Universal K-matrices for quantum Kac-Moody algebras

Appel, Andrea; Vlaar, Bart

doi:10.1090/ert/623

Universal K-matrices for quantum Kac-Moody algebras
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by Andrea Appel and Bart Vlaar

Represent. Theory 26 (2022), 764-824

DOI: https://doi.org/10.1090/ert/623

Published electronically: July 19, 2022

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Abstract:

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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