A decoupling property of some Poisson structures on Matn×d(C)×Matd×n(C) supporting GL(n,C)×GL(d,C) Poisson–Lie symmetry

Fairon, M. and Fehér, L. (2021) A decoupling property of some Poisson structures on Matn×d(C)×Matd×n(C) supporting GL(n,C)×GL(d,C) Poisson–Lie symmetry. Journal of Mathematical Physics, 62(3), 033512. (doi: 10.1063/5.0035935)

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Abstract

We study a holomorphic Poisson structure defined on the linear space S(n, d) ∶= Matn×d(C) × Matd×n(C) that is covariant under the natural left actions of the standard GL(n, C) and GL(d, C) Poisson–Lie groups. The Poisson brackets of the matrix elements contain quadratic and constant terms, and the Poisson tensor is non-degenerate on a dense subset. Taking the d = 1 special case gives a Poisson structure on S(n, 1), and we construct a local Poisson map from the Cartesian product of d independent copies of S(n, 1) into S(n, d), which is a holomorphic diffeomorphism in a neighborhood of 0. The Poisson structure on S(n, d) is the complexification of a real Poisson structure on Matn×d(C) constructed by the authors and Marshall, where a similar decoupling into d independent copies was observed. We also relate our construction to a Poisson structure on S(n, d) defined by Arutyunov and Olivucci in the treatment of the complex trigonometric spin Ruijsenaars–Schneider system by Hamiltonian reduction.

Item Type:Articles
Additional Information:The research of M.F. was supported by a Rankin–Sneddon Research Fellowship of the University of Glasgow. The work of L.F. was supported, in part, by the NKFIH funding agency under Grant No. K134946.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Fairon, Dr Maxime
Authors: Fairon, M., and Fehér, L.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Mathematical Physics
Publisher:AIP Publishing
ISSN:0022-2488
ISSN (Online):1089-7658
Published Online:24 March 2021
Copyright Holders:Copyright © 2021 The Authors
First Published:First published in Journal of Mathematical Physics 62(3): 033512
Publisher Policy:Reproduced in accordance with the publisher copyright policy

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