TY - JOUR

T1 - Universal K-matrices for quantum Kac-Moody algebras

AU - Appel, Andrea

AU - Vlaar, Bart

N1 - Funding Information:
Received by the editors September 7, 2020, and, in revised form, May 2, 2022, and June 2, 2022. 2020 Mathematics Subject Classification. Primary 81R10, 17B37; Secondary 17B67, 16T10. 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.
Publisher Copyright:
© 2022. American Mathematical Society

PY - 2022/7/19

Y1 - 2022/7/19

N2 - 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 UqLsl2.

AB - 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 UqLsl2.

UR - http://www.scopus.com/inward/record.url?scp=85135050309&partnerID=8YFLogxK

U2 - 10.1090/ert/623

DO - 10.1090/ert/623

M3 - Article

AN - SCOPUS:85135050309

SN - 1088-4165

VL - 26

SP - 764

EP - 824

JO - Representation Theory

JF - Representation Theory

ER -