Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals

TL;DR

This paper establishes an equivalence between two quantization approaches for symmetric spaces, linking KZ equations and coideals, and explores their invariants and applications to braid group representations.

Contribution

It proves the equivalence of two quantization methods for symmetric spaces and introduces invariants like reflection operators, with applications to braid group representations.

Findings

Equivalence between KZ-based and coideal-based quantizations.

Identification of reflection operators as invariants of quantization.

Different behaviors for Hermitian and non-Hermitian symmetric spaces.

Abstract

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez-Etingof cyclotomic Knizhnik-Zamolodchikov (KZ) equations and the other on the Letzter-Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno-Drinfeld type theorem on type B braid group representations defined by the monodromy of KZ-equations and by the Balagovi\'c-Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.