TY - JOUR

T1 - Factorization Problems on Rational Loop Groups, and the Poisson Geometry of Yang-Baxter Maps

AU - Li, Luen Chau

N1 - Funding Information:
The author would like to thank Vincent Caudrelier who introduced him to soliton collision problems in the context of the n-Manakov system and for bringing to his attention the references [, , , , ]. He also thanks the support of the Simons foundation through Grants #278994 and #585813. Last but not the least, the author acknowledges the referees for their careful reading of the manuscript, and for their helpful advice.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2022/3

Y1 - 2022/3

N2 - 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.

AB - 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.

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U2 - 10.1007/s11040-022-09419-4

DO - 10.1007/s11040-022-09419-4

M3 - Article

AN - SCOPUS:85125319220

SN - 1385-0172

VL - 25

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

IS - 1

M1 - 6

ER -