Cylindrical structure and universality for generalized QSP parameters

Establish the existence of a cylindrical structure for the quantum (pseudo-)symmetric pair (Uq(g), Uq(t)) for every parameter pair (y, o) in the set TEq of generalized parameters; moreover, characterize all invertible universal solutions of the generalized reflection equation (2.6) for Uq(g) in finite type as arising, up to gauge transformation, from such parameter choices.

Background

Quantum symmetric pairs (QSPs) are coideal subalgebras Uq(t) within quantum groups Uq(g) that encode symmetries compatible with the universal R-matrix. Standard parameter constraints ((y, o) ∈ L′q × E_q) ensure desirable structural properties, but generalized q-Onsager algebras motivate broader parameter sets TEq, allowing more flexible choices. The authors conjecture that these generalized parameters still yield cylindrical structures (a twist pair and a basic K-matrix solving a reflection equation) and that, in finite type, every invertible universal solution of the reflection equation comes from such QSPs up to gauge equivalence.

This problem aims to extend the known framework for constructing universal K-matrices to a larger parameter class and to provide a universality classification of all invertible universal solutions to the reflection equation in the finite-type setting.