The Origin of Trigonometric K-matrices - Part III

Explore the third part of a lecture series on the origin of trigonometric K-matrices, presented by Bart Vlaar at the Workshop on Integrable systems, exactly solvable models and algebras. Delve into the world of quantum integrability, characterized by R-matrices that solve the Yang-Baxter equation. Examine Drinfeld's observation on universal R-matrices of quantum affine algebras and their action on tensor products of finite-dimensional representations of quantum loop algebras. Investigate the resulting matrix-valued formal series R(z) and its rational dependence on the multiplicative spectral parameter z. Learn about K-matrices, solutions to the reflection equation (boundary Yang-Baxter equation), and their significance in quantum integrable systems with boundaries. Discover recent joint work with A. Appel proving the existence of a universal K-matrix and its application in a boundary analogue of Drinfeld's approach. Understand the role of suitable subalgebras as residual symmetries compatible with boundaries, leading to a "limitless supply" of trigonometric K-matrices. Explore the generalized reflection equation considered by Cherednik in 1992 and its implications for matrix-valued formal Laurent series K(z) in irreducible representations.