Cylindrical structures for Drinfeld-Jimbo quantum groups and the origin of trigonometric K-matrices

Abstract：

The study of spectral R-matrices, matrix solutions of the (parameter-dependent) Yang-Baxter equation, was a major motivation for the discovery of quantum groups. The quasitriangular structure of these bialgebras is the origin of large classes of spectral R-matrices. The Yang-Baxter equation has a type-B/cylindrical counterpart: the reflection equation. Its matrix solutions, spectral K-matrices, have been studied since the 1980s. Do they have a similar origin?

To answer this, in joint works with Andrea Appel we develop a general framework, in terms of braided tensor categories with additional structures. Concretely, take any Letzter-Kolb quantum symmetric pair: a Drinfeld-Jimbo quantum group together with a suitable coideal subalgebra (also known as i-quantum group). Building on works by Bao & Wang and Balagovic & Kolb, we show that a twisted intertwiner of the subalgebra satisfies a twisted reflection equation, acts on (integrable) category O modules, and endows this braided tensor category with a twisted cylinder braiding. In the case of affine quantum groups one can then indeed answer the above question, explaining large classes of so-called trigonometric K-matrices.