Factorization Problems on Rational Loop Groups, and the Poisson Geometry of Yang-Baxter Maps

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Abstract

The study of set-theoretic solutions of the Yang-Baxter equation, also known as Yang-Baxter maps, is historically a meeting ground for various areas of mathematics and mathematical physics. In this work, we study factorization problems on rational loop groups, which give rise to Yang-Baxter maps on various geometrical objects. We also study the symplectic and Poisson geometry of these Yang-Baxter maps, which we show to be integrable maps in the sense of having natural collections of Poisson commuting integrals. In a special case, the factorization problems we consider are associated with the N-soliton collision process in the n-Manakov system, and in this context we show that the polarization scattering map is a symplectomorphism.

Original languageEnglish (US)
Article number6
JournalMathematical Physics Analysis and Geometry
Volume25
Issue number1
DOIs
StatePublished - Mar 2022

All Science Journal Classification (ASJC) codes

  • Mathematical Physics
  • Geometry and Topology

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