Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models

Chalykh, O. and Fairon, M. (2017) Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models. Journal of Geometry and Physics, 121, pp. 413-437. (doi: 10.1016/j.geomphys.2017.08.006)

Abstract

We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case corresponds to the tadpole quiver and the Ruijsenaars–Schneider system and its variants, while for m > 1 we obtain new integrable systems that generalise the Ruijsenaars–Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs (Braverman et al. [32,34,35] and Kodera and Nakajima [36]), as well as in the context of the Macdonald theory (Chalykh and Etingof, 2013).

• Item Type: Articles
• Keywords: Quivers, noncommutative geometry, double poisson brackets, quasi-Hamiltonian reduction, Ruijsenaars-Schneider system.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Fairon, Dr Maxime
• Authors: Chalykh, O., and Fairon, M.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Geometry and Physics
• Publisher: Elsevier
• ISSN: 0393-0440
• ISSN (Online): 1879-1662
• Published Online: 18 August 2017
• Copyright Holders: Copyright © 2017 Elsevier B.V.
• First Published: First published in Journal of Geometry and Physics 121: 413-437
• Publisher Policy: Reproduced in accordance with the publisher copyright policy