Chalykh, O orcid.org/0000-0003-4529-2310 (2019) Quantum Lax Pairs via Dunkl and Cherednik Operators. Communications in Mathematical Physics, 369 (1). pp. 261-316. ISSN 0010-3616
Abstract
We establish a direct link between Dunkl operators and quantum Lax matrices L for the Calogero–Moser systems associated to an arbitrary Weyl group W (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix A so that L,A form a quantum Lax pair. Moreover, such an A can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of W, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D’Hoker–Phong and Bordner–Corrigan–Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic BCn case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Springer-Verlag GmbH Germany, part of Springer Nature 2019. This is a post-peer-review, pre-copyedit version of an article published in Communications in Mathematical Physics. The final authenticated version is available online at: https://doi.org/10.1007/s00220-019-03289-8 |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 19 Nov 2018 11:43 |
Last Modified: | 06 Feb 2020 01:38 |
Status: | Published |
Publisher: | Springer |
Identification Number: | 10.1007/s00220-019-03289-8 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:138764 |