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Explore the foundations of trigonometric K-matrices in this 52-minute lecture by Bart Vlaar at the Centre de recherches mathématiques (CRM). Delve into quantum integrability characterized by R-matrices and their connection to the Yang-Baxter equation. Examine Drinfeld's observation on universal R-matrices and their action on tensor products of finite-dimensional representations of quantum loop algebras. Investigate the development of K-matrices as solutions to the reflection equation in quantum integrable systems with boundaries. Learn about 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 how this research guarantees a "limitless supply" of trigonometric K-matrices and their relationship to finite-dimensional representations. Gain insights into the generalized reflection equation considered by Cherednik in 1992 and the rational dependence of K(z) on the spectral parameter z for irreducible representations.
