Quantum symmetric pairs, the reflection equation, and the beauty of the bar involution

Stefan Kolb (Newcastle University)

Tuesday 25th November, 2014 15:00-16:00 Maths 203


The theory of quantum groups was conceived about 30 years ago in order to give a conceptual mathematical interpretation of solutions of the quantum Yang-Baxter equation which appeared in various integrable models (XXZ, factorised scattering, spin chains).  For non-periodic boundary conditions the reflection equation enters the picture. Solutions of the reflection equation corresponding to Drinfeld-Jimbo quantised enveloping algebras appear within the theory of quantum symmetric pairs.  In Lusztig's approach to quantum groups, universal R-matrices are constructed as intertwiners of coproduct and bar involution. In a broader representation theoretic context it was recently observed by Bao & Wang and Stroppel & Ehrig that there exists a bar involution for certain quantum symmetric pairs. Bao & Wang's work further outlines a way to construct universal solutions of the reflection equation within this theory.

In this talk I will give an introduction to the theory quantum symmetric pairs and discuss recent joint work with M. Balagovic on the existence of the bar involution in this setting. Time permitting I will outline how this is expected to lead to universal solutions of the reflection equation.

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