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

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.

arXiv comment63 pages; v3: minor changes

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• Lectures on dualities ABC in representation theory
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