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<p>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 <i>N</i>-solitons of the <i>n</i>-Manakov system with a boundary, and in this context, the <i>N</i>-body polarization reflection map is a symplectomorphism.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月1日

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  • Luen-Chau Li,
  • Vincent Caudrelier

摘要

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.