Universal K-matrices for quantum Kac-Moody algebras
Andrea Appel, Bart Vlaar
TL;DR
This work develops a unified framework to construct universal K-matrices for quantum Kac–Moody algebras by introducing cylindrical bialgebras and twist-pair data. It extends Balagović–Kolb's finite-type constructions to symmetrizable Kac–Moody algebras via quantum pseudo-fixed-point subalgebras described by generalized Satake diagrams, yielding a family of universal K-matrices K_{Y,η} that interpolate between the quasi-K-matrix and the BK universal K-matrix. A key technical innovation is the use of Cartan-modified diagrammatic half-balances and auxiliary elements Ψ to obtain tractable coproduct identities and generalized reflection equations, enabling cylindrical representations on tensor products of integrable modules. In the affine rank-one case, the construction produces formal spectral K-matrices for U_qL sl2, giving intertwiners V(z)→V^ψ(1/z) and spectral-parameter reflection equations, thereby linking the theory to quantum integrable systems and boundary phenomena.
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 $U_qL\mathfrak{sl}_2$.