Trigonometric real form of the spin RS model of Krichever and Zabrodin

Fairon, M. , Fehér, L. and Marshall, I. (2021) Trigonometric real form of the spin RS model of Krichever and Zabrodin. Annales Henri Poincaré, 22(2), pp. 615-675. (doi: 10.1007/s00023-020-00976-4)

Abstract

We investigate the trigonometric real form of the spin Ruijsenaars--Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real form emerges from Hamiltonian reduction of an obviously integrable `free' system carried by a spin extension of the Heisenberg double of the $\U(n)$ Poisson--Lie group. The Poisson structure on the unreduced real phase space $\GL \times \C^{nd}$ is the direct product of that of the Heisenberg double and $d\geq 2$ copies of a $\U(n)$ covariant Poisson structure on $\C^n \simeq \R^{2n}$ found by Zakrzewski, also in 1995. We reduce by fixing a group valued moment map to a multiple of the identity, and analyze the resulting reduced system in detail. In particular, we derive on the reduced phase space the Hamiltonian structure of the trigonometric spin Ruijsenaars--Schneider system and we prove its degenerate integrability.

• Item Type: Articles
• Additional Information: Open access funding provided by University of Szeged, grant 5027.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Fairon, Dr Maxime
• Authors: Fairon, M., Fehér, L., and Marshall, I.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Annales Henri Poincaré
• Publisher: Springer
• ISSN: 1424-0637
• ISSN (Online): 1424-0661
• Published Online: 21 November 2020
• Copyright Holders: Copyright © 2020 The Authors
• First Published: First published in Annales Henri Poincaré 22(2): 615-675
• Publisher Policy: Reproduced under a Creative Commons License