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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Universal K-matrices for quantum Kac-Moody algebras
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by Andrea Appel and Bart Vlaar PDF
Represent. Theory 26 (2022), 764-824 Request permission


We introduce the notion of a cylindrical bialgebra, which is a quasitriangular bialgebra $H$ endowed with a universal K-matrix, i.e., a universal solution of a generalized reflection equation, yielding an action of cylindrical braid groups on tensor products of its representations. We prove that new examples of such universal K-matrices arise from quantum symmetric pairs of Kac-Moody type and depend upon the choice of a pair of generalized Satake diagrams. In finite type, this yields a refinement of a result obtained by Balagović and Kolb, producing a family of non-equivalent solutions interpolating between the quasi-K-matrix originally due to Bao and Wang and the full universal K-matrix. Finally, we prove that this construction yields formal solutions of the generalized reflection equation with a spectral parameter in the case of finite-dimensional representations over the quantum affine algebra $U_qL\mathfrak {sl}_{2}$.
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Additional Information
  • Andrea Appel
  • Affiliation: Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy
  • MR Author ID: 1269493
  • ORCID: 0000-0002-6446-9305
  • Email:
  • Bart Vlaar
  • Affiliation: Department of Mathematics, Heriot–Watt University, Edinburgh EH14 4AS, United Kingdom; and Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
  • MR Author ID: 1025642
  • ORCID: 0000-0002-0792-720X
  • Email:
  • Received by editor(s): September 7, 2020
  • Received by editor(s) in revised form: May 2, 2022, and June 2, 2022
  • Published electronically: July 19, 2022
  • Additional Notes: The first author was supported in part by the ERC Grant 637618 and the Programme FIL of the University of Parma co-sponsored by Fondazione Cariparma. The second author was supported in part by the EPSRC Grant EP/R009465/1.
  • © Copyright 2022 American Mathematical Society
  • Journal: Represent. Theory 26 (2022), 764-824
  • MSC (2020): Primary 81R10, 17B37; Secondary 17B67, 16T10
  • DOI:
  • MathSciNet review: 4454332