices
MaRDI profile type: Publication
full work available at URL: j.aim.2003.11.013
OpenAlex ID: W2019736143
cites work: Twisted quantum affine algebras
cites work: Q3819194
cites work: Weyl group extension of quantized current algebras
cites work: Integral presentations for the universal \(R\)-matrix
cites work: On correlation functions of Drinfeld currents and shuffle algebras
cites work: Q3840129
cites work: The uniqueness theorem for the universal \(R\)-matrix
cites work: Quantum affine (super)algebras \(U_ q(A_ 1^{(1)})\) and \(U_ q(C(2)^{(2)})\).
DOI: 10.1016/J.AIM.2003.11.013
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description: scientific article; zbMATH DE number 2136508