Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry

Abstract

The study of the set-theoretic solutions of the reflection equation, also known as reflection maps, is closely related to that of the Yang–Baxter maps. In this work, we construct reflection maps on various geometrical objects, associated with factorization problems on rational loop groups and involutions. We show that such reflection maps are smoothly conjugate to the composite of permutation maps, with corresponding reduced Yang–Baxter maps. In the case when the reduced Yang–Baxter maps are independent of parameters, the latter are just braiding operators. We also study the symplectic and Poisson geometry of such reflection maps. In a special case, the factorization problems are associated with the collision of N-solitons of the n-Manakov system with a boundary, and in this context, the N-body polarization reflection map is a symplectomorphism.

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Item Type:	Article
Authors/Creators:	Li, L.-C. Caudrelier, V. https://orcid.org/0000-0003-0129-6758
Copyright, Publisher and Additional Information:	© 2025 The Author(s). This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.
Dates:	Accepted: 12 June 2025 Published (online): 1 July 2025
Institution:	The University of Leeds
Academic Units:	The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds)
Depositing User:	Symplectic Publications
Date Deposited:	16 Jun 2025 13:06
Last Modified:	07 Aug 2025 15:25
Published Version:	https://link.springer.com/article/10.1007/s00023-0...
Status:	Published online
Publisher:	EMS Press
Identification Number:	10.1007/s00023-025-01599-3
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:227838

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Reflection Maps Associated with Involutions and Factorization Problems, and Their Poisson Geometry

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