Quantum affine algebras and spectral k-matrices

Abstract: The Yang-Baxter equation (YBE) and the reflection equation (RE) are two fundamental symmetries in mathematics arising from particles moving along a line or a half-line. The quest for constant solutions of YBE (R-matrices) is at the very origin of the Drinfeld-Jimbo quantum groups and their universal R-matrix. Similarly, constant solutions of RE (k-matrices) naturally appear in the context of quantum symmetric pairs (QSP).

In joint work with Bart Vlaar, we construct a discrete family of universal k-matrices associated to an arbitrary quantum symmetric Kac-Moody pair as operators on category O integrable representations. This generalises previous results by Balagovic-Kolb and Bao-Wang valid for finite-type QSP. In this talk, I will explain how, in affine type, this construction gives rise to parameter-dependent operators (spectral k-matrices) on finite-dimensional representations of quantum loop algebras solving the same RE introduced by Cherednik and Sklyanin in the 1980s in the context of quantum integrability near a boundary.

category theoryoperator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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