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

Luen-Chau Li, Vincent Caudrelier

TL;DR

The paper develops a geometric framework for reflection maps arising from involutive factorization problems in the rational loop group $K_{ ext{rat}}$, linking them to parametric Yang-Baxter maps and their reductions. It shows that reflection maps are smoothly conjugate to composites of permutation maps with reduced Yang-Baxter maps, and that parameter-free reductions reduce to braiding operators; it further establishes symplectic and Poisson properties of these maps at the level of projectors, complex projective spaces, and Poisson-Lie groups via Dirac reduction. A physically meaningful $N$-body polarization reflection map for the $n$-Manakov system on the half-line is identified and shown to be a symplectomorphism, connecting soliton-boundary interactions to symplectic geometry. Finally, a general Poisson-Lie group framework is developed to produce Poisson reflection maps satisfying a reflection equation, with concrete realization on the rational loop group through Dirac reductions, highlighting the unifying role of involutions and reduction in integrable systems.

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.

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

TL;DR

The paper develops a geometric framework for reflection maps arising from involutive factorization problems in the rational loop group , linking them to parametric Yang-Baxter maps and their reductions. It shows that reflection maps are smoothly conjugate to composites of permutation maps with reduced Yang-Baxter maps, and that parameter-free reductions reduce to braiding operators; it further establishes symplectic and Poisson properties of these maps at the level of projectors, complex projective spaces, and Poisson-Lie groups via Dirac reduction. A physically meaningful -body polarization reflection map for the -Manakov system on the half-line is identified and shown to be a symplectomorphism, connecting soliton-boundary interactions to symplectic geometry. Finally, a general Poisson-Lie group framework is developed to produce Poisson reflection maps satisfying a reflection equation, with concrete realization on the rational loop group through Dirac reductions, highlighting the unifying role of involutions and reduction in integrable systems.

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.
Paper Structure (10 sections, 22 theorems, 230 equations)

This paper contains 10 sections, 22 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parametric Yang-Baxter maps
  4. The rank 1 case.
  5. Yang-Baxter maps and $K_{\text{rat}}$
  6. Dirac submanifolds, Poisson involutions, and Dirac reduction
  7. Reflection maps at the level of projectors
  8. The rank $1$ case
  9. Reflection maps and Poisson Lie groups
  10. Conclusion

Key Result

Theorem 2.1

(a) $g_{\alpha, P}\in {K_{\text{rat}}}.$ (b) ${K_{\text{rat}}}$ is generated by the simple elements, i.e., every $g\in {K_{\text{rat}}}$ can be factorized into a product of simple elements.

Theorems & Definitions (48)

  • Theorem 2.1: U
  • Theorem 2.2: L1
  • Theorem 2.3: L1
  • Definition 2.4
  • Theorem 2.5: L1
  • Corollary 2.6: L1
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 38 more