Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik-Zamolodchikov equations and Letzter-Kolb coideals
Kenny De Commer, Sergey Neshveyev, Lars Tuset, Makoto Yamashita
TL;DR
The paper develops a formal, multiplier-algebra framework to quantize irreducible symmetric spaces of compact type via two parallel approaches: cyclotomic Knizhnik–Zamolodchikov (KZ) equations and Letzter–Kolb coideals. It proves that these two paths are categorically equivalent as ribbon twist-braided quasi-coactions, with the cyclotomic KZ data producing universal associators $\Psi_{\mathrm{KZ},s;\mu}$ and the LK coideals furnishing corresponding $K$-matrices, culminating in Kohno–Drinfeld-type theorems for type B braid group representations. A sharp dichotomy emerges between non-Hermitian and Hermitian symmetric pairs: the former exhibits quasi-coaction rigidity, while the latter admits a one-parameter family of nonequivalent quasi-coactions linked to Poisson structures and Cayley transforms. The results unify deformation quantization, dynamical r-matrices, and quantum symmetric-pair theory into a coherent framework, with concrete ramifications for AIII-type examples and potential extensions to broader coadjoint orbits. The work thereby provides deep structural invariants (ribbon braids, K-matrices) and a universal picture for the quantization of symmetric spaces, with implications for representation theory and quantum symmetry in geometric settings.
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 Balagović-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.