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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 Balagovi\'c and Kolb, producing a family of non-equivalent solutions
interpolating between the quasi-K-matrix 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
$U_qL\mathfrak{sl}_2$.