Universal K-matrices for quantum Kac-Moody algebras

TL;DR

This paper introduces universal K-matrices for quantum Kac-Moody algebras, expanding the theory of reflection equations and braid group actions with new constructions from quantum symmetric pairs.

Contribution

It defines cylindrical bialgebras with universal K-matrices and constructs new examples from quantum symmetric pairs, refining previous results and providing solutions to reflection equations.

Findings

New universal K-matrices from quantum symmetric pairs of Kac-Moody type.

Refinement of finite type solutions interpolating known quasi-K-matrices.

Formal solutions of the generalized reflection equation with spectral parameter.

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 Balagovi\'c 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 $U_qL\mathfrak{sl}_2$.