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

Autor:	Li, Luen-Chau, Caudrelier, Vincent
Rok vydání:	2023
Předmět:	Mathematical Physics Mathematics - Symplectic Geometry Nonlinear Sciences - Exactly Solvable and Integrable Systems 16J25 37J39 37K25 35Q55 35C08
Druh dokumentu:	Working Paper
Popis:	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. Comment: 38 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2312.05164 Zobrazit plný text záznamu View this record from Arxiv