The reflection equation, coideal subalgebras and Satake diagrams

Bart Vlaar (University of York)

Thursday 23rd February, 2017 16:00-17:00 Maths 522

Abstract

It is well-known that many R-matrices, i.e. matrix solutions of the quantum Yang-Baxter equation (YBE), are associated to representations of affine quantum groups (deformed universal enveloping algebras of affine Lie algebras). Restrictions of these representations to certain coideal subalgebras are known to be related to the reflection equation (RE), which involves the aforementioned R-matrix and an additional object called K-matrix. It appeared in the 1980s in work by Cherednik, Sklyanin and others in the study of quantum integrable systems with boundaries.

A particularly useful class of coideal subalgebras can be defined in terms of generalized Satake diagrams, largely following the ideas of Letzter and Kolb. In joint work with Vidas Regelskis, the corresponding solutions of the RE in the vector representation of affine quantum groups of classical (A, B, C, D) Lie type have been found by solving a suitable intertwining equation. In other words, this is a boundary version of Jimbo’s 1986 paper in which he obtained explicit formulas for R-matrices using their intertwining property. It appears that all matrix solutions of the RE involving these R-matrices can be obtained by this method.

The underlying Satake diagram governs how many nonzero entries a particular solution of the RE may have, how many free parameters it may depend on and how it may be related to another solution through similarity transformations. Furthermore, there are potential applications to finding RTT-presentations and Schur-Weyl dualities for coideal subalgebras.

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