School of Mathematics, Statistics and Physics

Seminar Item

Pure Maths (Algebra and Geometry) - Makoto Yamashita (Oslo)

Categorical quantization of symmetric spaces and reflection equation

Date/Time: Monday 19 October, 1.30pm - Algebra and Geometry Seminar

Venue: Zoom

Abstract: 

A fundamental result of Drinfeld says that the representation category of a simple Lie algebra has 1-dimensional parameter space of deformation as tensor category, which is concretely encoded in the eigenvalues of compatible braiding. Our result is an analogue of that for irreducible compact symmetric pairs. This amounts to adding the Reflection Equation besides the Yang-Baxter Equation, which is a powerful guiding principle to quantize Poisson homogeneous spaces into actions of quantum groups.

Our classification leads to a Kohno-Drinfeld type result comparing the type B braid group representations from the cyclotomic Knizhnik-Zamolodchikov equations on the one hand, and the universal R-matrix and the universal K-matrix of Balagovic and Kolb on the other. The proof relies on elementary but curious combination of Lie algebra cohomology, Hochschild cohomology, and formality machinery from noncommutative geometry inspired by works of Calaque and Brochier.

Based on joint works with Kenny De Commer, Sergey Neshveyev, and Lars Tuset (arXiv:1712.08047 / CMP ’19 and arXiv:2009.06018).