Geometric Representation Theory Seminar | Cylindrical structures for Drinfeld-Jimbo quantum groups and the origin of trigonometric K-matrices

Abstract 

The study of R-matrices, matrix solutions of the spectral (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 R-matrices. The Yang-Baxter equation has a "twisted type-B/cylindrical" counterpart: the reflection equation. Its matrix solutions, known as K-matrices, have been studied since the 1980s. Is there an analogous origin for these solutions?

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 (quantized enveloping algebra of a Kac-Moody algebra) together with a suitable subalgebra (also known as i-quantum group). Further to works by Bao & Wang and Balagovic & Kolb, a twisted intertwiner of the subalgebra satisfies a reflection equation, acts on (integrable) category O modules and endows this braided tensor category with a twisted cylinder braiding. For affine quantum groups one can develop the parallel with R-matrices much further and account for large classes of so-called trigonometric K-matrices.

About the speaker 

Dr. Bart Vlaar has joined BIMSA in September 2022 as an Associate Research Fellow. His research interests are in algebra and representation theory and applications in mathematical physics. He obtained a PhD in Mathematics from the University of Glasgow. Previously, he has held postdoctoral positions in Amsterdam, Nottingham, York and Heriot-Watt University. Before coming to BIMSA he visited the Max Planck Institute of Mathematics in Bonn.