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

Kenny De Commer; Makoto Yamashita; Sergey Neshveyev; Lars Tuset

doi:10.1017/fmp.2023.11

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

Kenny De Commer, Makoto Yamashita, Sergey Neshveyev, Lars Tuset

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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 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.

Original language	English
Article number	e14
Pages (from-to)	1-79
Number of pages	79
Journal	Forum of Mathematics: Pi
Volume	11
Issue number	14
DOIs	https://doi.org/10.1017/fmp.2023.11
Publication status	Published - 2 May 2023

Bibliographical note

Funding Information:
The work of K.DC. was supported by the FWO grants G025115N and G032919N. The work of S.N. and M.Y. was partially supported by the NFR funded project 300837 ‘Quantum Symmetry’. M.Y. also acknowledges support by Grant for Basic Science Research Projects from The Sumitomo Foundation and JSPS Kakenhi 18K13421 at an early stage of collaboration.

Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press.

Copyright:
Copyright 2023 Elsevier B.V., All rights reserved.

Keywords

Quantum groups
Tensor categories
Deformation quantization

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10.1017/fmp.2023.11Licence: CC BY

95185336Final published version, 1.15 MBLicence: CC BY

https://arxiv.org/abs/2009.06018Licence: CC BY

Cite this

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title = "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 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.",

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note = "Funding Information: The work of K.DC. was supported by the FWO grants G025115N and G032919N. The work of S.N. and M.Y. was partially supported by the NFR funded project 300837 {\textquoteleft}Quantum Symmetry{\textquoteright}. M.Y. also acknowledges support by Grant for Basic Science Research Projects from The Sumitomo Foundation and JSPS Kakenhi 18K13421 at an early stage of collaboration. Publisher Copyright: {\textcopyright} The Author(s), 2023. Published by Cambridge University Press. Copyright: Copyright 2023 Elsevier B.V., All rights reserved.",

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N2 - 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 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.

AB - 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 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.

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

Abstract

Bibliographical note

Keywords

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FWOAL900: Quantum symmetric spaces, operator algebras and quantum cluster algebras

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